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Using infinite matrices to approximate functions of class Lip\((\alpha ,p)\) using trigonometric polynomials. (English) Zbl 1106.42001

Summary: Given a function \(f\) in the class \(\text{Lip}(\alpha,p)\) (\(0<\alpha\leq 1,p\geq 1\)), P. Chandra [Trigonometric approximation of functions in \(L_p\)-norm, J. Math. Anal. Appl. 275, 13–26 (2002; Zbl 1011.42001)] approximated such an \(f\) by using trigonometric polynomials, which are the \(n\)th terms of either certain weighted mean or Nörlund mean transforms of the Fourier series representation for \(f\). He showed that the degree of its approximation is \(O(n^{-\alpha})\). In this paper we obtain the same degree of approximation for a more general class of lower triangular matrices, and deduce some of the results of P. Chandra, (loc. cit) as corollaries.

MSC:

42A10 Trigonometric approximation
41A25 Rate of convergence, degree of approximation

Citations:

Zbl 1011.42001
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References:

[1] Agnew, R. P., A genesis for Cesáro methods, Bull. Amer. Math. Soc., 51, 90-94 (1945) · Zbl 0060.16001
[2] Chandra, P., Trigonometric approximation of functions in \(L_p\)-norm, J. Math. Anal. Appl., 275, 13-26 (2002) · Zbl 1011.42001
[3] Hausdorff, F., Summationsmethoden und Momentfolgen, Math. Z., 9, 74-109 (1921)
[4] Quade, E. S., Trigonometric approximation in the mean, Duke Math. J., 3, 529-542 (1937) · Zbl 0017.20501
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