On Approximation Properties for Non-linear Integral Operators

TL;DR

This paper studies the pointwise convergence of a family of non-linear integral operators with non-negative kernels, applicable on finite intervals and the entire real line, contributing to the understanding of their approximation properties.

Contribution

It provides new results on the convergence behavior of non-linear integral operators with parameter-dependent kernels for functions in L1 spaces.

Findings

Established conditions for pointwise convergence on finite intervals.

Extended convergence results to the whole real line.

Analyzed the impact of kernel properties on approximation accuracy.

Abstract

We investigate the problem of pointwise convergence of the family of non-linear integral operators: \begin{equation} L_\lambda(f,x) = \int_a^b \sum_{m=1}^N f^m(t) K_{\lambda ,m}(x,t) dt, \end{equation} where $\lambda $ is a real parameters, $K_{\lambda ,m}(x,t)$ is non-negative kernel and $f$ is the function in $L_{1}(a,b)$. We consider two cases where $% (a,b)$ is a finite interval and when is the whole real axis.

Equations

$$L_{\lambda }(f,x)=\underset{a}{\overset{b}{\int }}\sum _{m=1}^N{f}^m(t)K_{\lambda ,m}(x,t)dt,$$

$$T_{w}f(s)={\int }_G{K}_w(t-s,f(t))dt,\text{ s}\in G$$

$$L_{\lambda }(f,x)=\sum _{m=1}^N\underset{a}{\overset{b}{\int }}{f}^m(t)K_{\lambda ,m}(x,t)dt$$

$$\underset{\lambda \to \infty }{\lim }{L}_{\lambda }(f,x_0)=\sum _{m=1}^N{C}_m{f}^m(x_0)$$

$$L_{\lambda }(f,x_0)-\sum _{m=1}^N{C}_m{f}^m(x_0)$$

$$\left|L_{\lambda }(f,x_0)-\sum _{m=1}^N{C}_m{f}^m(x_0)\right|$$

$$I_{11}(x_0,\lambda ,m)=\sum _{m=1}^N\underset{a}{\overset{x_0-\delta }{\int }}\left|f^m(t)-f^m(x_0)\right|K_{\lambda ,m}(x_0,t)dt.$$

$$I_{11}(x_0,\lambda ,m)\le \sum _{m=1}^N{K}_{\lambda ,m}(x_0,x_0-\delta )\left\{\underset{a}{\overset{x_0-\delta }{\int }}\left|f^m(t)\right|dt+x\underset{a}{\overset{x_0-\delta }{\int }}\left|f^m(x_0)\right|dt\right\}$$

$$\le \sum _{m=1}^N{K}_{\lambda ,m}(x_0,x_0-\delta )\left\{{\Vert f^m\Vert }_{L_1(a,b)}+\left|f^m(x_0)\right|(b-a)\right\}$$

$$I_{14}(x_0,\lambda ,m)$$

$$\underset{x_0}{\overset{x_0+h}{\int }}\left|f(t)-f(x_0)\right|dt<\epsilon h$$

$$\underset{x_0-h}{\overset{x_0}{\int }}\left|f(t)-f(x_0)\right|dt<\epsilon h$$

$$F(t)=\underset{x_0}{\overset{t}{\int }}\left|f(u)-f(x_0)\right|du.$$

$$F(t)\le \epsilon (t-x_0).$$

$$\left|f^m(t)-f^m(x_0)\right|\le \le \left|f(t)-f(x_0)\right|M$$

$$I_{13}(x_0,\lambda ,m)$$

$$\left|I_{13}(x_0,\lambda ,m)\right|\le M\sum _{m=1}^N\left\{F(x_0+\delta ,x_0)K_{\lambda ,m}(x_0+\delta ,x_0)+\underset{x_0}{\overset{x_0+\delta }{\int }}F(t)d\left(-K_{\lambda ,m}(x_0,t)\right)\right\}.$$

$$\left|I_{13}(x_0,\lambda ,m)\right|\le M\sum _{m=1}^N\left\{\epsilon \delta K_{\lambda ,m}(x_0+\delta ,x_0)+\epsilon \underset{x}{\overset{x+\delta }{\int }}(t-x_0)d\left(-K_{\lambda ,m}(x_0,t)\right)\right\}.$$

