Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

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Abstract:

Using the following notation: C is the space of continuous bounded functions f equipped with the norm $left| f right| = mathop {sup }limits_{x in mathbb{R}} left| {fleft( x right)} right|$ , V is the set of functions f such that ${mathop Vlimits_{ - infty }^infty} (f) < infty$ , the set E consists of f infin

CV and possesses the following property:

${mathop {sup }limits_{f in E}} left| f right| < infty ,;;;;{mathop {sup }limits_{f in E}} {mathop Vlimits_{ - infty }^infty} (f) < infty ,;;;{mathop {lim }limits_{h to 0 + }} {mathop {sup }limits_{f in E}} (f,h) = 0,$

$$A = left{ {D:{mathbb{R}} to {mathbb{R}}{kern 1pt} } right.{kern 1pt} |{kern 1pt} D$$

is summable on each finite interval, $left. {intlimits_{ to - infty }^{ to + infty } {D = 1} } right},$ we establish some assertions similar to the following theorem: Let $$D in A,;x in mathbb{R},;alpha ,;h >0$$

$$D in A,;x in mathbb{R},;alpha ,;h >0$$

$varphi (u) = intlimits_u^{ to + infty } {D(t)dt,;;;;l_{k,h} (f) = frac{1}{h}} intlimits_{kh}^{(k + 1)h} {f(t)dt.}$

Then for f infin

V the series

$U_{alpha ,h} (f,x) = f( - infty ) + sumlimits_{k in {mathbb{Z}}} {(l_{k,h} (f); - ;l_{k - 1,h} (f))varphi left( {frac{{kh - x}}{alpha }} right)}$

uniformly converges with respect to $$x in {mathbb{R}}$$

and the following equality holds:

${mathop {lim }limits_{alpha ,h to 0{text{ + }}}} {mathop {sup }limits_{f in E}} left| {f - U_{alpha ,h} (f)} right| = 0.$

This theorem develops some results obtained by Zubov relative to the approximation of probability distributions. Bibliography: 4 titles.

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