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Degree of approximation of functions belonging to Lip\(\alpha\) class and weighted \((L^r,\xi(t))\) class by product summability method. (English) Zbl 1399.42005
Summary: A good amount of work has been done on degree of approximation of functions belonging to \(\text{Lip}\alpha\), \(\text{Lip}(\alpha,r)\), \(\text{Lip}(\xi(t),r)\) and \(W(L^r,\xi(t))\) classes using Cesàro and (generalized) Nörlund single summability methods by a number of researchers like G. Alexits [International Series of Monographs on Pure and Applied Mathematics. Vol. 20, 350 (1961; Zbl 0098.27403)], B. N. Sahney and D. S. Goel [Ranchi Univ. Math. J. 4, 50–53 (1973; Zbl 0296.41008)], K. Qureshi and H. K. Neha [Ganita 41, No. 1–2, 37–42 (1990; Zbl 0856.42005)], K. Quershi [Tamkang J. Math. 12, 35–38 (1981; Zbl 0502.42002), Indian J. Pure Appl. Math. 13, 898–903 (1982; Zbl 0498.42002)], P. Chandra [J. Math. Anal. Appl. 275, No. 1, 13–26 (2002; Zbl 1011.42001)], H. H. Khan [Indian J. Pure Appl. Math. 5, 132–136 (1974; Zbl 0308.41010)], L. Leindler [J. Math. Anal. Appl. 302, No. 1, 129–136 (2005; Zbl 1057.42004)] and B. E. Rhoades [Tamkang J. Math. 34, No. 3, 245–247 (2003; Zbl 1039.42001)]. But till now no work seems to have been done so far in the direction of present work. Therefore, in present paper, two quite new results on degree of approximation of functions \(f\in\text{Lip}\alpha\) and \(f\in W(L^r,\xi(t))\) class by \((E,1)(C,1)\) product summability means of Fourier series have been obtained.

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