Direct inverse and saturation theorems for linear combination of convolution type of linear positive operators

Abstract

In chapter 1, we will review the results already obtained by various authors. We newlinewill give a brief introduction of preliminary concepts and definitions, number of newlinenotations , definitions and conventions to be used. newline newline In chapter 2, we have estimated some direct results for the even positive newlineconvolution integrals, newline(and#119864;(and#119899;,and#120578;)and#119891;)(and#119909;) = (and#119891; and#8727; and#119892;(and#119899;))(and#119909;) = newline1 newline2and#120587; newlineand#8747; and#119891;(and#119905;)and#119892;(and#119899;) newline(and#119909; and#8722; and#119905;)and#119889;and#119905; 2and#120587; newline0 newline (and#119899; and#8712; and#119873; ; and#119909; and#8712; and#119877;) newlinewhere , a kernel and#120578; = (and#119892;(and#119899;)) newlineand#119899;=1 newlineand#8734; newline, is a sequence of positive even normalized newlinetrigonometric polynomial and and#119892;(and#119899;) newline(and#119905;) is a non-negative trigonometric polynomial of newlinedegree atmost and#119899; on and#119862;2and#120587;, Banach space of 2and#120587; - periodic functions. newline Here, positive kernels and#120578; = (and#119892;(and#119899;)) newlineand#119899;=1 newlineand#8734; newlineare of finite oscillations of degree 2and#119896;. newline newline The method of linear combination is used for improving the order of newlineapproximation.The properties of Central factorial numbers, inverse formulas, mixed newlinealgebraic trigonometric formula are used throughout the work. newline newline In chapter 3, we have studied some inverse and saturation results for the family newlineof linear positive convolution operators, newline(and#119864;(and#119899;,and#120578;)and#119891;)(and#119909;) = (and#119891; and#8727; and#119892;(and#119899;))(and#119909;) = newline1 newlineand#120587; newlineand#8747; and#119891;(and#119905;)and#119892;(and#119899;) newline(and#119909; and#8722; and#119905;)and#119889;and#119905; and#120587; newlineand#8722;and#120587; newline, and#119899; and#8712; and#119873; and and#119909; and#8712; and#119877; newline iii newlinewhere and#119862;2and#120587; is the space of 2and#120587;-periodic functions with norm, newline and#8214;and#119891;and#8214; and#8788; and#119898;and#119886;and#119909;|and#119891;(and#119909;)| newlinewith kernel and#951;= (and#119892;and#119899; newline(and#119909;)) newlineand#119899;gt0 newlineand#8834;and#119871;2and#120587; newline1 newlinedepending upon the parameters and#119899; gt 0 and and#119899; and#8594; and#8734;. newline Here, kernel and#120578; = {and#119892;(and#119899;)} newlineand#119899;=1 newlineand#8734; newlinebe a sequence of even trigonometric polynomials of newlinedegree atmost and#119898;(and#119899;) = and#119874;(and#119899;) , which are normalized by, newline newline1 newlineand#120587; newlineand#8747; and#119892;(and#119899;)(and#119905;)and#119889;and#119905; = 1 newlineand#120587; newlineand#8722;and#120587; newline newlineand#119892;(and#119899;)(and#119909;) = newline1 newline2 newline+ and#8721; and#120588;(and#119896;,and#119899;) cos and#119896;and#119909; and#119898;(and#119899;) newlineand#119896;=1 newline newline newline Here, the operators are uniformly bounded, newlineand#8214;(and#119864;(and#119899;,and#120578;)and#119891;) newlinequot newlineand#8214; and#8804; and#119860;and#8214;and#119891;and#8214; newlineand satisfy Bernstein type inequality, newline and#8214;(and#119864;(and#119899;,and#120578;)and#119891;) newlinequot newlineand#8214; and#8804; and#119860;and#120593;(and#119899;) newline2and#8214;(and#119864;(and#119899;,and#120578;)and#119891;)and#8214; newline The Bernstein inequality have been used for proving inverse theorems. newline newline Then we have obtained some linear combinations and#120594; of Jackson De La vallee newlinePoussin kernel given by, newline and#120594;and#119899; newline(and#119909;) = newline9 newline4 newlineand#119875;and#119886;1and#119899; newline2 newline(and#119909;) and#8722; newline4 newline3 newlineand#119875;and#119886;2and#119899; newline2 newline(and#119909;) + newline1 newline12 newlineand#119875;and#119886;3and#119899; newline3 newline(and#119909;) , and#119909; and#8712; and#119877; newlineand linear combination and#120594;and#771; of Jacks