Approximation by trigonometric polynomials to functions in L_p-norm
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Let n(f) denote the n-th term of the (C; 1) transform the partial sums of Fourier series of a 2 periodic function f. Quade, in [3], has shown that if f is a function in the class Lip( ; p), (0 < 1, p 1), then kfnkp = O(n )for either p > 1 and 0 < 1 or p = 1 and 0 < < 1. He also showed that if p = = 1, then kf nk1 = O(n1log(n + 1)).The work of Quade [3] was improved by Chandra [2] for trigonometric polynomials. In this paper, we extend some results of Chandra to more general C-method obtained by deleting a set of rows from the Cesáro matrix C1.
Abstract
Let n(f) denote the n-th term of the (C; 1) transform the partial sums of Fourier series of a 2 periodic function f. Quade, in [3], has shown that if f is a function in the class Lip( ; p), (0 < 1, p 1), then kfnkp = O(n )for either p > 1 and 0 < 1 or p = 1 and 0 < < 1. He also showed that if p = = 1, then kf nk1 = O(n1log(n + 1)).The work of Quade [3] was improved by Chandra [2] for trigonometric polynomials. In this paper, we extend some results of Chandra to more general C-method obtained by deleting a set of rows from the Cesáro matrix C1.
참고문헌 (5)
- D. H. Armitage, "A new type of Cesáro mean", Analysis, 9, pp.195~ (1989)
- P. Chandra, "Trigonometric approximation of functions in Lp-norm", Journal Of Mathematical Analysis and Applications, 275, pp.13~26 (2002)
- E. S. Quade, "Trigonometric approximation in the mean", Duke Math. J, 3, pp.529~542 (1937)
- J. A. Osikiewicz, "Equivalance results for Cesáro submethods", Analysis, 20, pp.35~43 (2000)
- A. Zygmund, "Trigonometric Series, Vol. I" (1959), Cambridge University Press