P. Junghanns, U. Weber

Local Theory of a Collocation Method for Cauchy Singular Integral Equations on an Interval

Peter Junghanns, Uwe Weber Technische Universität Chemnitz - Zwickau Fakultät für Mathematik D-09107 Chemnitz Germany

peter.junghanns@mathematik.tu-chemnitz.de uwe.weber@mathematik.tu-chemnitz.de

Abstract We consider a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polynomials, where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra methods, and numerical results are given.

1 Introduction

The subject of the present paper is the investigation of a collocation method based on weighted polynomials for the approximate solution of singular integral equations on (−1, 1) of the type

a(x)u(x) + \frac{b(x)}{πi} ∫_{−1}^{1} \frac{u(t)}{t − x} dt = f(x),  x ∈ (−1, 1),  (1.1)

where u is the unknown function and a, b, f are given. All functions involved are assumed to be complex-valued, and we require that b(±1) = 0.

A lot of attention has been paid to investigating polynomial collocation and quadrature methods for this and similar types of equations (see [PS, Chapter 9]). These are essentially based on special mapping properties of the operator AμI, where A denotes the operator on the left-hand side of (1.1), and μ is a generalized Jacobi weight depending on the coefficients a and b, which are required to satisfy a Hölder condition.

We are going to give a somewhat different approach using weighted polynomials as ansatz functions (a finite section (Galerkin) method with the same type of ansatz functions was investigated in [JRW]). Also in the case of variable coefficients a and b there will always occur pure Jacobi weights, which are easier to cope with than generalized Jacobi weights (for instance, the recurrence coefficients of the corresponding orthogonal polynomials, which are needed in the computer implementation of our method, are explicitly known), and we will always use the same (Chebyshev) collocation points independently of the coefficients. This is an advantage if a Newton method for a nonlinear singular integral equation results in a sequence of linear equations of type (1.1), the coefficients of which are different in each step. Furthermore, we can (apart from the additional assumption b(±1) = 0) admit arbitrary piecewise continuous coefficients. Coefficients with jumps can occur, for example, when considering seepage problems for channels or dams with corners ([Ju2]). Finally we mention that one can introduce a scale of Sobolev-like spaces with respect to a system of weighted orthogonal polynomials analogous to [BHS] and can under suitable smoothness conditions on the coefficients obtain error estimates for the collocation method in the norms of these Sobolev-like spaces, which are continuously embedded in certain spaces of weighted continuous functions. The results we achieved here, however, still need to be improved.

We will prove the strong convergence of the approximation operators associated with our collocation method, and will investigate the stability of this method using Banach algebra techniques, which have proved to be an efficient tool in stability analysis (see for example [Si], [JS], [HRS], [JRW]). We obtain necessary and sufficient stability conditions in the case of continuous coefficients and a sufficient condition if the coefficients are piecewise continuous.

In Section 2 we give some notations and define the numerical method we are going to deal with, Section 3 provides some basic facts on Banach algebra techniques we will need for the stability analysis, Section 4 is concerned with the strong convergence of some operator sequences, and in Section 5 we will use Banach algebra techniques to derive the main result concerning the stability. Finally, we make some remarks concerning the computer implementation and present numerical results in Section 6.

2 Some notations and preliminaries

We consider equation (1.1) in the weighted Lebesgue space L_{σ}^{2} := L_{σ}^{2}(−1, 1) of all (classes of) measurable functions u : (−1, 1) → C for which

∥u∥_{σ}^{2} := ∫_{−1}^{1} |u(x)|^{2}σ(x) dx

is finite, equipped with the inner product

(u, v)_{σ} := ∫_{−1}^{1} u(x)\overline{v(x)}σ(x) dx,

which turns L_{σ}^{2} into a Hilbert space. Here σ is a Jacobi weight defined by σ(x) = v^{α,β}(x) := (1 − x)^{α}(1 + x)^{β} satisfying the conditions

− 1 < α, β < 1. (2.2)

These conditions guarantee the boundedness of the Cauchy singular integral operator S defined by

(Su)(x) = \frac{1}{πi} ∫_{−1}^{1} \frac{u(t)}{t − x} dt (2.3)

on L_{σ}^{2} ([GK], Theorem I.4.1). The coefficients a, b are assumed to belong to the algebra P C of all piecewise continuous functions. The latter is defined as the closure (in the space of all bounded functions, equipped with the supremum norm) of the set of those functions being continuous on [−1, 1] with the possible exception of a finite number of jumps in (−1, 1), where the value of the function coincides with the left-sided limit. Note that P C-functions possess finite one-sided limits at all points. Under these assumptions, the operator on the left-hand side of (1.1), which in the following will be briefly referred to as aI +bS, is bounded on L_{σ}^{2}.

Let φ denote the Chebyshev weight of second kind,

φ(x) = \sqrt{1 − x^{2}},

and let U_{n} be the orthonormal polynomial of degree n (with positive leading coefficient) with respect to the inner product (., .)_{φ}. For these polynomials we have the well-known trigonometric representation

U_{n}(cos s) = \sqrt{\frac{2}{π}} \frac{sin(n + 1)s}{sin s}.

If σ is a Jacobi weight satisfying (2.2), we define the function

w_{σ} := \sqrt{σφ}.

Obviously, w_{σ^{−1}}I is an isometric isomorphism from L_{φ}^{2} onto L_{σ}^{2}. Thus, the functions

ũ_{n} := w_{σ^{−1}}U_{n},  n = 0, 1, 2, …,  (2.4)

form an orthonormal basis in L_{σ}^{2}, because the same is true for U_{n} in the space L_{φ}^{2}. For the approximation method we want to apply to (1.1), the functions (2.4) will be used as ansatz functions. We need two sequences of projections with respect to the system {ũ_{n}}_{n=0}^{∞}. The first are the Fourier projections P_{n}^{σ} given by

P_{n}^{σ} ∑_{k=0}^{∞} ξ_{k}ũ_{k} := ∑_{k=0}^{n−1} ξ_{k}ũ_{k}.

Evidently, P_{n}^{σ} converges strongly to the identity operator I as n → ∞. The second is the weighted interpolation operator L̃_{n}^{σ} assigning to a Riemann integrable function f the uniquely determined weighted polynomial of degree less than n (that is, the element of X_{n} := span \{ũ_{k}\}_{k=0}^{n−1}) that coincides with f in the collocation points x_{kn}^{φ} = cos \frac{kπ}{n+1} (k = 1, …, n), which are the zeros of U_{n}. Thus we can write

L̃_{n}^{σ} = w_{σ^{−1}}L_{n}^{φ}(w_{σ^{−1}})^{−1}I,

where L_{n}^{φ} is the usual (polynomial) Lagrangian interpolation operator with respect to these nodes. A class of functions f for which ∥L̃_{n}^{σ}f − f∥_{σ} → 0 will be described in Corollary 4.6.

We now consider a collocation method which replaces equation (1.1) by the discrete approximate equations

L̃_{n}^{σ}(aI + bS)v_{n} = L̃_{n}^{σ}f, (2.5)

where v_{n} ∈ span {ũ_{k}}_{k=0}^{n−1} is sought. Our main concern is the applicability of this method to the original equation. In the following, we are going to define in a somewhat more general situation what we understand by this concept: Let X be a Banach space, let {P_{n}} ⊂ L(X) (where L(X, Y) means the set of all linear bounded operators between two Banach spaces X and Y, and L(X) := L(X, X)) be a sequence of projections with P_{n} → I (strongly), X_{n} = im P_{n}, A ∈ L(X) and A_{n} ∈ L(X_{n}), where we require that A_{n}P_{n} converges strongly to A.

Definition 2.1 The projection method

A_{n}u_{n} = P_{n}f (2.6)

is said to be applicable to the equation

Au = f (2.7)

if the following conditions are fulfilled:

(i) The equations (2.6) have a unique solution for all sufficiently large n.

(ii) Their solutions u_{n} converge to a solution of (2.7) for n → ∞.

Definition 2.2 The sequence {A_{n}} is said to be stable if there is an n_{0} such that A_{n} is invertible for all n ≥ n_{0} and sup_{n≥n_{0}} ∥A_{n}^{−1}P_{n}∥ < ∞.

The problem of the applicability of (2.6) to (2.7) can be reduced to the stability of {A_{n}} by the following lemma.

Lemma 2.1 ([HRS], Prop. 1.1) Let A_{n}P_{n} → A strongly. Then (2.6) is applicable to (2.7) if and only if A is invertible and the sequence {A_{n}} is stable.

In our concrete situation we have X = L_{σ}^{2}, P_{n} = P_{n}^{σ}, A = aI + bS and A_{n} = L̃_{n}^{σ}A|_{X_{n}}. The strong convergence of these operators will be dealt with in section 4.

Remark 2.2 Note that in the following we consider the projection method (2.6) rather than (2.5). If, however, (2.6) is applicable to (1.1) and the right-hand side f satisfies ∥f − L̃_{n}^{σ}f∥_{σ} → 0 (n → ∞), the solutions v_{n} of (2.5) converge to the solution u of (1.1).

Proof. We have

∥v_{n} − u∥ ≤ ∥A_{n}^{−1}L̃_{n}^{σ}f − A_{n}^{−1}A_{n}P_{n}^{σ}u∥ + ∥P_{n}^{σ}u − u∥
≤ ∥A_{n}^{−1}P_{n}^{σ}∥ (∥L̃_{n}^{σ}f − f∥ + ∥Au − A_{n}P_{n}^{σ}u∥) + ∥P_{n}^{σ}u − u∥ ⟶ 0

3 Banach algebra techniques

3.1 Basic facts

In this subsection we compile some basic facts that will be used later on to investigate the stability problem for our approximation method by Banach algebra techniques.

Definition 3.1 Let B, C be unital Banach algebras, J ⊂ B a closed two-sided ideal. A unital homomorphism W : B → C is called J -lifting, if W (J) is a closed two-sided ideal in C and W |_{J} is an isomorphism between J and W (J).

The following theorem is usually called ‘lifting theorem’, its first version appears in [Si].

Theorem 3.1 (see [HRS], Theorem 1.8) Let B, C_{t} be unital Banach algebras, where t belongs to an arbitrary index set T, let J_{t} be closed two-sided ideals in B and let W_{t} : B → C_{t} be J_{t}-lifting homomorphisms. By J we denote the smallest closed two-sided ideal in B that contains all J_{t}, t ∈ T. Then an element b ∈ B is invertible in B if and only if W_{t}(b) is invertible in C_{t} for all t ∈ T and the coset b + J is invertible in the quotient algebra B/J.

The invertibility of the coset b + J is usually investigated by the help of local principles.

Definition 3.2 Let B be a unital Banach algebra. A subset M ⊂ B is called a localizing class if 0 ∉ M and if for all a_{1}, a_{2} ∈ M there exists an element a ∈ M such that

aa_{j} = a_{j}a = a (j = 1, 2).

In the following let M be a localizing class. Two elements x, y ∈ B are called M-equivalent (in symbols: x ∼^{M} y), if

inf_{a∈M} ∥a(x − y)∥ = inf_{a∈M} ∥(x − y)a∥ = 0.

Further, x ∈ B is called M-invertible if there exist a_{1}, a_{2} ∈ M, z_{1}, z_{2} ∈ B such that

z_{1}xa_{1} = a_{1}, a_{2}xz_{2} = a_{2}.

A system {M_{t}}_{t∈T} of localizing classes (T is an arbitrary index set) is said to be covering, if for each system {a_{t}}_{t∈T}, a_{t} ∈ M_{t}, there exists a finite subsystem a_{t_{1}}, …, a_{t_{n}} such that a_{t_{1}} + ⋯ + a_{t_{n}} is invertible in B.

Now we can formulate the local principle of Gohberg and Krupnik:

Theorem 3.2 ([GK], Theorem XII.1.1) Let B be a unital Banach algebra, {M_{t}}_{t∈T} a covering system of localizing classes in B, x ∈ B and x ∼^{M_{t}} x_{t} for all t ∈ T. Further, assume that x commutes with all elements from ⋃_{t∈T} M_{t}. Then x is invertible in B if and only if x_{t}

is M_{t}-invertible for all t ∈ T.

Another local principle is due to Allan and Douglas:

Theorem 3.3 ([BS], Theorem 1.34) Let B be a unital Banach algebra and C ⊂ B a closed central subalgebra (that is, all elements of C commute with all elements of B) that contains the unit element. For every maximal ideal t of C we introduce the local ideal J_{t} as the smallest closed two-sided ideal of B that contains t. Then

(i) An element x ∈ B is invertible in B if and only if the cosets x + J_{t} are invertible in B/J_{t} for all t.