$$\left|I_{13}(x_0,\lambda ,m)\right|$$

$$G(t)={\int }_t^x\left|f(y)-f(x)\right|dy.$$

$$dG(t)=-\left|f(t)-f(x_0)\right|dt.$$

$$G(t)\le \epsilon \left|x_0-t\right|$$

$$I_{12}(x_0,\lambda ,m)\le M\sum _{m=1}^N\underset{x_0-\delta }{\overset{x_0}{\int }}\left|f(t)-f(x_0)\right|K_{\lambda ,m}(x_0,t)dt.$$

$$\left|I_{12}(x_0,\lambda ,m)\right|\le M\sum _{m=1}^N\left[-\underset{x_0-\delta }{\overset{x_0}{\int }}{K}_{\lambda ,m}(x_0,t)dG(t)\right].$$

$$\left|I_{12}(x,\lambda ,m)\right|\le M\sum _{m=1}^N\left\{G(x-\delta K_{\lambda ,m}(x-\delta ,x)+\underset{x-\delta }{\overset{x}{\int }}G(t)d_t(K_{\lambda ,m}(x,t))\right\}.$$

$$\left|I_{12}(x_0,\lambda ,m)\right|\le M\sum _{m=1}^N\left\{\epsilon \delta K_{\lambda ,m}(x_0,x_0-\delta )+\epsilon \underset{x_0-\delta }{\overset{x_0}{\int }}(x_0-t)d_t(K_{\lambda ,m}(x_0,t))\right\}.$$

$$\left|I_{12}(x_0,\lambda ,m)\right|\le \epsilon M\sum _{m=1}^N\underset{a}{\overset{b}{\int }}{K}_{\lambda ,m}(x_0,t)dt.$$

$$\left|I_{12}(x_0,\lambda ,m)\right|+\left|I_{13}(x_0,\lambda ,m)\right|\le 2\epsilon M\sum _{m=1}^N\underset{a}{\overset{b}{\int }}{K}_{\lambda ,m}(x_0,t)dt.$$

$$\underset{\lambda \to \infty }{\lim }{L}_{\lambda }(f,x_0)=\sum _{m=1}^N{C}_m{f}^m(x_0).$$

$$\underset{\lambda \to \infty }{\lim }\underset{-\infty }{\overset{x-\delta }{\int }}{K}_{\lambda ,m}(t,x)dt=0$$

Full text

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where $\lambda$ is a real parameters, $K_{\lambda,m}(x,t)$ is non-negative kernel and $f$ is the function in $L_{1}(a,b)$. We consider two cases where $(a,b)$ is a finite interval and when is the whole real axis..

The auhor wish to thank Prof. Dr. A. D. Gadjiev for the useful suggestions in the matter.

1.

Introduction

In $\left[6\right]$ the concept of singularity was extended to cover the case of nonlinear integral operators,

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the assumption of linearity of the operators being replaced by an assumption of a Lipschitz condition for $K_{w}$ with respect to the second variable. Recently, Swiderski and Wachnicki $\left[7\right]$ investigated the pointwise convergence of the operators $T_{w}f$ in $L_{p}(-\pi,\pi)$ and $L_{p}(R)$ at the points, where $x_{0}$ is a point of continuity and a Lebesque point of $f.$

In $\left[2\right],$ Karsli studied both the pointwise convergence and the rate of pointwise convergence of above operators on a $\mu-generelized$  Lebesque point to $f\in L_{1}(a,b)$ as $(x,\lambda)\rightarrow(x_{0},\lambda_{0}).$In $\left[1\right],$ it is studied the rate of convergence at a point $x,$ which has a discontinuity of  the first kinds as $\lambda\rightarrow\lambda_{0}.$In $\left[4\right]$ they obtained convergence results and rate of approximation for functions belonging to BV-spaces by mean of nonlinear convolution integral operators.

