Convergence in Phi-Variation and Rate of Approximation for Nonlinear Integral Operators Using Summability Process
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Convergence in Phi-Variation and Rate of Approximation for Nonlinear Integral Operators Using Summability Process ˙ ˙ ASLAN ISMA IL Abstract. We investigate the approximation properties of nonlinear integral operators of the convolution type. In this approximation, we use functions of bounded variation based on the appropriate functionals. To get more general results, we consider Bell-type summability methods in the approximation. Moreover, we examine the rate of approximation. Then, using summability methods, we obtain a characterization for absolute continuity. Our examples at the end of the paper clearly demonstrate why we used summability methods rather than convergence in the conventional sense. Mathematics Subject Classification. 26A45, 40A25, 41A25, 41A35, 47G10. Keywords. Summability process, nonlinear integral operators, convolution-type integral operators, bounded ϕ-variation, rate of approximation.
1. Introduction Convolution-type integral operators, which constitute the main tools of this paper, have remarkable applications in image processing, signal processing, ultrasound diagnostics, approximation theory, etc. (see [2,11,12,16–18,20– 22,28,33,39,43]). On the other hand, summability method is commonly held method to cope with the lack of convergence. Although there are many summability types, such as Ces`aro method [23], almost convergence [31], and order summability [25,26], Bell’s method (A−summability) (see [14,15]) is quite general and it contains all those ones. Furthermore, in some particular case, it works in the acceleration of rate of convergence (see [19,27,41,45]). Taking regular families of matrices enables us to preserve the classical convergence. While there are numerous applications of this method in positive linear operators (see [8,25,26,29,38,42]), in contrast, there are very few studies in the nonlinear setting (see [9,10]). 0123456789().: V,-vol
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It is known that bounded variation theory was first introduced by Jordan in [24], and it has been generalized by Wiener, Young, and Love in [32,44,46,47]. Later, using their idea, Musielak and Orlicz obtained a new concept which is known in the literature as Musielak–Orlicz ϕ-variation (see [37]). This is a strict generalization of classical Jordan variation and many properties of the Jordan variation are kept. Some results about classical variation, ϕ-variation, and other variants can be found in [1,4,6,7,13,30,34– 36,48]. Since there is no integral representation of ϕ-variation for absolutely continuous functions, we will need some auxiliary results given in [3]. The starting point of this study is based on the papers in which Angeloni and Vinti investigated the approximation properties of convolution-type nonlinear integral operators by means of functions of bounded ϕ-variation in [3,5]. Now, in this paper, our goal is, using Bell’s method, to get more general results than Angeloni and Vinti’s one in [3,5]. We also improve and generalize our recent paper for classical jordan variation in [10] without pe
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