(ii) ⋂_{t} J_{t} is contained in the radical of B.

Remark 3.1 (see [PS], proof of Theorem 1.21) If B, C are C^{∗}-algebras, there is a close relation between the two local principles. Let M(C) denote the maximal ideal space of C. For t ∈ M(C) we define M_{t} := {a ∈ C : 0 ≤ (Ga)(s) ≤ 1, (Ga)(s) ≡ 1 in some neighbourhood of t}, where G : C → C(M(C)) denotes the Gelfand map. Then {M_{t}}_{t∈M(C)} forms a covering system of localizing classes in B, and the local ideals occurring in the principle of Allan and Douglas can be described by J_{t} = {x ∈ B : x ∼^{M_{t}} 0}.

3.2 Application to stability analysis

We want to investigate the applicability of the approximation method (2.6) to equation (2.7). In all what follows we assume that A_{n}P_{n} converges strongly to A, which allows us to reduce the problem to the question if {A_{n}} is stable. Furthermore, we specify X to be a Hilbert space.

By E we denote the set of all operator sequences {A_{n}P_{n}}, where A_{n} ∈ L(X_{n}) and sup_{n} ∥A_{n}P_{n}∥ < ∞. Endowed with componentwise algebraic operations and the norm

∥{B_{n}}∥_{E} = sup_{n} ∥B_{n}∥, E becomes a C^{∗}-algebra. The set N := {{C_{n}} ∈ E : ∥C_{n}∥ → 0} is a closed ideal in E. In the sequel we will use the notation GB for the set of all invertible elements of a Banach algebra B. The following well-known result identifies the question of stability with an invertibility problem.

Lemma 3.2 ([HRS], Proposition 1.2) A sequence {A_{n}} ∈ E is stable if and only if {A_{n}} + N ∈ G(E/N).

We introduce a further family of operators W_{n} ∈ L(X), where we assume that W_{n} = W_{n}^{∗} converges weakly to 0, W_{n}P_{n} = W_{n} and W_{n}^{2} = P_{n}. Let A denote the set of all sequences {A_{n}} ∈ E for which A_{n}, A_{n}^{∗}, Ã_{n} := W_{n}A_{n}W_{n} and Ã_{n}^{∗} are strongly convergent.

Lemma 3.3 The set A is a C^{∗}-algebra.

Proof. Evidently, A is a linear space. If {A_{n}} ∈ A, then so is {A_{n}^{∗}} (note that {W_{n}A_{n}^{∗}W_{n}} = {(W_{n}A_{n}W_{n})^{∗}}). Let {A^{(n)}} be a fundamental sequence in A, where A^{(n)} = {A_{k}^{(n)}}_{k=1}^{∞}. If ε > 0 is given, we have

∥A_{k}^{(n)} − A_{k}^{(m)}∥ < ε

for all k if m, n are large enough. Hence, there is a sequence {A_{k}} ∈ L(X) such that ∥A_{k}^{(n)} − A_{k}∥ → 0 (n → ∞) uniformly with respect to k. If we choose x ∈ X, we can estimate

∥(A_{k} − A_{l})x∥ ≤ ∥A_{k} − A_{k}^{(n)}∥ ∥x∥ + ∥A_{l} − A_{l}^{(n)}∥ ∥x∥ + ∥(A_{k}^{(n)} − A_{l}^{(n)})x∥.

The first two terms can be made arbitrarily small by the choice of n, and the third one goes to zero if n is fixed and k, l → ∞, since {A_{k}^{(n)}}_{k=1}^{∞} ∈ A. Hence, A_{k} is strongly convergent. The sequences A_{k}^{∗}, Ã_{k} and Ã_{k}^{∗} are treated in the same way (note that ∥Ã_{k}^{(n)} − Ã_{k}^{(m)}∥ ≤ const ∥A_{k}^{(n)} − A_{k}^{(m)}∥ because of the weak convergence of W_{n}).

In the following, the coset {A_{n}} + N of a sequence {A_{n}} ∈ E will be denoted by ˆ {A_{n}}. The ideal N is contained in A, and thus Â := A/N is a C^{∗}-subalgebra of Ê := E/N and is therefore inverse-closed (that means Â ∩ GÊ = GÂ). Hence, the stability of {A_{n}} ∈ A is equivalent to ˆ {A_{n}} ∈ GÂ.

In all what follows, K(X, Y) denotes the space of all compact linear operators between two Banach spaces X and Y. If X = Y, we briefly write K(X) instead of K(X, X).

Lemma 3.4 ([PS], 1.1.h) Let X, Y, Z, V be Banach spaces, {A_{n}} ⊂ L(Z, V), {B_{n}} ⊂ L(X, Y) such that A_{n} → A ∈ L(Z, V), B_{n}^{∗} → B^{∗} ∈ L(Y^{∗}, X^{∗}) strongly. If K ∈ K(Y, Z), then ∥A_{n}KB_{n} − AKB∥_{L(X,V)} → 0 (n → ∞). If K ∈ K(Y, Z) and B_{n} ⇀ B ∈ L(X, Y) (weakly), then KB_{n} → KB strongly.

Let J_{0} := { ˆ {P_{n}KP_{n}} : K ∈ K(X)}, J_{1} := { ˆ {W_{n}KW_{n}} : K ∈ K(X)}. By Lemma 3.4, J_{0}, J_{1} are subsets of Â.

Lemma 3.5 ([Si], Satz 2) J_{0}, J_{1} are closed ideals in Â, and the smallest closed ideal containing J_{0} and J_{1} equals

J = {{P_{n}K_{1}P_{n} + W_{n}K_{2}W_{n}} + N : K_{1}, K_{2} ∈ K(X)}.

Proof. We show that I_{0} := {{P_{n}KP_{n} +C_{n}} : K ∈ K(X), ∥C_{n}∥ → 0} is a closed ideal in A, whence the corresponding property of J_{0} in Â follows immediately. Obviously, I_{0} is a linear space. Let {B_{n}} ⊂ I_{0} be a fundamental sequence, B_{n} = {A_{k}^{(n)}}_{k=1}^{∞}, A_{k}^{(n)} = P_{k}T^{(n)}P_{k} + C_{k}^{(n)}, where T^{(n)} ∈ K(X), ∥C_{k}^{(n)}∥ → 0 (k → ∞). We have A_{k}^{(n)} → T^{(n)} (strongly), and hence for ε > 0 the relation ∥T^{(n)} − T^{(m)}∥ ≤ sup_{k} ∥A_{k}^{(n)} − A_{k}^{(m)}∥ < ε holds for n, m large enough.

Therefore, T^{(n)} converges uniformly to some T ∈ K(X). Besides, we can estimate

∥C_{k}^{(n)} − C_{k}^{(m)}∥ ≤ ∥P_{k}(T^{(n)} − T^{(m)})P_{k}∥ + ∥A_{k}^{(n)} − A_{k}^{(m)}∥ < ε

for sufficiently large m, n independently of k. Thus, there exists a sequence {C_{k}} with ∥C_{k}^{(n)} − C_{k}∥ → 0 (n → ∞), and since ∥C_{k}∥ ≤ ∥C_{k}^{(n)} − C_{k}∥ + ∥C_{k}^{(n)}∥, we have {C_{k}} ∈ N. If we put B := {P_{k}T P_{k} + C_{k}}, we have

∥B_{n} − B∥_{A} ≤ const ∥T^{(n)} − T ∥ + sup_{k} ∥C_{k}^{(n)} − C_{k}∥ ⟶ 0 (n → ∞),

which proves the closedness of I_{0}. To show (for instance) that I_{0} is a left ideal, let {A_{k}} ∈ A, A_{k} → A, {B_{k}} = {P_{k}T P_{k} + C_{k}} ∈ I_{0}. Then

\{A_{k}\}\{B_{k}\} = \{P_{k}A_{k}T P_{k} + A_{k}C_{k}\} = \{P_{k}AT P_{k} + \underbrace{P_{k}(A_{k} − A)T P_{k} + A_{k}C_{k}}_{∈N}\} ∈ I_{0}

(cf. Lemma 3.4). The proof for J_{1}, J is analogous. Obviously, J is contained in every ideal that contains J_{0} and J_{1}, which completes the proof.

For ˆ {A_{n}} ∈ Â we define

W_{0} ˆ {A_{n}} := s− lim_{n→∞} A_{n}, W_{1} ˆ {A_{n}} := s− lim_{n→∞} W_{n}A_{n}W_{n}.

Note that W_{0}, W_{1} are correctly defined. Evidently, W_{i} (i = 0, 1) are continuous ∗-homomorphisms from Â into L(X) (remember that ∥s− lim A_{n}∥ ≤ lim inf ∥A_{n}∥). Further, it is easy to see that W_{i}|_{J_{i}} is an isomorphism between J_{i} and the ideal K(X) of all compact linear operators on X, in other words, W_{i} is J_{i}-lifting. If we apply Theorem 3.1 with T = {0, 1} to this situation, we obtain the original version of the lifting theorem:

Theorem 3.4 ([Si], Satz 3) Let {A_{n}} ∈ A, A_{n} → A, Ã_{n} → Ã strongly. Then {A_{n}} is stable if and only if A, Ã ∈ GL(X) and ˆ {A_{n}} + J ∈ G(Â/J).

Remark 3.6 The third condition of the preceding theorem can be written in the equivalent form {A_{n}} + I ∈ G(A/I), with the ideal I = {{P_{n}K_{1}P_{n} + W_{n}K_{2}W_{n} + C_{n}} : K_{1}, K_{2} ∈ K(X), ∥C_{n}∥ → 0}.

In the following two sections we are going to apply the tools from this section to the stability analysis of the collocation method. There we choose P_{n} = P_{n}^{σ} and introduce the operators W_{n} = W_{n}^{σ} defined by

W_{n}^{σ} ∑_{k=0}^{∞} α_{k}ũ_{k} = ∑_{k=0}^{n−1} α_{n−1−k}ũ_{k},

which obviously possess the properties required above. The approximation operators under consideration here are A_{n} = L̃_{n}^{σ}(aI + bS)P_{n}^{σ}, a, b ∈ P C, in the space X = L_{σ}^{2}. We will show that, under the additional condition b(±1) = 0, we have {A_{n}} ∈ A, compute Ã and apply Theorem 3.4 (with Remark 3.6) to the stability problem. The invertibility of the coset {A_{n}} + I will be investigated by the local principle of Gohberg and Krupnik.

4 Strong convergence of the operator sequences

The strong convergence of A_{n} = L̃_{n}^{σ}(aI + bS)P_{n}^{σ} and of A_{n}^{∗} will be shown for Riemann integrable coefficients. Unfortunately, the authors did not succeed in proving the convergence of Ã_{n} and Ã_{n}^{∗} without the additional condition b ∈ P C and b(±1) = 0.

4.1 Strong convergence of A_{n}

First we give sufficient conditions for the weighted interpolation polynomial L̃_{n}^{σ}f to converge in the L_{σ}^{2}-norm. For this end, we provide some material from [Fr] concerning the convergence of Gaussian quadrature rules and Lagrangian interpolation operators.

Consider a Jacobi weight v. Let x_{kn}^{v} (k = 1, …, n) be the zeros of the orthogonal polynomial of degree n related to v, and L_{n}^{v} the Lagrangian interpolation operator with respect to x_{kn}^{v}. By Q_{n}^{v} we denote the Gaussian quadrature rule

Q_{n}^{v}f := ∫_{−1}^{1}(L_{n}^{v}f)(x)v(x) dx = ∑_{k=1}^{n} A_{kn}^{v}f(x_{kn}^{v}).

Lemma 4.1 ([Fr], Hilfssatz III.1.5) Let g : (−1, 1) → [0, ∞), g^{(2ν)}(x) ≥ 0 for all x ∈ (−1, 1) and ν = 0, 1, 2, …, and let ∫_{−1}^{1} g(x)v(x) dx < ∞. Then

Q_{n}^{v}g ≤ ∫_{−1}^{1} g(x)v(x) dx.

Lemma 4.2 ([Fr], Satz III.1.4) Let f be bounded on (−1, 1). Then

Q_{n}^{v}f → ∫_{−1}^{1} f(x)v(x) dx (n → ∞)

provided that this integral exists in the Riemann sense.

Lemma 4.3 (cf. [Fr], Satz III.1.6b) Assume that f is bounded on every compact subinterval of (−1, 1) and the (improper) Riemann integral ∫_{−1}^{1} f(x)v(x) dx exists. Suppose that there exist functions g_{−1}, g_{1} satisfying the conditions of Lemma 4.1 and the relations

lim_{x→−1+0} \frac{f(x)}{g_{−1}(x)} = lim_{x→1−0} \frac{f(x)}{g_{1}(x)} = 0. (4.1)

Then Q_{n}^{v}f → ∫_{−1}^{1} f(x)v(x) dx (n → ∞).

