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Luther, Uwe : Weakly Singular Integral Operators in Weighted $L^\infty$--Spaces

Luther, Uwe : Weakly Singular Integral Operators in Weighted $L^\infty$--Spaces

Author(s):
Luther, Uwe
Title:
Weakly Singular Integral Operators in Weighted $L^\infty$--Spaces
Electronic source:
application/postscript
application/pdf
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 12, 2003
Mathematics Subject Classification:
45A05 [ Linear integral equations ]
45E99 [ None of the above, but in this section ]
Abstract:
We study integral operators on $(-1,1)$ with kernels $k(x,t)$ which may have weak singularities in $(x,t)$ with $x\in N_1$, $t\in N_2$, or $x=t$, where $N_1,N_2$ are sets of measure zero. It is shown that such operators map weighted $\fL^\infty$--spaces into certain weighted spaces of smooth functions, where the degree of smoothness is as higher as smoother the kernel $k(x,t)$ as a function in $x$. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors of best weighted uniform approximation by algebraic polynomials.
Keywords:
Weakly singular integral operators, Weighted spaces of continuous functions, Approximation spaces
Language:
English
Publication time:
12 / 2003