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 Cesàro 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 Cesàro sums and the Cesàro means of order of the series . Applications of the Cesàro transformation can be found in
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