Proof. Splitting the interval, we can restrict ourselves to the case f(x) = 0 in some neighbourhood of 1, g := g_{−1}. Let ε > 0 be arbitrary and δ > 0 such that |f(x)| ≤ εg(x) for −1 < x ≤ −1 + δ. Lemma 4.2 yields

lim_{n→∞} ∑_{x_{kn}^{v}≥−1+δ} A_{kn}^{v}f(x_{kn}^{v}) = ∫_{−1+δ}^{1} f(x)v(x) dx.

Furthermore,

| ∑_{x_{kn}^{v}<−1+δ} A_{kn}^{v}f(x_{kn}^{v})| ≤ ε ∑_{k=1}^{n} A_{kn}^{v}g(x_{kn}^{v}) ≤ ε ∫_{−1}^{1} g(x)v(x) dx

by Lemma 4.1, and

|∫_{−1}^{1+δ} f(x)v(x) dx| ≤ ε ∫_{−1}^{1} g(x)v(x) dx.

Remark 4.4 If v = v^{γ,δ} with γ, δ satisfying (2.2) we can choose g_{−1} = (1 + x)^{−1−δ+ε}, g_{1}(x) = (1 − x)^{−1−γ+ε} with some ε > 0. Hence, the Gaussian quadrature rule converges if f is locally Riemann integrable and satisfies

|f(x)| ≤ const(1 − x)^{−1−γ+ε}(1 + x)^{−1−δ+ε}.

Lemma 4.5 ([Fr], Satz III.2.1) Assume the hypotheses of Lemma 4.3 are fulfilled with

|f|^{2} instead of f and lim_{x→−1+0} g_{−1}(x) = lim_{x→1−0} g_{1}(x) = ∞. Then lim_{n→∞} ∥L_{n}^{v}f − f∥_{L_{v}^{2}} = 0.

Proof. According to [Fr], Satz III.4.3, the polynomials are dense in L_{v}^{2}. (This remains true in the complex case, since real and imaginary part can be approximated separately.) Let ε > 0, and let p be a polynomial with ∥p − f∥_{L_{v}^{2}} < ε. For n > deg p we have

\begin{aligned}∥f − L_{n}^{v}f∥^{2} & ≤ & 2(∥f − p∥^{2} + ∥L_{n}^{v}(p − f)∥^{2}) \\ & < & 2(ε^{2} + Q_{n}^{v}(|p − f|^{2})) \end{aligned}

(note that the Gaussian quadrature rule with n nodes is exact for polynomials of degree less than 2n). Given r > 0, in a suitable neighbourhood of − 1 the relation

|p(x) − f(x)|^{2} ≤ 2(|p(x)|^{2} + |f(x)|^{2}) < rg_{−1}(x)

holds (as well as the analogous relation in a neighbourhood of 1). Thus, (4.1) is satisfied with |p − f|^{2} instead of f. We further have

∫_{−1}^{1} |p(x) − f(x)|^{2}v(x) dx ≤ 2 ∫_{−1}^{1}(|p(x)|^{2} + |f(x)|^{2})v(x) dx < ∞.

Lemma 4.3 now yields Q_{n}^{v}(|p − f|^{2}) → ∫_{−1}^{1} |p(x) − f(x)|^{2}v(x) dx < ε^{2}.

Corollary 4.6 Let σ = v^{α,β}. If f is locally Riemann integrable on (−1, 1) and

|f(x)| ≤ const(1 − x)^{(−1−α)/2+ε}(1 + x)^{(−1−β)/2+ε}

with some ε > 0, then

∥L̃_{n}^{σ}f − f∥_{σ} → 0 (n → ∞).

Proof. Using the isometric isomorphism w_{σ^{−1}}I : L_{φ}^{2} → L_{σ}^{2}, we have

∥L̃_{n}^{σ}f − f∥_{σ} = ∥L_{n}^{φ}w_{σ^{−1}}^{−1}f − w_{σ^{−1}}^{−1}f∥_{φ}.

Now the assertion follows from Lemma 4.5 and Remark 4.4.

In the sequel we will investigate the behaviour of the operators A_{n}. First we do this for the multiplication operator A = aI separately.

Proposition 4.1 Let a be Riemann integrable. Then L̃_{n}^{σ}aP_{n}^{σ} → aI strongly on L_{σ}^{2}. Furthermore, we have the estimation ∥L̃_{n}^{σ}aP_{n}^{σ}∥_{L(L_{σ}^{2})} ≤ ∥a∥_{∞}.

Proof. First we show the convergence on the dense subset span {ũ_{m}}_{m=0}^{∞}. Clearly, the functions aũ_{m} satisfy the conditions of Corollary 4.6. Thus, for n > m we have ∥A_{n}ũ_{m} − Aũ_{m}∥_{σ} = ∥L̃_{n}^{σ}aũ_{m}−aũ_{m}∥_{σ} → 0. For showing the uniform boundedness, let u ∈ L_{σ}^{2} and write P_{n}^{σ}u = w_{σ^{−1}}q_{n} with a polynomial q_{n} of degree less than n. If we again note the exactness of the Gaussian quadrature rule, we have

∥L̃_{n}^{σ}aP_{n}^{σ}u∥_{σ}^{2} = ∥L_{n}^{φ}aq_{n}∥_{φ}^{2} = Q_{n}^{φ}(|aq_{n}|^{2})
≤ ∥a∥_{∞}^{2}Q_{n}^{φ}(|q_{n}|^{2}) = ∥a∥_{∞}^{2}∥q_{n}∥_{φ}^{2} = ∥a∥_{∞}^{2}∥P_{n}^{σ}u∥_{σ}^{2} ≤ ∥a∥_{∞}^{2}∥u∥_{σ}^{2}.

Now the assertion follows from the Banach-Steinhaus theorem.

Let ϱ be a Jacobi weight and μ ∈ (0, 1). By H_{0}^{μ}(ϱ) we denote the Banach space of all functions f for which ϱf ∈ C^{0,μ}[−1, 1] and (ϱf)(±1) = 0. The norm in this space is defined

by ∥f∥_{H_{0}^{μ}(ϱ)} := ∥ϱf∥_{C^{0,μ}}, where, as usually, ∥g∥_{C^{0,μ}} = ∥g∥_{∞} + sup_{x≠y} \frac{|g(x) − g(y)|}{|x − y|^{μ}}.

Lemma 4.7 ([GK], Theorem I.6.2) Let ϱ = v^{γ,δ}, μ ∈ (0, 1) and μ < γ, δ < μ + 1. Then the Cauchy singular integral operator S defined by (2.3) is bounded on H_{0}^{μ}(ϱ).

Proposition 4.2 The operators L̃_{n}^{σ}SP_{n}^{σ} converge to S on span {ũ_{m}}_{m=0}^{∞}.

Proof. Let ϱ = v^{γ,δ}, where we choose γ, δ such that

max \{0, \frac{α}{2} − \frac{1}{4}\} < γ < \frac{1 + α}{2}, max \{0, \frac{β}{2} − \frac{1}{4}\} < δ < \frac{1 + β}{2} ,

and let

0 < μ < min \{γ, δ, γ + \frac{1}{4} − \frac{α}{2}, δ + \frac{1}{4} − \frac{β}{2} \} .

Then we have μ < γ, δ < μ + 1 and ũ_{m} ∈ H_{0}^{μ}(ϱ). The latter relation follows from

ϱũ_{m}(x) = U_{m}(x)(1 − x)^{γ+1/4−α/2}(1 + x)^{δ+1/4−β/2},

the exponents being greater than μ. By Lemma 4.7, we also have Sũ_{m} ∈ H_{0}^{μ}(ϱ), which means that h := ϱSũ_{m} ∈ C^{0,μ}. Thus, we can estimate

|(Sũ_{m})(x)| = |(ϱ^{−1}h)(x)| = |h(x)(1 − x)^{−γ}(1 + x)^{−δ}|
≤ const (1 − x)^{−(1+α)/2+ε}(1 + x)^{−(1+β)/2+ε}

if ε > 0 is small enough. Corollary 4.6 now gives the assertion.

In all what follows we exclude the cases α = \frac{1}{2} and β = \frac{1}{2} since they bring about some technical difficulties in the proofs that we could only partially overcome. Some remarks concerning these cases will be given in a separate subsection.

Lemma 4.8 (comp. [PS], 9.7 and 9.9) Assume that (2.2) is satisfied. Let g, b̃ ∈ C^{0,η}[−1, 1] (η ∈ (0, 1)) be real-valued functions for which g(x) − ib̃(x) ≠ 0 for all x ∈ [−1, 1]. Let λ, ν be integers such that α_{0} := λ + g̃(1), β_{0} := ν − g̃(−1) ∈ (−1, 1), where g(x)−ib̃(x) = \sqrt{b̃^{2}(x) + g^{2}(x)}e^{iπg̃(x)} with a continuous function g̃. Then there exists a positive function c ∈ C^{0,η}[−1, 1] such that the operator (gI + iSb̃I)v^{α_{0},β_{0}}cI transforms every polynomial of degree n into a polynomial of degree n − κ, where κ = −(λ + ν). (If n < κ, this is to be understood in the sense that a polynomial of negative degree is identically zero.)

If, as we assumed, α ≠ \frac{1}{2} and β ≠ \frac{1}{2}, we can choose b̃ ≡ 1 and g ∈ C^{1} such that λ := \frac{1}{4} − \frac{α}{2} − g̃(1) and ν := \frac{1}{4} − \frac{β}{2} + g̃(−1) are integers, whence we get the mapping properties described in Lemma 4.8 for the operator (gI + iS)w_{σ^{−1}}cI with some positive function c, c ∈ C^{0,η} for all η ∈ (0, 1). Furthermore, we can always achieve κ ≥ −1 (we need the latter relation to guarantee the exactness of the Gaussian quadrature rule): Evidently, we always have g̃(x) ∈ (−1, 0).

• In case α_{0} := \frac{1}{4} − \frac{α}{2}, β_{0} := \frac{1}{4} − \frac{β}{2} < 0 we choose g̃(1) = α_{0}, g̃(−1) = −1 − β_{0} and have κ = 1.

• If α_{0} , β_{0} > 0, let g̃(1) = α_{0} − 1, g̃(−1) = −β_{0}.

• Finally, take g̃(1) = α_{0}, g̃(−1) = −β_{0} if α_{0} < 0 < β_{0}, and g̃(1) = α_{0} − 1, g̃(−1) = −1 − β_{0} if β_{0} < 0 < α_{0}.

Our proof of the uniform boundedness of L̃_{n}^{σ}SP_{n}^{σ} will be based on the following decomposition of the operator S:

S = igI − ic^{−1}(gI + iS)cI + c^{−1}(cS − ScI). (4.2)

Here g ∈ C^{1} and c are the same as in Lemma 4.8. In particular, c ∈ C^{0,η} for all η ∈ (0, 1). Now we are going to estimate the three summands of L̃_{n}^{σ}SP_{n}^{σ} separately.

By virtue of Proposition 4.1 we have ∥L̃_{n}^{σ}gP_{n}^{σ}∥ ≤ ∥g∥_{∞}, so we are done with the first term.

Lemma 4.9 ([Ne], Theorem 9.25) Let v, v^{∗} be Jacobi weights with vv^{∗} ∈ L^{1}. Let l ∈ N be fixed and q a polynomial with deg q ≤ ln. Then

∑_{k=1}^{n} A_{kn}^{v}|q(x_{kn}^{v})|v^{∗}(x_{kn}^{v}) ≤ const ∫_{−1}^{1} |q(x)|v(x)v^{∗}(x) dx,

the constant being independent of n and q.

Proposition 4.3 We have the estimation

∥L̃_{n}^{σ}c^{−1}(gI + iS)cP_{n}^{σ}∥_{L(L_{σ}^{2})} ≤ const ∥c^{−1}∥_{∞} ∥gcI + iScI∥_{L(L_{σ}^{2})}.