The aim of the article is to obtain pointwise convergence results for a family of non-linear operators of the form

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where $K_{\lambda,m}(x,t)$ is a family of kernels depending on $\lambda.$We study convergence of the family (1) at every Lebesque point of the function $f$ in the spaces of $L_{1}(a,b)$ and $L_{1}(-\infty,\infty).$

Now we give the following definition.

a) $K_{\lambda,m}(x,t)$ is a non-negative function defined for all $t$ on $(a,b)$ and $\lambda\in\Lambda.$

b) As function of $t$, $K_{\lambda,m}(x,t)$ is non-decreasing on $\left[a,x\right]$ and non-increasing on $\left[x,b\right]$ for any fixed $x$

c)For any fixed $x,$ $\underset{\lambda\rightarrow\infty}{\lim}\mathop{\displaystyle\int}\limits_{a}^{b}K_{\lambda,m}(x,t)dt=C_{m}$.

d ) For every $m,\underset{\lambda\rightarrow\infty}{1\leq m\leq N\text{ and every }y\neq x\text{ , }\lim}K_{\lambda,m}(x,y)=0.$

2. Main Result

We are going to prove the family of  non-linear integral operators (1) with the positive kernel convergence to the functions $f\in L_{1}(a,b)$

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holds at every $x_{0}-$lebesque point of $f$ function

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and in view of a)

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It is suffcient to show that terms on right hand side of the last inequality tend to zero as $\lambda\rightarrow\infty.$By property c) ,it is clear that $I_{2}(x_{0},\lambda)$ tends to zero as $\lambda\rightarrow\infty.$

Now we consider $I_{1}(x_{0},\lambda).$ For any fixed $\delta>0$, we can write $I_{1}(x_{0},\lambda)$ as follow.

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Firstly we shall calculate $I_{11}(x_{0},\lambda,m)$, that’s

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By the condition b), we have

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and

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In the same way, we can estimate $I_{14}(x_{0},\lambda,m)$.From property b)

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On the other hand, Since $x_{0}$ is a lebesque point of $f,$ for every $\varepsilon>0,$ there exists a $\delta>0$ such that

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and

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for all $0<h\leq\delta.$Now let’s define a new function as follows,

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Then from (5), for $t-x_{0}\leq\delta$ we have

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Also, since $f$ is bounded, there exists $M>0$ such that

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is satisfied. Therefore, we can estimate $I_{13}(x_{0},\lambda,m)$ as follows.

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We apply integration by part, then we obtain the following result.

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Since $K_{\lambda,m}$ is decreasing on $\left[x_{0},b\right]$, it is clear that $-K_{\lambda,m}$ is increasing.Hence its differential is positive. Therefore, we can wirte

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Integration by parts again we have the following inequality

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Now, we can use similar method for evaluation $I_{12}(x_{0},\lambda,m).$ Let

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Then, the statement

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is satisfied. For $x_{0}-t\leq\delta$, by using (6),it can be written as follows

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Hence, we get

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Then, we shall write

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By integration of parts, we have

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From (6), we obtain

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By using integration of parts again, we find

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Combined (7) and (8), we get

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Finally,from (3), (4) and (9),the terms on right hand side of these inequalitys tend to 0 as $\lambda\rightarrow\infty$. That’s

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Thus, the proof is completed.

In this theorem, specially it may be $a=-\infty$ and $b=\infty$. In this case, we can give the following theorem.

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and

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then the statement

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is satisfied at almost every $x\in R$.

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It is clear that $A_{2}(x,\lambda)\rightarrow 0$ as $\lambda\rightarrow\infty.$

For a fixed $\delta>0,$ we divide the integral $A_{1}(x,\lambda)$ of the form

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$A_{12}(x,\lambda,m)$ and $A_{13}(x,\lambda,m)$ integrals are calculated as above the proof.For proof, it is sufficent to show that $A_{11}(x,\lambda,m)$ and $A_{14}(x,\lambda,m)$ tend to zero as $\lambda\rightarrow\infty.$

Firstly, we consider $A_{11}(x,\lambda,m).$ Since $f$ is bounded and by the property b),this integration is written the form

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In addition to, we obtain the inequality

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According to the conditions d), (10) and (11),we find that $A_{11}(x,\lambda,m)+A_{14}(x,\lambda,m)\rightarrow 0$ as $\lambda\rightarrow\infty.$This completes the proof.