On certain multidimensional nonlinear integrals

Year 2020, Volume: 69 Issue: 2, 1356 - 1367, 31.12.2020

Özge Özalp Güller , Gumrah Uysal

https://doi.org/10.31801/cfsuasmas.762646

Abstract

The aim of the paper is to obtain generalized convergence results for nonlinear multidimensional integrals of the form:

L_{η}(ω;x)=((ηⁿ)/(Ω_{n-1}))∫_{D}K(η|t-x|,ω(t))dt.

We will prove pointwise convergence of the family L_{η}(ω;x) as η→∞ at a fixed point x∈D which represents any generalized Lebesgue point of function ω∈L₁(D), where D is an open bounded subset of Rⁿ. Moreover, we will consider the case D=Rⁿ.

The aim of the paper is to obtain generalized convergence results for nonlinear multidimensional integrals of the form:

L_{η}(ω;x)=((ηⁿ)/(Ω_{n-1}))∫_{D}K(η|t-x|,ω(t))dt.

We will prove pointwise convergence of the family L_{η}(ω;x) as η→∞ at a fixed point x∈D which represents any generalized Lebesgue point of function ω∈L₁(D), where D is an open bounded subset of Rⁿ. Moreover, we will consider the case D=Rⁿ.

Keywords

Taylor expansion, Generalized Lebesgue point, Pointwise convergence

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