Proof. According to Lemma 4.8, q_{n−κ} := (gI + iS)cP_{n}^{σ}u is a polynomial of degree less than n − κ for all u ∈ L_{σ}^{2}. Now we have

∥L̃_{n}^{σ}c^{−1}(gI + iS)cP_{n}^{σ}u∥_{σ}^{2} = ∥L_{n}^{φ}c^{−1}(w_{σ^{−1}})^{−1}q_{n−κ}∥_{φ}^{2}

= ∑_{k=1}^{n} A_{kn}^{φ}|c^{−1}(x_{kn}^{φ})|^{2}(w_{σ^{−1}}(x_{kn}^{φ}))^{−2}|q_{n−κ}(x_{kn}^{φ})|^{2}

≤ ∥c^{−1}∥_{∞}^{2} ∑_{k=1}^{n} A_{kn}^{φ}(w_{σ^{−1}}(x_{kn}^{φ}))^{−2}|q_{n−κ}(x_{kn}^{φ})|^{2}

≤ const ∥c^{−1}∥_{∞}^{2} ∫_{−1}^{1} |q_{n−κ}(x)|^{2}(w_{σ^{−1}}(x))^{−2}φ(x) dx

= const ∥c^{−1}∥_{∞}^{2} ∥q_{n−κ}∥_{σ}^{2}

≤ const ∥c^{−1}∥_{∞}^{2} ∥gcI + iScI∥_{L(L_{σ}^{2})}^{2} ∥u∥_{σ}^{2}, where we used Lemma 4.9 with v = φ, v^{∗} = w_{σ^{−1}}^{−2} for the estimation in the fourth line.

The following lemma is a generalization of [Ju1, Lemma 2.3]. Lemma 4.10 Let σ = v^{α,β} and 0 < γ < m := \frac{1−max\{α,β,0\}}{2}. Then

∫_{−1}^{1} |\frac{1}{|t − x|^{γ}} − \frac{1}{|t − y|^{γ}} |^{2} σ^{−1}(t) dt ≤ const |x − y|^{2λ}

for all λ ∈ (0, 1) with λ + γ < m. Proof. Let λ_{0} := λ + γ. We have

|t − x|^{−γ} − |t − y|^{−γ} = \frac{|t − x|^{λ_{0}−γ} − |t − y|^{λ_{0}−γ}}{|t − x|^{λ_{0}}} + \frac{|t − x|^{λ_{0}−γ} − |t − y|^{λ_{0}−γ}}{|t − y|^{λ_{0}}}

+\frac{|t − y|^{2λ_{0}−γ} − |t − x|^{2λ_{0}−γ}}{|t − x|^{λ_{0}}|t − y|^{λ_{0}}} .

Hence, we can estimate

||t − x|^{−γ} − |t − y|^{−γ}|^{2}

≤ 3 (|t − x|^{λ_{0}−γ} − |t − y|^{λ_{0}−γ})^{2} (\frac{1}{|t − x|^{2λ_{0}}} + \frac{1}{|t − y|^{2λ_{0}}} )

+3\frac{(|t − y|^{2λ_{0}−γ} − |t − x|^{2λ_{0}−γ})^{2}}{|t − x|^{2λ_{0}}|t − y|^{2λ_{0}}}

≤ const [|x − y|^{2(λ_{0}−γ)} (\frac{1}{|t − x|^{2λ_{0}}} + \frac{1}{|t − y|^{2λ_{0}}} ) + \frac{|x − y|^{4λ_{0}−2γ}}{|t − x|^{2λ_{0}}|t − y|^{2λ_{0}}} ]

= const |x − y|^{2λ} [\frac{1}{|t − x|^{2λ_{0}}} + \frac{1}{|t − y|^{2λ_{0}}} + |\frac{1}{t − x} − \frac{1}{t − y} |^{2λ_{0}} ]

≤ const |x − y|^{2λ} (\frac{1}{|t − x|^{2λ_{0}}} + \frac{1}{|t − y|^{2λ_{0}}} ) .

Obviously, we have σ^{−1} ∈ L^{p} for p < \frac{1}{max\{α,β,0\}}. Then for the adjoint exponent q the relation q > \frac{1}{1−max\{α,β,0\}} = \frac{1}{2m} holds. Since λ_{0} < m, we can guarantee 2λ_{0}q < 1. Hence, the Hölder inequality gives

∫_{−1}^{1} \frac{σ^{−1}(t)}{|t − x|^{2λ_{0}}} dt ≤ ∥σ^{−1}∥_{L^{p}} (∫_{−1}^{1} \frac{dt}{|t − x|^{2λ_{0}q}} )^{\frac{1}{q}} ≤ const

independently of x, and the lemma is proved.

Lemma 4.11 Let σ = v^{α,β} and c ∈ C^{0,η} with η > \frac{1+max\{α,β,0\}}{2}. Then K := cS − ScI ∈ K(L_{σ}^{2}, C^{0,λ}) for some λ > 0.

Proof. Choose 1−η < γ < \frac{1−max\{α,β,0\}}{2} and put k(t, x) := \frac{c(t)−c(x)}{t−x}|t−x|^{γ}. If λ ≤ η−(1−γ), we have k ∈ C^{0,λ} in both variables, uniformly with respect to the other ([Mu], §5). Moreover, we require γ + λ < \frac{1−max\{α,β,0\}}{2}. Let now u ∈ L_{σ}^{2}. We can write

|(Ku)(x) − (Ku)(y)| ≤ ∫_{−1}^{1} |\frac{k(t, x)}{|t − x|^{γ}} − \frac{k(t, y)}{|t − y|^{γ}} | |u(t)| dt

≤ (∫_{−1}^{1} |\frac{k(t, x)}{|t − x|^{γ}} − \frac{k(t, y)}{|t − y|^{γ}} |^{2} σ^{−1}(t) dt)^{\frac{1}{2}} ∥u∥_{σ},

and, using Lemma 4.10, the following estimation holds:

∫_{−1}^{1} |\frac{k(t, x)}{|t − x|^{γ}} − \frac{k(t, y)}{|t − y|^{γ}} |^{2} σ^{−1}(t) dt ≤

≤ ∫_{−1}^{1} (|\frac{k(t, x) − k(t, y)}{|t − x|^{γ}} | + |k(t, y)| |\frac{1}{|t − x|^{γ}} − \frac{1}{|t − y|^{γ}} |)^{2} σ^{−1}(t) dt

≤ 2 ∫_{−1}^{1} (|\frac{k(t, x) − k(t, y)}{|t − x|^{γ}} |^{2} + |k(t, y)|^{2} |\frac{1}{|t − x|^{γ}} − \frac{1}{|t − y|^{γ}} |^{2}) σ^{−1}(t) dt

≤ const |x − y|^{2λ} ∫_{−1}^{1} \frac{σ^{−1}(t) dt}{|t − x|^{2γ}} + const |x − y|^{2λ}

≤ const |x − y|^{2λ}.

Hence, all functions in {Ku : ∥u∥_{σ} ≤ 1} uniformly satisfy a Hölder condition with the exponent λ. It remains to show the uniform boundedness of these functions. Using Lemma 2.4 from [CJLM], we get

|(Ku)(x)| ≤ \frac{1}{π} ∫_{−1}^{1} |\frac{c(t) − c(x)}{t − x}| |u(t)| dt

≤ const (∫_{−1}^{1} \frac{σ^{−1}(t) dt}{|t − x|^{2(1−η)}} )^{\frac{1}{2}} ∥u∥_{σ}
≤ const (1 − x)^{−α^{+}/2}(1 + x)^{−β^{+}/2}∥u∥_{σ},

where α^{+} := max{0, α}, β^{+} := max{0, β}. In particular, ∥u∥_{σ} ≤ 1 implies |(Ku)(0)| ≤ const, which together with the uniform Hölder condition results in ∥Ku∥_{∞} ≤ const. Thus, we have K ∈ L(L_{σ}^{2}, C^{0,λ}) for some λ > 0, and the assertion follows if we note that the embedding C^{0,λ} ⊂ C^{0,λ′} is compact for λ > λ′.

Using Lemma 4.11 and Corollary 4.6, we can now estimate the third summand:

∥L̃_{n}^{σ}c^{−1}KP_{n}^{σ}∥_{L(L_{σ}^{2})} ≤ ∥L̃_{n}^{σ}∥_{L(C^{0,λ},L_{σ}^{2})} ∥c^{−1}∥_{C^{0,λ}} ∥K∥_{L(L_{σ}^{2},C^{0,λ})} ≤ const.

Thus, we have proved the uniform boundedness of L̃_{n}^{σ}SP_{n}^{σ}. If we note the obvious identity L̃_{n}^{σ}bSP_{n}^{σ} = L̃_{n}^{σ}bP_{n}^{σ} L̃_{n}^{σ}SP_{n}^{σ}, we can summarize the results of this subsection as follows:

Theorem 4.1 Let a, b be Riemann integrable on [−1, 1]. Then the operators A_{n} = L̃_{n}^{σ}(aI + bS)P_{n}^{σ} converge strongly to aI + bS on L_{σ}^{2}.

4.2 The cases α = \frac{1}{2}, β = \frac{1}{2}

If one of the numbers α_{0} := \frac{1}{4} − \frac{α}{2}, β_{0} := \frac{1}{4} − \frac{β}{2} is zero, the decomposition (4.2), on which our proof of the uniform boundedness of L̃_{n}^{σ}SP_{n}^{σ} was based, is not possible. We cannot choose b̃ ≡ 1 in Lemma 4.8 since in this case α_{0} − g̃(1) (or β_{0} + g̃(−1), respectively) cannot be an integer. We can, however, to some extent overcome the difficulties connected with this fact if we estimate the whole term L̃_{n}^{σ}bSP_{n}^{σ} instead of considering L̃_{n}^{σ}SP_{n}^{σ} separately.

Assume for instance α_{0} = 0, β_{0} ≠ 0. In this case we can proceed as follows. Let b be a function satisfying the following conditions:

\{ \begin{matrix}b ∈ C^{0,η} for some η > \frac{1+max\{α,β,0\}}{2} \\ b(1) = 0 \\ b(−1) ≠ 0. \end{matrix} (4.3)

(If β_{0} = 0, we would also require b(−1) = 0). For the following argument we can assume without loss of generality that b is real-valued and b(−1) < 0. Now we choose some g ∈ C^{1}, g(1) = 1 (that is, g̃(1) = 0) such that the operator (gI +iSbI)w_{σ^{−1}}cI possesses the mapping properties described in Lemma 4.8 with κ ≥ −1. If β_{0} > 0, we can choose g such that g̃(−1) = 1 − β_{0}, whence κ = −1, if β_{0} < 0 we choose g̃(−1) = −β_{0} and have κ = 0. (If we had β_{0} = 0, we would simply take g ≡ 1, which means κ = 0.)

Instead of (4.2) we now use the decomposition

bS = SbI + K = igI − ic^{−1}(gI + iSbI)cI + c^{−1}(cS − ScI)bI + K, where K = bS − SbI ∈ K(L_{σ}^{2}, C^{0,μ}) for some μ > 0 (compare Lemma 4.11). This enables us to show in the same way as before that A_{n} = L̃_{n}^{σ}bSP_{n}^{σ} and A_{n}^{∗} converge strongly if b satisfies (4.3). In particular, we have ∥L̃_{n}^{σ}(1 − x)^{η}SP_{n}^{σ}∥ ≤ const if η > \frac{1+max\{α,β,0\}}{2}. Proposition 4.4 Let α_{0} = 0, β_{0} ≠ 0. Let b ∈ P C and b(x) = o((1 − x)^{η}) for x → 1 with some η > \frac{1+max\{α,β,0\}}{2}. Then ∥L̃_{n}^{σ}bSP_{n}^{σ}∥ ≤ const.

Proof. Let χ be Riemann integrable, and let b̃ ∈ P_{1}, which denotes the set of all polynomials vanishing in 1. Then L̃_{n}^{σ}χb̃SP_{n}^{σ} = L̃_{n}^{σ}χ(1 + k)^{k}P_{n}^{σ} L̃_{n}^{σ}b̃_{1}SP_{n}^{σ} with some nonnegative integer k, where b̃_{1} satisfies (4.3). Hence, the uniform boundedness of L̃_{n}^{σ}bSP_{n}^{σ} continues to hold if b is the product of a polynomial from P_{1} with a Riemann integrable function.

