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