<p>This paper studies the degenerate kernel approximation theory for the second-kind Fredholm integral equations in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> spaces (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1 \leqslant p \leqslant \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>p</mi> <mo>⩽</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>). By introducing the mixed norm of kernel functions to construct an adaptive analytical tool, the research framework is extended from the classical <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> space to the more general <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> spaces. Based on the degenerate kernel approximation method, through refined norm estimation of iterated kernels and convergence analysis of the resolvent kernel series, the resolvent kernel representation theory for the solution of integral equations is established. This weakens the strong constraints on the growth of kernel functions and improves the applicability of the theory to non-compact intervals and weakly decaying kernel scenarios. On this basis, two types of error estimations are proposed: dual resolvent kernel error estimate and single resolvent kernel error estimate, which clearly characterize the error relationship between the approximate solution and the exact solution. This research provides a unified framework for the analysis of solutions to integral equations with different regularity characteristics and improves the system of integral equation approximation theory. Furthermore, the degenerate kernel approximation theory and error estimation results established in this paper can be directly extended to integral equations in high-dimensional Euclidean spaces, and their analytical framework and conclusions remain valid in high-dimensional cases.</p>

Degenerate kernel approximation and error estimation for integral equations in $L^p$ spaces

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发表日期：

2025年11月4日

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Sheng-Ya Feng,
Der-Chen Chang

摘要

This paper studies the degenerate kernel approximation theory for the second-kind Fredholm integral equations in $L^p$ $L^{p}$ spaces ( $1 \leqslant p \leqslant \infty $ $1 ⩽ p ⩽ \infty$ ). By introducing the mixed norm of kernel functions to construct an adaptive analytical tool, the research framework is extended from the classical $L^2$ $L^{2}$ space to the more general $L^p$ $L^{p}$ spaces. Based on the degenerate kernel approximation method, through refined norm estimation of iterated kernels and convergence analysis of the resolvent kernel series, the resolvent kernel representation theory for the solution of integral equations is established. This weakens the strong constraints on the growth of kernel functions and improves the applicability of the theory to non-compact intervals and weakly decaying kernel scenarios. On this basis, two types of error estimations are proposed: dual resolvent kernel error estimate and single resolvent kernel error estimate, which clearly characterize the error relationship between the approximate solution and the exact solution. This research provides a unified framework for the analysis of solutions to integral equations with different regularity characteristics and improves the system of integral equation approximation theory. Furthermore, the degenerate kernel approximation theory and error estimation results established in this paper can be directly extended to integral equations in high-dimensional Euclidean spaces, and their analytical framework and conclusions remain valid in high-dimensional cases.

Degenerate kernel approximation and error estimation for integral equations in \(L^p\) spaces

摘要