Now let b be as in the hypothesis with a finite number of jumps. Then (1 − x)^{−η}b is piecewise continuous and can therefore be approximated uniformly by a piecewise polynomial b̃ = ∑_{j=1}^{m} χ_{j}b̃_{j}, where b̃_{j} ∈ P_{1}, χ_{j} is the characteristic function of [x_{j}, x_{j+1}] and −1 = x_{1} < x_{2} < … < x_{m+1} = 1. (Note that P_{1} is dense in C[x_{j}, x_{j+1}] for all j = 1, … m due to the Stone-Weierstraß theorem.) Then we have

∥L̃_{n}^{σ} ((1 − x)^{η}b̃ − b) SP_{n}^{σ}∥ ≤ ∥L̃_{n}^{σ} (b̃ − (1 − x)^{−η}b) P_{n}^{σ}∥ ∥L̃_{n}^{σ}(1 − x)^{η}SP_{n}^{σ}∥
≤ const ∥b̃ − (1 − x)^{−η}b∥_{∞} ,

which can be made as small as desired by the choice of b̃. If we note the fact that the set of all sequences from E which, together with their adjoint operators, are strongly convergent is a closed subalgebra of E (compare the proof of Lemma 3.3), we get the assertion for b with a finite number of jumps. If b ∈ P C is arbitrary, b(x) = o((1 − x)^{η}), we approximate (1 − x)^{−η}b uniformly by a function b̃ with the same properties and only finitely many jumps and repeat the same arguments as above to get the assertion for b.

Remark 4.12 If α_{0} = β_{0} = 0, we would have to require b = o((1 − x^{2})^{η}) for x → ±1 in the preceding proposition.

4.3 Strong convergence of A_{n}^{∗}

In all what follows we identify the dual space of L_{σ}^{2} with L_{σ}^{2} itself and consider A_{n}^{∗} as an element of L(L_{σ}^{2}). As an auxiliary relation, we deduce a formula for the Fourier coefficients of L̃_{n}^{σ}f: We have L̃_{n}^{σ}f = ∑_{k=0}^{n−1} α_{k}ũ_{k}, where, because of the exactness of the Gaussian quadrature rule,

\begin{matrix}α_{k} & = & ∑_{s=0}^{n−1} α_{s} \overbrace{∑_{j=1}^{n} Ã_{jn}^{σ}ũ_{k}(x_{jn}^{φ})ũ_{s}(x_{jn}^{φ})}^{=δ_{ks}} \\ & = & ∑_{j=1}^{n} Ã_{jn}^{σ}ũ_{k}(x_{jn}^{φ}) ∑_{s=0}^{n−1} α_{s}ũ_{s}(x_{jn}) \\ & = & ∑_{j=1}^{n} Ã_{jn}^{σ}ũ_{k}(x_{jn}^{φ})f(x_{jn}^{φ}), \end{matrix} (4.4)

where Ã_{jn}^{σ} := w_{σ^{−1}}^{−2}(x_{jn}^{φ})A_{jn}^{φ} and δ_{ks} denotes the Kronecker symbol.

Now we compute (L̃_{n}^{σ}aP_{n}^{σ})^{∗} with Riemann integrable a. For u = ∑_{k=0}^{∞} u_{k}ũ_{k}, v = ∑_{k=0}^{∞} v_{k}ũ_{k} we have due to (4.4)

(L̃_{n}^{σ}aP_{n}^{σ}u, v)_{σ} =

= ∑_{k=0}^{n−1} \overline{v_{k}} (∑_{j=1}^{n} Ã_{jn}^{σ}a(x_{jn}^{φ}) ∑_{s=0}^{n−1} u_{s}ũ_{s}(x_{jn}^{φ}) ũ_{k}(x_{jn}^{φ}))

= ∑_{s=0}^{n−1} u_{s}\overline{(∑_{j=1}^{n} Ã_{jn}^{σ}\overline{a}(x_{jn}^{φ}) ∑_{k=0}^{n−1} v_{k}ũ_{k}(x_{jn}^{φ}) ũ_{s}(x_{jn}^{φ}))}

= (u, L̃_{n}^{σ}\overline{a}P_{n}^{σ}v)^{σ},

that means

(L̃_{n}^{σ}aP_{n}^{σ})^{∗} = L̃_{n}^{σ}\overline{a}P_{n}^{σ},

which is strongly convergent due to Proposition 4.1.

Since we have (L̃_{n}^{σ}bSP_{n}^{σ})^{∗} = (L̃_{n}^{σ}SP_{n}^{σ})^{∗} (L̃_{n}^{σ}bP_{n}^{σ})^{∗}, we can now restrict ourselves to investigating (L̃_{n}^{σ}SP_{n}^{σ})^{∗}. This will be done again by using a three-term decomposition according to (4.2). The multiplication operator was already considered. To deal with (gI +iS)cI (we can obviously neglect the factor c^{−1}), we note that in Lemma 4.8 there is always κ ∈ {−1, 0, 1}, which implies that

q_{n−κ} := (gI + iS)cP_{n}^{σ}u

is always a polynomial of degree at most n. If u, v ∈ L_{σ}^{2} and Ã_{jn}^{σ} are as above, we can write

(L̃_{n}^{σ}(gI + iS)cP_{n}^{σ}u, v)_{σ} =

= ∑_{j=1}^{n} Ã_{jn}^{σ}q_{n−κ}(x_{jn}^{φ})\overline{(P_{n}^{σ}v)(x_{jn}^{φ})}

= (w_{σ^{−1}}q_{n−κ}, L̃_{n}^{σ}w_{σ^{−1}}^{−1}P_{n}^{σ}v)_{σ}

= (w_{σ^{−1}}(gI + iS)cP_{n}^{σ}u, L̃_{n}^{σ}w_{σ^{−1}}^{−1}P_{n}^{σ}v)_{σ}
= (u, P_{n}^{σ}c(gI + iS)^{∗}w_{σ^{−1}} L̃_{n}^{σ}w_{σ^{−1}}^{−1}P_{n}^{σ}v)_{σ}.

(Note that gI + iS ∈ L(L_{σ}^{2}) and w_{σ^{−1}} L̃_{n}^{σ}f ∈ L_{σ}^{2}.) Hence, we have

(L̃_{n}^{σ}(gI + iS)cP_{n}^{σ})^{∗} = P_{n}^{σ}c(gI + iS)^{∗}w_{σ^{−1}} L̃_{n}^{σ}w_{σ^{−1}}^{−1}P_{n}^{σ}.

Lemma 4.13 (Cf. [MR], Cor. 3.2 and the following remark; [DT], Th. 6.2.1) Let u, w be Jacobi weights satisfying

\frac{u}{\sqrt{wφ}}, \frac{\sqrt{wφ}}{u} ∈ L^{2}.

If f is a function with f′φu ∈ L^{2}, then the following estimation holds:

∥u(L_{n}^{w}f − f)∥_{L^{2}} ≤ \frac{const}{n}∥f′φu∥_{L^{2}}.

Lemma 4.14 The operators P_{n}^{σ}c(gI + iS)^{∗}w_{σ^{−1}} L̃_{n}^{σ}w_{σ^{−1}}^{−1}P_{n}^{σ} converge strongly in L_{σ}^{2}.

Proof. Since the uniform boundedness is trivial in view of Lemma 4.3, we only have to show the convergence on span {ũ_{m}}_{m=0}^{∞}. First we consider the term w_{σ^{−1}} L̃_{n}^{σ}w_{σ^{−1}}^{−1}I. Let m ≥ n. We can write

∥w_{σ^{−1}} L̃_{n}^{σ}w_{σ^{−1}}^{−1}ũ_{m} − ũ_{m}∥_{σ} = ∥L̃_{n}^{σ}U_{m} − U_{m}∥_{φ} = ∥φ^{1/2}w_{σ^{−1}}(L_{n}^{φ}w_{σ^{−1}}^{−1}U_{m} − w_{σ^{−1}}^{−1}U_{m})∥_{L^{2}}.

We now apply Lemma 4.13 with f = w_{σ^{−1}}^{−1}U_{m}, u = φ^{1/2}w_{σ^{−1}} and w = φ, which allows us to estimate the last expression by

\frac{const}{n} ∥(w_{σ^{−1}}^{−1}U_{m})′ φ^{1/2}w_{σ^{−1}} φ∥_{L^{2}} ≤ \frac{const}{n} ⟶ 0 (n → ∞).

Since P_{n}^{σ} → I and (gI + iS)^{∗} ∈ L(L_{σ}^{2}), the assertion follows.

Due to Lemma 4.11, we have K := cS −ScI ∈ K(L_{σ}^{2}, C^{0,λ}) for some λ > 0. Furthermore, (P_{n}^{σ})^{∗} = P_{n}^{σ} → I in L_{σ}^{2} and L̃_{n}^{σ} → E (cf. Corollary 4.6), where E denotes the continuous embedding of C^{0,λ} into L_{σ}^{2}. Thus, if we write again K instead of EK, Lemma 3.4 gives

∥L̃_{n}^{σ}KP_{n}^{σ} − K∥_{L(L_{σ}^{2})} ⟶ 0 (n → ∞),

and hence

∥(L̃_{n}^{σ}KP_{n}^{σ})^{∗} − K^{∗}∥_{L(L_{σ}^{2})} ⟶ 0 (n → ∞).

Thus, we have proved the following theorem:

Theorem 4.2 If a, b are Riemann integrable, then (L̃_{n}^{σ}(aI +bS)P_{n}^{σ})^{∗} is strongly convergent.

4.4 Strong convergence of Ã_{n}

First we consider the case of a multiplication operator A = aI with a Riemann integrable function a. The following lemma together with Proposition 4.1 shows the convergence of Ã_{n} in this case.

Lemma 4.15 We have W_{n}^{σ}L̃_{n}^{σ}aW_{n}^{σ} ≡ L̃_{n}^{σ}aP_{n}^{σ}.

Proof. It is sufficient to show the identity on span {ũ_{m}}. For n > m we have (compare relation (4.4))

W_{n}^{σ}L̃_{n}^{σ}aW_{n}^{σ}ũ_{m} = W_{n}^{σ}L̃_{n}^{σ}aũ_{n−1−m}

= W_{n}^{σ} ∑_{k=0}^{n−1} (∑_{j=1}^{n} Ã_{jn}^{σ}ũ_{k}(x_{jn}^{φ})ũ_{n−1−m}(x_{jn}^{φ})a(x_{jn}^{φ})) ũ_{k}

= ∑_{k=0}^{n−1} (∑_{j=1}^{n} Ã_{jn}^{σ}ũ_{n−1−k}(x_{jn}^{φ})ũ_{n−1−m}(x_{jn}^{φ})a(x_{jn}^{φ})) ũ_{k} =: ∑_{k=0}^{n−1} β_{km}ũ_{k}.

If we remember that x_{jn}^{φ} = cos \frac{jπ}{n+1} (j = 1, …, n) and A_{jn}^{φ} = π \frac{1−(x_{jn}^{φ})^{2}}{n+1} , we get

β_{km} = \frac{2}{n + 1} ∑_{j=1}^{n} sin \frac{(n − k)jπ}{n + 1} sin \frac{(n − m)jπ}{n + 1} a(x_{jn}^{φ})

= \frac{2}{n + 1} ∑_{j=1}^{n} sin \frac{(k + 1)jπ}{n + 1} sin \frac{(m + 1)jπ}{n + 1} a(x_{jn}^{φ})

= β_{n−1−k,n−1−m},

which means W_{n}^{σ}L̃_{n}^{σ}aW_{n}^{σ}ũ_{m} = L̃_{n}^{σ}aP_{n}^{σ}ũ_{m}.

Let V := xI − iw_{σ^{−1}}Sw_{σ}I, that is, (V u)(x) = xu(x) − iw_{σ^{−1}}(x)(Sw_{σ}u)(x) for u ∈ L_{σ}^{2}. The following lemma is a generalization of a result from [RR] (cf. also [PS, Th. 4.123]), which is formulated there for the case α = β = 0.

Lemma 4.16 ([RR], see also [PS], Theorem 4.123) V is a shift operator with respect to the system {ũ_{n}}_{n=0}^{∞}, more precisely, the relation

V ∑_{k=0}^{∞} α_{k}ũ_{k} = ∑_{k=0}^{∞} α_{k}ũ_{k+1}

holds. The adjoint operator, which satisfies

V^{∗} ∑_{k=0}^{∞} α_{k}ũ_{k} = ∑_{k=0}^{∞} α_{k+1}ũ_{k},

is given by

V^{∗} = xI + iw_{σ^{−1}}Sw_{σ}I.

Proof. For the shift property of V compare the proof of [PS, Theorem 4.123]. To verify the representation of V^{∗}, note that σ^{1/2}I : L_{σ}^{2} → L^{2} is an isometric isomorphism and S^{∗} = S in L^{2}.

Lemma 4.17 ([Lu], Lemma 3.10, cf. also [Mu], §5) Assume that

b ∈ C^{p,η}[−1, 1] (0 < η ≤ 1)  and  b^{(j)}(±1) = 0 (j = 0, …, p).

