%0 Journal Article
%T An Estimate of the Rate of Convergence of the Fourier Series in the Generalized H£¿lder Metric by Delayed Arithmetic Mean
%A L. Nayak
%A G. Das
%A B. K. Ray
%J International Journal of Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/171675
%X We study the rate of convergence problem of the Fourier series by Delayed Arithmetic Mean in the generalized H£¿lder metric space which was earlier introduced by Das, Nath, and Ray and obtain a sharper estimate of Jackson's order. 1. Definition Let be a -periodic function such that . Let the Fourier series of at be given by Let Let be the th partial sum of (1). Then it is known ([1], page 50) that where is known as Dirichlet¡¯s kernel. Let denote the Banach space of all -periodic continuous functions defined on under the supremum norm. The space with reduces to defined over . We write when the norm has been taken with respect to throughout the paper. The quantities and are, respectively, called the modulus of continuity and integral modulus of continuity of . It is known ([1], page 45) that and both tend to zero as . It was Pr£¿ssdorf [2] who first studied the degree of approximation problems of the Fourier series in space in the H£¿lder metric. Generalizing the H£¿lder metric, Leindler [3] introduced the space given by where is a modulus of continuity; that is, is a positive nondecreasing continuous function on with the following property: (i) ,(ii) ,(iii) . Further Leindler [3] has introduced the following metric on space: In the case , the space reduces to space (the norm being replaced by ) which is introduced by Pr£¿ssdorf [2]. It is known that [2] The degree of approximation problem in space has been studied by Leindler [3], Totik [4, 5], Mazhar and Totik [6], and Mazhar [7, 8]. The space was further generalized by Das et al. [9] as follows. For , , we write where is a modulus of continuity. If then we say that Lip . We define It can be seen that is a norm in . To prove the completeness of the space we use the completeness of . If we put , then reduces to space (with the norm replaced by ) which is introduced earlier by Das et al. [10]. If as , then exists and is 0 everywhere and is constant. Given the spaces and , if is nondecreasing, then since For , if we put and , then (12) reduces to the following: Note that the space is the familiar space introduced earlier by Pr£¿ssdorf [2]. 1.1. The Ces¨¤ro Transformation Let be an infinite series and let denote the sequence of its th partial sums. Then the series is said to be summable to the sum (finite), if (see [1], page 76) where and are defined by the following formulae: where , . From the definition of and it follows that ([1], page 77) The numbers and are called, respectively, the Ces¨¤ro sums and the Ces¨¤ro means of order of the series . Applications of the Ces¨¤ro transformation can be found in
%U http://www.hindawi.com/journals/ijanal/2014/171675/