Let further v = v^{−γ,−δ} and λ := η − max{γ, δ, 0} > 0. Then

bv ∈ C^{p,λ}[−1, 1]  and  (bv)^{(j)}(±1) = 0 (j = 0, …, p).

We introduce the notations

P C_{0} := {b ∈ P C[−1, 1] : b(±1) = 0} ,
C_{0} := {b ∈ C[−1, 1] : b(±1) = 0}

and

C_{0,0}^{1,η} := {b ∈ C^{1,η}[−1, 1] : b(±1) = b′(±1) = 0} .

Proposition 4.5 Let b ∈ P C_{0}. Then W_{n}^{σ}L̃_{n}^{σ}bSW_{n}^{σ} converges strongly to −bw_{σ}^{−1}Sw_{σ}I.

Proof. First let b ∈ C_{0,0}^{1,η}, where η > max\{\frac{1}{4} + \frac{α}{2}, \frac{1}{4} + \frac{β}{2} , 0\}. We use the following decomposition of bS (cf. Lemma 4.16):

\begin{matrix}bS & = & bw_{σ}^{−1}Sw_{σ}I + \underbrace{bS − SbI + (Sbw_{σ}^{−1}I − bw_{σ}^{−1}S)w_{σ}I}_{=:K} \\ & = & ibφ^{−1}(xI − V^{∗}) + K. \end{matrix} (4.5)

Using Lemma 4.15, we can manage the first two summands (note that bφ^{−1} is continuous):

iW_{n}^{σ}L̃_{n}^{σ}bφ^{−1}xW_{n}^{σ} = iL̃_{n}^{σ}bφ^{−1}xP_{n}^{σ} ⟶ ibφ^{−1}xI

and

−iW_{n}^{σ}L̃_{n}^{σ}bφ^{−1}V^{∗}W_{n}^{σ} = −iW_{n}^{σ}L̃_{n}^{σ}bφ^{−1}W_{n}^{σ} · W_{n}^{σ}V^{∗}W_{n}^{σ}
= −iL̃_{n}^{σ}bφ^{−1}P_{n}^{σ} · P_{n}^{σ}V ⟶ −ibφ^{−1}V.

The multiplication operator w_{σ}I is an isometric isomorphism from L_{σ}^{2} onto L_{φ^{−1}}^{2}. Since b ∈ C_{0,0}^{1,η}, we get bφ^{−1} ∈ C^{1} from Lemma 4.17, and Lemma 4.11 gives us Sbw_{σ}^{−1}I − bw_{σ}^{−1}S ∈ K(L_{φ^{−1}}^{2}, C^{0,λ}) as well as bS − SbI ∈ K(L_{σ}^{2}, C^{0,λ}) with some λ > 0. Hence, K ∈ K(L_{σ}^{2}, C^{0,λ}). Since W_{n}^{σ} converges weakly to 0, we have KW_{n}^{σ} → 0 strongly in L(L_{σ}^{2}, C^{0,λ}) by virtue of Lemma 3.4. Moreover, we have ∥W_{n}^{σ}L̃_{n}^{σ}∥ ≤ ∥W_{n}^{σ}∥∥L̃_{n}^{σ}∥_{L(C^{0,λ},L_{σ}^{2})} ≤ const, which results in W_{n}^{σ}L̃_{n}^{σ}KW_{n}^{σ} → 0 (strongly). Thus, we can conclude

W_{n}^{σ}L̃_{n}^{σ}bSW_{n}^{σ} → ibφ^{−1}(xI − V) = −bw_{σ}^{−1}Sw_{σ}I (b ∈ C_{0,0}^{1,η}).

If χ is an arbitrary Riemann integrable function and b ∈ C_{0,0}^{1,η}, we get (using Lemma 4.15)

W_{n}^{σ}L̃_{n}^{σ}χbSW_{n}^{σ} = W_{n}^{σ}L̃_{n}^{σ}χW_{n}^{σ} · W_{n}^{σ}L̃_{n}^{σ}bSW_{n}^{σ} → −χbw_{σ}^{−1}Sw_{σ}I. (4.6)

Now we consider the general case b ∈ P C_{0}. Without loss of generality we can restrict ourselves to investigating functions with a finite number of jumps, that is, we can write b = ∑_{j=1}^{m} χ_{j}b_{j}, where b_{j} ∈ C_{0}, χ_{j} = χ_{[x_{j},x_{j+1}]} is the characteristic function of the subinterval [x_{j}, x_{j+1}], and −1 = x_{1} < x_{2} < … < x_{m} < x_{m+1} = 1 is an arbitrary partition of [−1, 1].

Now we approximate b by piecewise C_{0,0}^{1,η}-functions: Let ε > 0 and choose numbers y_{1} ∈ (−1, x_{2}), y_{m+1} ∈ (x_{m}, 1) such that

|b_{1}(x)| < \frac{ε}{2} for x ≤ y_{1}, |b_{m}(x)| < \frac{ε}{2} for x ≥ y_{m+1}.

Let (for instance) b̃_{0}(x) = b̃_{m+1}(x) := \frac{ε}{2}(1 − x)^{2}(1 + x)^{2}. Then b̃_{0}, b̃_{m+1} ∈ C_{0,0}^{1,η},

|b_{1}(x) − b̃_{0}(x)| < ε for x ∈ [−1, y_{1}], |b_{m}(x) − b̃_{m+1}(x)| < ε for x ∈ [y_{m+1}, 1].

Put y_{j} := x_{j} (j = 2, …, m), y_{0} := −1, y_{m+2} := 1. For j = 1, …, m we choose functions b̃_{j} ∈ C_{0,0}^{1,η} such that

|b_{j}(x) − b̃_{j}(x)| < ε, x ∈ [y_{j}, y_{j+1}].

If we define χ̃_{j} := χ_{[y_{j},y_{j+1}]}, b̃:= ∑_{j=0}^{m+1} χ̃_{j}b̃_{j}, we obviously have ∥b − b̃∥_{∞} < ε. Since

∥{L̃_{n}^{σ}(b − b̃)SP_{n}^{σ}}∥_{E} ≤ ∥L̃_{n}^{σ}(b − b̃)P_{n}^{σ}∥ ∥L̃_{n}^{σ}SP_{n}^{σ}∥ ≤ const ∥b − b̃∥_{∞} < ε

and since A is a closed subalgebra of E, we can conclude from (4.6) and Proposition 4.6 from the following subsection that W_{n}^{σ}L̃_{n}^{σ}bSW_{n}^{σ} is strongly convergent. The continuity of the homomorphism W_{1} and the relation

∥(b − b̃)w_{σ}^{−1}Sw_{σ}I∥ ≤ const ∥b − b̃∥_{∞}

imply that W_{1} {L̃_{n}^{σ}bSP_{n}^{σ}} = −bw_{σ}^{−1}Sw_{σ}I for b ∈ P C_{0} with finitely many jumps. An arbitrary P C_{0}-function can be approximated uniformly by such functions, and we can repeat the same arguments as above to get the assertion.

Remark 4.18 If we have one of the special cases from subsection 4.2, say α = \frac{1}{2}, we require b(x) = o((1 − x)^{ξ}), ξ > \frac{1+max\{α,β,0\}}{2} and choose b̃ in the preceding proof such that ∥(b − b̃)(1 − x)^{−ξ}∥_{∞} becomes small. Thus we can estimate ∥{L̃_{n}^{σ}(b − b̃)SP_{n}^{σ}}∥_{E} ≤ ∥(b − b̃)(1 − x)^{−ξ}∥_{∞} ∥L̃_{n}^{σ}(1 − x)^{ξ}SP_{n}^{σ}∥ < ε.

Remark 4.19 (cf. [GK], Theorem IX.4.1) If a ∈ P C, b ∈ P C_{0}, the operator Ã = aI − bw_{σ}^{−1}Sw_{σ}I is invertible in L_{σ}^{2} if and only if A = aI + bS is so.

4.5 Strong convergence of Ã_{n}^{∗}

Proposition 4.6 If a is Riemann integrable and b ∈ P C_{0}, then Ã_{n}^{∗} = (W_{n}^{σ}L̃_{n}^{σ}(aI+bS)W_{n}^{σ})^{∗} is strongly convergent.

Proof. For the multiplication operator we have

(W_{n}^{σ}L̃_{n}^{σ}aW_{n}^{σ})^{∗} = (L̃_{n}^{σ}aP_{n}^{σ})^{∗} = L̃_{n}^{σ}\overline{a}P_{n}^{σ} ⟶ \overline{a}I

by Lemma 4.15.

The investigation of bS will again be based on (4.5), where we first assume b ∈ C_{0,0}^{1,η}. The expression bφ^{−1}xI is already covered by the preceding arguments. Further we have

(W_{n}^{σ}L̃_{n}^{σ}bφ^{−1}V^{∗}W_{n}^{σ})^{∗} = (W_{n}^{σ}L̃_{n}^{σ}bφ^{−1}W_{n}^{σ} · W_{n}^{σ}V^{∗}W_{n}^{σ})^{∗}
= W_{n}^{σ}V W_{n}^{σ}L̃_{n}^{σ}\overline{b}φ^{−1}P_{n}^{σ} = V^{∗}P_{n}^{σ}L̃_{n}^{σ}\overline{b}φ^{−1}P_{n}^{σ} ⟶ V^{∗}\overline{b}φ^{−1}I.

Since K ∈ K(L_{σ}^{2}, C^{0,λ}) and hence ∥L̃_{n}^{σ}K − K∥_{L(L_{σ}^{2})} → 0, we also have ∥(L̃_{n}^{σ}K)^{∗} − K^{∗}∥ → 0. Thus, we can write

(W_{n}^{σ}L̃_{n}^{σ}KW_{n}^{σ})^{∗} = W_{n}^{σ} ((L̃_{n}^{σ}K)^{∗} − K^{∗}) W_{n}^{σ} + W_{n}^{σ}K^{∗}W_{n}^{σ},

the first summand uniformly and the second strongly converging to 0 (compare Lemma 3.4). If χ is Riemann integrable (in particular a characteristic function of a subinterval) and b ∈ C_{0,0}^{1,η}, we have

(W_{n}^{σ}L̃_{n}^{σ}χbSW_{n}^{σ})^{∗} = (W_{n}^{σ}L̃_{n}^{σ}bSW_{n}^{σ})^{∗}(W_{n}^{σ}L̃_{n}^{σ}χW_{n}^{σ})^{∗},

which allows us to apply the preceding reflections. By approximation we can finally get the assertion for arbitrary b ∈ P C_{0} (compare the proof of Proposition 4.5).

5 Application of the local principle and main result

Having shown {A_{n}} = {L̃_{n}^{σ}(aI + bS)P_{n}^{σ}} ∈ A and having computed Ã, we are left with investigating the invertibility of the coset {A_{n}} + I ∈ A/I (comp. Theorem 3.4), which will be done by the local principle of Gohberg and Krupnik.

For t ∈ [−1, 1] let

m_{t} := {f ∈ C[−1, 1] : 0 ≤ f(x) ≤ 1, f(x) ≡ 1 in some neighbourhood of t}

and define

M_{t} := {{L̃_{n}^{σ}fP_{n}^{σ}} + I : f ∈ m_{t}}.

Lemma 5.1 (comp. [JS], Lemma 2.6) {M_{t}}_{t∈[−1,1]} is a covering system of localizing classes in A/I.

Now we have to show that {A_{n}} + I commutes with all elements of ⋃_{t∈[−1,1]} M_{t}. For this

end, we consider an analogous problem in the space L^{2}(T), where T is the unit circle {t ∈ C : |t| = 1}. If f ∈ L^{2}(T), f(t) = ∑_{k=−∞}^{∞} f_{k}t^{k}, we introduce the operators

(P_{n}^{T}f)(t) := ∑_{k=−n−1}^{n} f_{k}t^{k},

(W_{n}^{T}f)(t) := f_{−1}t^{−n−1} + ⋯ + f_{−n−1}t^{−1} + f_{n} + ⋯ + f_{0}t^{n}

and consider the algebra A^{T} and the ideal (in A^{T}) I^{T} related to these operator sequences and defined analogously to A and I.

Let M_{n} be the Multhopp interpolation operator that assigns to every Riemann integrable function f on T the polynomial (M_{n}f)(t) = ∑_{k=−n−1}^{n} α_{k}t^{k} coinciding with f in the nodes e^{(ikπ)/(n+1)} (k = −n − 1, …, n). We remind that ∥M_{n}f − f∥_{L^{2}(T)} → 0 for all Riemann integrable f. We further introduce the projections (P f)(t) = ∑_{k=0}^{∞} f_{k}t^{k}, Q = I − P and (T f)(t) = \frac{1}{2}(f(t) − f(t^{−1})). Note that T is the orthogonal projection onto the subspace of all odd functions (that means, the space of all f ∈ L^{2}(T) for which f(t) = −f(t̄) for all t ∈ T). If a is a complex-valued function on [−1, 1], we define a function â on T by â(e^{iφ}) := a(cos φ).

Finally, we use the following mapping F from L_{σ}^{2} onto the subspace of all odd functions:

(F u)(t) := \{ \begin{matrix}\sqrt{π}u(ℜt)w_{σ}(ℜt), & ℑt > 0 \\ −(F u)(t̄), & ℑt < 0 \\ 0, & t = ±1, \end{matrix}

where ℜt and ℑt denote the real and the imaginary part of a complex number t, respectively. Using the formula

U_{n}(x) = \frac{(x + i\sqrt{1 − x^{2}})^{n+1} − (x − i\sqrt{1 − x^{2}})^{n+1}}{\sqrt{2π}i\sqrt{1 − x^{2}}} ,

we obtain

(F ũ_{n})(t) = \frac{t^{n+1} − t^{−n−1}}{\sqrt{2}i} ,

which shows that F is an isometric isomorphism between L_{σ}^{2} and the space of all odd functions in L^{2}(T). The following lemma summarizes the transformation of some operators we are interested in. Lemma 5.2 The following identities hold: L̃_{n}^{σ} = F^{−1}M_{n}F, bI = F^{−1}b̂F, P_{n}^{σ} = F^{−1}P_{n}^{T}F P_{n}^{σ}, V^{∗} = F^{−1}(t^{−1}P + tQ)F, P_{n}^{σ}F^{−1}T P_{n}^{T} = P_{n}^{σ}F^{−1}T, P_{n}^{T}F P_{n}^{σ} = F P_{n}^{σ}, P_{n}^{σ}F^{−1}T W_{n}^{T} = W_{n}^{σ}F^{−1}T (tP + QtI), W_{n}^{T}F P_{n}^{σ} = t^{−1}F W_{n}^{σ}.

Lemma 5.3 ([JS], Lemma 2.5) If f ∈ C(T), then the sequences {QM_{n}fP P_{n}^{T}} and {P M_{n}fQP_{n}^{T}} belong to I^{T}.

Lemma 5.4 Let f ∈ C[−1, 1]. Then the sequence {L̃_{n}^{σ}fV^{∗}P_{n}^{σ} − V^{∗}L̃_{n}^{σ}fP_{n}^{σ}} belongs to I.

Proof. Using Lemma 5.2, we can transform the sequence under consideration as follows: {L̃_{n}^{σ}fV^{∗}P_{n}^{σ} − V^{∗}L̃_{n}^{σ}fP_{n}^{σ}} (5.7)

= {F^{−1} [M_{n}f̂(t^{−1}P + tQ)P_{n}^{T} − (t^{−1}P + tQ)M_{n}f̂P_{n}^{T}] F P_{n}^{σ}}

We will show that the term in brackets belongs to I^{T}. Note that we can insert the operator M_{n} before the expression (t^{−1}P + tQ)M_{n}f̂P_{n}^{T} since the space im M_{n} is left invariant by t^{−1}P + tQ. We have

{M_{n}f̂tQP_{n}^{T} − M_{n}tQM_{n}f̂P_{n}^{T}}

= {M_{n}tP_{n}^{T}(M_{n}f̂QP_{n}^{T} − QM_{n}f̂P_{n}^{T})}
= {M_{n}tP_{n}^{T}}{P_{n}^{T}M_{n}f̂QP_{n}^{T} − QM_{n}f̂P P_{n}^{T}} ∈ I^{T}

according to Lemma 5.3. (Note that {M_{n}tP_{n}^{T}} ∈ A^{T} ([JS]).) Analogously,

{M_{n}t^{−1}P M_{n}f̂P_{n}^{T} − M_{n}f̂t^{−1}P P_{n}^{T}} = {M_{n}t^{−1}P_{n}^{T}}{P M_{n}f̂QP_{n}^{T} − QM_{n}f̂P P_{n}^{T}} ∈ I^{T}.

Hence, there are operators K_{1}, K_{2} ∈ K(L^{2}(T)) such that (5.7) equals

{P_{n}^{σ}F^{−1}T (P_{n}^{T}K_{1}P_{n}^{T} + W_{n}^{T}K_{2}W_{n}^{T} + C_{n})F P_{n}^{σ}},

where ∥C_{n}∥ → 0. (We insert the projection T to be able to consider the three summands individually, since it is not guaranteed that each of them maps into im F.) If we use the relations given in Lemma 5.2, we see that the latter expression equals

{P_{n}^{σ}F^{−1}T K_{1}F P_{n}^{σ} + W_{n}^{σ}F^{−1}T (tP + QtI)K_{2}t^{−1}F W_{n}^{σ} + P_{n}^{σ}F^{−1}T C_{n}F P_{n}^{σ}},

which is obviously an element of I.

Proposition 5.1 Let a ∈ P C, b ∈ P C_{0}. The coset {L̃_{n}^{σ}(aI + bS)P_{n}^{σ}} + I commutes with all elements of ⋃_{t∈[−1,1]} M_{t}.

Proof. In the case of the multiplication operator A = aI, the assertion is a consequence of

L̃_{n}^{σ}aP_{n}^{σ}L̃_{n}^{σ}fP_{n}^{σ} = L̃_{n}^{σ}fP_{n}^{σ}L̃_{n}^{σ}aP_{n}^{σ} = L̃_{n}^{σ}afP_{n}^{σ}.

As for the singular integral operator, we start with considering coefficients of the form χb with a Riemann integrable function χ and b ∈ C_{0,0}^{1,η}, where η > max\{\frac{1}{4} + \frac{α}{2}, \frac{1}{4} + \frac{β}{2} , 0\}. Equation (4.5) and Lemma 4.11 give

χbS = χK + iχbφ^{−1}(xI − V^{∗})

with χK ∈ K(L_{σ}^{2}, R), where R denotes the Banach space of all Riemann integrable functions on [−1, 1], endowed with the supremum norm. Consequently, the commutator L̃_{n}^{σ}fP_{n}^{σ} L̃_{n}^{σ}χKP_{n}^{σ} − L̃_{n}^{σ}χKP_{n}^{σ} L̃_{n}^{σ}fP_{n}^{σ} converges uniformly to fχK − χKfI ∈ K(L_{σ}^{2}) and is therefore of the form

P_{n}^{σ}(fχK − χKfI)P_{n}^{σ} + C_{n}, ∥C_{n}∥ → 0,

which is contained in I.

If we abbreviate c := iχbφ^{−1}, it remains to consider

L̃_{n}^{σ}fcV^{∗}P_{n}^{σ} − L̃_{n}^{σ}cV^{∗}L̃_{n}^{σ}fP_{n}^{σ} = L̃_{n}^{σ}cP_{n}^{σ}(L̃_{n}^{σ}fV^{∗}P_{n}^{σ} − V^{∗}L̃_{n}^{σ}fP_{n}^{σ}),

and Lemma 5.4 shows that this expression is in I. Hence, the assertion is true for coefficients of the form χb, where b ∈ C_{0,0}^{1,η}, and χ is a characteristic function of a subinterval. Arbitrary P C_{0}-coefficients can be approximated in the supremum norm by sums of such functions (compare the proof of Lemma 4.5). Finally, we take into account that ∥{L̃_{n}^{σ}bSP_{n}^{σ}} + I∥ ≤ const ∥b∥_{∞} and that the ideal I is closed, which completes the proof of the proposition.

Now we are able to give local representatives for {A_{n}} + I.

Lemma 5.5 Let τ ∈ [−1, 1], a, a_{τ} ∈ P C, b, b_{τ} ∈ P C_{0} such that

a_{τ}(τ ± 0) = a(τ ± 0),  b_{τ}(±0) = b(τ ± 0).  (5.8)

Then {L̃_{n}^{σ}(aI + bS)P_{n}^{σ}} + I and {L̃_{n}^{σ}(a_{τ}I + b_{τ}S)P_{n}^{σ}} + I are M_{τ}-equivalent. If further b_{τ} = χ_{[−1,τ]}b_{1} +χ_{[τ,1]}b_{2} with b_{1}, b_{2} ∈ C_{0,0}^{1,η}, η > max\{\frac{1}{4} + \frac{α}{2}, \frac{1}{4} + \frac{β}{2} , 0\}, then \{L̃_{n}^{σ}((a_{τ} + ib_{τ}φ^{−1}x)I − ib_{τ}φ^{−1}V^{∗})P_{n}^{σ}} + I is M_{τ}- equivalent to both cosets.

Proof. Let f ∈ m_{τ}. We have

∥{L̃_{n}^{σ}fP_{n}^{σ}}{L̃_{n}^{σ}((a − a_{τ})I + (b − b_{τ})S)P_{n}^{σ}}∥_{A/I}
≤ ∥L̃_{n}^{σ}f(a − a_{τ})P_{n}^{σ} + L̃_{n}^{σ}f(b − b_{τ})P_{n}^{σ} L̃_{n}^{σ}SP_{n}^{σ}∥_{L(L_{σ}^{2})}
≤ ∥f(a − a_{τ})∥_{∞} + const ∥f(b − b_{τ})∥_{∞},

which can be made arbitrarily small by a suitable choice of f. Thus, we have proved the first assertion (note Proposition 5.1). For the second, formula (4.5) shows

bS = ib_{τ}φ^{−1}(xI − V^{∗}) + χ_{[−1,t]}K_{1} + χ_{[t,1]}K_{2}

with K_{1}, K_{2} ∈ K(L_{σ}^{2}, C^{0,λ}) and therefore L̃_{n}^{σ}(χ_{[−1,t]}K_{1} + χ_{[t,1]}K_{2})P_{n}^{σ} ∈ I.

Remark 5.6 If we have one of the special cases with respect to the exponents of the weight σ considered in Subsection 4.2, we slightly modify the preceding lemma and its proof. If for instance α = \frac{1}{2}, we require that b(x) = o((1 − x)^{ξ}) for x → 1, ξ > \frac{1+max\{α,β,0\}}{2}, which ensures ∥L̃_{n}^{σ}(1−x)^{ξ}SP_{n}^{σ}∥ ≤ const. Then we estimate as follows: ∥L̃_{n}^{σ}f(b−b_{τ})P_{n}^{σ} L̃_{n}^{σ}SP_{n}^{σ}∥ ≤ ∥f(1 − x)^{−ξ}(b − b_{τ})∥_{∞} ∥L̃_{n}^{σ}(1 − x)^{ξ}SP_{n}^{σ}∥.

In the following we are going to give stability conditions. If the coefficients of the singular integral operator are continuous, we can obtain a very general result.

Theorem 5.1 Let a, b ∈ C[−1, 1] with b(±1) = 0, and let A = aI + bS. Then the sequence {A_{n}} = {L̃_{n}^{σ}AP_{n}^{σ}} is stable if and only if A is invertible in L_{σ}^{2}.

Proof. Due to Theorem 3.4 and Remark 4.19, we only have to consider the invertibility of the coset {A_{n}}+I. Let τ ∈ [−1, 1]. Note that the invertibility of A implies a^{2}(τ)−b^{2}(τ) ≠ 0 ( [GK, Th. IX.4.1]). We choose a_{τ} ∈ C, b_{τ} ∈ C_{0,0}^{1,η} such that (5.8) is fulfilled, and additionally a_{τ}^{2}(x) − b_{τ}^{2}(x) ≠ 0 for all x ∈ [−1, 1]. Then, because of Lemma 5.5, an M_{τ}-equivalent local representative of {A_{n}} + I is given by {A_{n}^{t}} + I, where (cf. Lemma 5.2)

A_{n}^{τ} = L̃_{n}^{σ} ((a_{τ} + ib_{τ}φ^{−1}x)I − ib_{τ}φ^{−1}V^{∗}) P_{n}^{σ}
= F^{−1}M_{n} ((â_{τ} + ib̂_{τ}φ̂^{−1}x̂)I − ib̂_{τ}φ̂^{−1}(t^{−1}P + tQ)) P_{n}^{T}F P_{n}^{σ}
= F^{−1} [M_{n}(c_{τ}P + c̃_{τ}Q)P_{n}^{T}] F P_{n}^{σ}

with c_{τ}(s) = â_{τ} + ib̂_{τ}φ̂^{−1}(x̂ − s^{−1}) and c̃_{τ}(s) = c_{τ}(s^{−1}), and hence

c_{τ}(x + iy) = \{ \begin{matrix}a_{τ}(x) − b_{τ}(x), & y ≥ 0 \\ a_{τ}(x) + b_{τ}(x), & y < 0. \end{matrix}

Let

B_{n}^{τ} = L̃_{n}^{σ} ((\frac{a_{τ}}{a_{τ}^{2} − b_{τ}^{2}} − i\frac{b_{τ}}{a_{τ}^{2} − b_{τ}^{2}} φ^{−1}x) I + i\frac{b_{τ}}{a_{τ}^{2} − b_{τ}^{2}} φ^{−1}V^{∗}) P_{n}^{σ}
= F^{−1} [M_{n}(c_{τ}^{−1}P + c̃_{τ}^{−1}Q)P_{n}^{T}] F P_{n}^{σ}.

Evidently, B_{n}^{τ} ∈ A. The proof of [JS, Theorem 2.1] now yields

A_{n}^{τ}B_{n}^{τ} = F^{−1} [M_{n}(c_{τ}P + c̃_{τ}Q)P_{n}^{T}] [M_{n}(c_{τ}^{−1}P + c̃_{τ}^{−1}Q)P_{n}^{T}] F P_{n}^{σ}
= F^{−1}[P_{n}^{T} + C_{n}^{T}]F P_{n}^{σ} = P_{n}^{σ} + C_{n},

where {C_{n}^{T}} ∈ I^{T}, {C_{n}} ∈ I (cf. the proof of Lemma 5.4). Analogously, B_{n}^{τ}A_{n}^{τ} = P_{n}^{σ} + C′_{n}, {C′_{n}} ∈ I, and Theorem 3.2 yields the assertion.

As for arbitrary piecewise continuous coefficients, we restrict ourselves to giving a sufficient stability condition in a special case.

Proposition 5.2 Let a ∈ P C, b ∈ P C_{0} such that A = aI + bS is invertible in L_{σ}^{2} and that moreover for the one-sided limits

|a(τ ± 0) + ib(τ ± 0)tφ^{−1}(τ)| > |b(τ ± 0)φ^{−1}(τ)|

holds for all τ ∈ [−1, 1]. Then {A_{n}} = {L̃_{n}^{σ}AP_{n}^{σ}} is stable.

Proof. Again we only have to show the invertibility of the coset {A_{n}} + I. For t ∈ [−1, 1] choose a_{τ} ∈ P C, b_{1}, b_{2} ∈ C_{0,0}^{1,η} (η > max\{\frac{1}{4} + \frac{α}{2}, \frac{1}{4} + \frac{β}{2} , 0\}) such that a_{τ} and b_{τ} := χ_{[−1,t]}b_{1} + χ_{[t,1]}b_{2} satisfy (5.8) and

|a_{τ}(s ± 0) + ib_{τ}(s ± 0)sφ^{−1}(s)| > |b_{τ}(s ± 0)φ^{−1}(s)| (5.9)

for all s ∈ [−1, 1]. Then {A_{n}^{τ}} + I, where A_{n}^{τ} is defined as in the proof of Theorem 5.1, is an M_{τ}-equivalent local representative of {A_{n}}. We have

A_{n}^{τ} = (L̃_{n}^{σ}(a_{τ} + ib_{τ}φ^{−1}x)P_{n}^{σ}) (P_{n}^{σ} − iL̃_{n}^{σ} \underbrace{b_{τ}φ^{−1}(a_{τ} + ib_{τ}φ^{−1}x)^{−1}}_{=:c_{τ}} P_{n}^{σ} V^{∗}P_{n}^{σ})

(for the invertibility of a_{τ} + ib_{τ}φ^{−1}x in L^{∞} note (5.9)). Since ∥{L̃_{n}^{σ}c_{τ}P_{n}^{σ}}∥_{A} ≤ ∥c_{τ}∥_{∞} < 1 and ∥V^{∗}P_{n}^{σ}∥ = 1, the sequence {P_{n}^{σ} − iL̃_{n}^{σ}c_{τ}P_{n}^{σ} V^{∗}P_{n}^{σ}} is invertible in A, which of course also implies the invertibility of {A_{n}^{τ}} + I in A/I.

6 Implementation and numerical results

6.1 Some remarks on the implementation

For practical computations, we write the weighted polynomial v_{n} that solves (2.5) in the form

v_{n}(x) = w_{σ^{−1}}(x) ∑_{k=0}^{n−1} ξ_{k}P_{k}^{(σ)}(x),

where P_{k}^{(σ)} denotes the monic orthogonal polynomial of degree k with respect to the weight w_{σ^{−1}}, and we solve the system of linear equations

∑_{k=0}^{n−1} \underbrace{[(aw_{σ^{−1}}P_{k}^{(σ)}) (x_{jn}^{φ}) + b(x_{jn}^{φ}) (Sw_{σ^{−1}}P_{k}^{(σ)}) (x_{jn}^{φ})]}_{=: a_{jk}} ξ_{k} = f(x_{jn}^{φ}), j = 1, …, n (6.10)

to determine the ξ_{k}, k = 0, …, n − 1.

It is well-known that the orthogonal polynomials satisfy a three-term recurrence formula of the form

P_{k+1}^{(σ)}(x) = (x − α_{k})P_{k}^{(σ)}(x) − β_{k}P_{k−1}^{(σ)}(x),  j = 0, 1, 2, …,  (6.11)

where P_{−1}^{(σ)} ≡ 0 and P_{1}^{(σ)} ≡ 1. In the case of Jacobi weights, there are explicit formulas for the α_{k} and β_{k}. This allows us to compute the matrix coefficients a_{jk} recursively. We have

a_{j,k+1} = (x_{jn}^{φ} − α_{k})a_{jk} − β_{k}a_{j,k−1}, k = 1, …, n − 1,

with the initial values

a_{j0} = (aw_{σ^{−1}}) (x_{jn}^{φ}) + b(x_{jn}^{φ})ϱ_{0}(x_{jn}^{φ})

and

a_{j1} = (x_{jn}^{φ} − α_{0}) (aw_{σ^{−1}}) (x_{jn}^{φ}) + b(x_{jn}^{φ}) (β_{0} + (x_{jn}^{φ} − α_{0})ϱ_{0}(x_{jn}^{φ}))
= (x_{jn}^{φ} − α_{0})a_{j0} + b(x_{jn}^{φ})β_{0},

where β_{0} := \frac{1}{πi} ∫_{−1}^{1} w_{σ^{−1}}(t) dt and ϱ_{0}(x) := (Sw_{σ^{−1}}) (x). The computation of ϱ_{0} is based upon formula (2.2) from [GW]. If (6.10) is solved, one can efficiently compute the values of v_{n} from the coefficients ξ_{j} using (6.11).

6.2 Numerical examples

In the following examples, we approximated the error ∥u − v_{n}∥_{σ} by the quadrature rule \sqrt{Q_{m}^{φ}(|u − v_{n}|^{2}σφ^{−1})} with m = 256. We always chose a ≡ 1 and α = β = 0.

Example 1 b(x) = i\sqrt{1 − x^{2}}, f(x) = 1 + \frac{\sqrt{1 − x^{2}}}{π} ln \frac{1 − x}{1 + x}, u ≡ 1.

Example 2 b(x) = i\sqrt{1 − x^{2}}, f(x) = |x| + x\frac{\sqrt{1 − x^{2}}}{π} ln \frac{(1 + x)(1 − x)}{x^{2}}, u(x) = |x|.

Example 3 b(x) = i\sqrt{1 − x^{2}}, f(x) = sgn x + \frac{\sqrt{1 − x^{2}}}{π} ln \frac{(1 + x)(1 − x)}{x^{2}}, u(x) = sgn x.

\sqrt{Q_{m}^{φ}(|u − v_{n}|^{2}σφ^{−1})}

n Ex. 1  Ex. 2  Ex. 3
8 7.99 E-3 8.04 E-3 4.98 E-2
16 4.04 E-3 4.12 E-3 3.56 E-2
32 2.01 E-3 2.04 E-3 2.52 E-2
64 9.49 E-4 9.59 E-4 1.77 E-2
128 3.65 E-4 3.68 E-4 1.18 E-2
256 2.65 E-6 2.07 E-5 5.78 E-3
512 1.02 E-7 1.78 E-6 4.28 E-4

We also considered some examples with b(±1) ≠ 0, which are not covered by the theoretical results of this paper. One is inclined to conjecture that in this case the sequence A_{n} is stable if and only if the operators A = aI + bS and Ã = aI − bw_{σ}^{−1}Sw_{σ}I are invertible in L_{σ}^{2}. In Example 4 this is the case, whereas in Example 5 the operator A is invertible but Ã is not. In both cases the approximate solutions seem to converge, but in Example 5 the convergence is somewhat slower in despite of the same smoothness of the input data.

Example 4 b = −\frac{i}{10}, f(x) = |x| − \frac{x}{10π} ln \frac{(1 + x)(1 − x)}{x^{2}}, u(x) = |x|.

Example 5 b = −i, f(x) = |x| − \frac{x}{π} ln \frac{(1 + x)(1 − x)}{x^{2}}, u(x) = |x|.

\sqrt{Q_{m}^{φ}(|u − v_{n}|^{2}σφ^{−1})}

n Ex. 4  Ex. 5
8 8.33 E-3 8.36 E-3
16 4.21 E-3 4.46 E-3
32 2.07 E-3 2.27 E-3
64 9.67 E-4 1.09 E-3
128 3.70 E-4 4.61 E-4
256 1.63 E-5 1.45 E-4
512 1.26 E-6 8.80 E-6

References

[BS] A. Böttcher, B. Silbermann: Analysis of Toeplitz Operators. Berlin, Heidelberg, New York: Springer-Verlag 1990.

[BHS] D. Berthold, W. Hoppe, B. Silbermann: A fast algorithm for solving the generalized airfoil equation. JCAM 43 (1992), 185–219.

[CJLM] M. R. Capobianco, P. Junghanns, U. Luther, G. Mastroianni: Weighted uniform convergence of the quadrature method for Cauchy singular integral equations. Singular Integral Operators and Related Topics, ed. by A. Böttcher and I. Gohberg, Operator Theory Advances and Applications, Vol. 90, Birkhäuser Verlag, 1996, 153– 181.

[DT] Z. Ditzian, V. Totik: Moduli of Smoothness. Berlin, Heidelberg, New York: Springer Verlag 1987.

[Fr] G. Freud: Orthogonale Polynome. Berlin: Deutscher Verlag der Wissenschaften 1969.

[GK] I. Gohberg, N. Krupnik: One-dimensional Linear Singular Integral Equations. Basel, Boston, Berlin: Birkhäuser Verlag 1992.

[GW] W. Gautschi, J. Wimp: Computing the Hilbert transform of a Jacobi weight function. BIT 27 (1987), 203–215.

[HRS] R. Hagen, S. Roch, B. Silbermann: Spectral Theory of Approximation Methods for Convolution Equations. Basel, Boston, Berlin: Birkhäuser Verlag 1994.

[JRW] P. Junghanns, S. Roch, U. Weber: Finite sections for singular integral operators by weighted Chebyshev polynomials. IEOT 21 (1995).

[Ju1] P. Junghanns: Product integration for the generalized airfoil equation, in: Beiträge zur Angewandten Analysis und Informatik (ed. E. Schock), Shaker Verlag, Aachen 1994, 171–188.

[Ju2] P. Junghanns: On the numerical solution of nonlinear singular integral equations. ZAMM 76(2) (1996), 152–166.

[JS] P. Junghanns, B. Silbermann: Local theory of the collocation method for singular integral equations. IEOT 7 (1984), 791–807.

[Lu] U. Luther: Kollokations- und Quadraturformelverfahren für singuläre Integralgleichungen bzgl. modifizierter Systeme von Nullstellen orthogonaler Polynome. Master’s Thesis, TU Chemnitz-Zwickau 1995.

[Ne] P. Nevai: Orthogonal polynomials. Mem. Amer. Math. Soc. 213 (1979).

[MR] G. Mastroianni, M. G. Russo: Lagrange interpolation in weighted Besov spaces, manuscript.

[Mu] N. I. Muskhelishvili: Singuläre Integralgleichungen. Berlin: Akademieverlag 1965.

[PS] S. Prössdorf, B. Silbermann: Numerical Analysis for Integral and Related Operator Equations. Basel, Boston, Berlin: Birkhäuser Verlag 1991.

[RR] M. Rosenblum, J. Rovnyak: Hardy classes and operator theory. Oxford University Press, New York, Clarendon Press, Oxford 1985.

[Si] B. Silbermann: Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren. Math. Nachr. 104 (1981), 137–146.