Abstract
Vuĉkoviĉ [Maths. Zeitchr. 89, 192 (1965)] and Kathal [Riv. Math. Univ. Parma, Italy 10, 33-38 (1969)] have studied summability of Fourier series by Karamata (K λ ) summability method. In present paper, for the first time, we study the degree of approximation of function f ∈ Lip (α,r) and f ∈ W(L r ,ξ(t)) by Kλ-summability means of its Fourier series and conjugate of function and by Kλ-summability means of its conjugate Fourier series and establish four quite new theorems. MSC: primary 42B05; 42B08; 42A42; 42A30; 42A50.
Highlights
The method Kl was first introduced by Karamata [1] and Lotosky [2] reintroduced the special case l = 1
Nörlund means of the Fourier series has been studied by Alexits [8], Sahney and Goel
2π-periodic, belonging to W (Lr, ξ (t)) its degree of approximation by Kl-summability means on its Fourier series is given by sn − f r = O
Summary
The method Kl was first introduced by Karamata [1] and Lotosky [2] reintroduced the special case l = 1. Vuĉkoviĉ [4] applied this method for summability of Fourier series. Working in the same direction, Ojha [6], Tripathi and Lal [7] have studied Kl-summability of Fourier series under different conditions. The degree of approximation of a function f Î Lip a by Cesàro and Nörlund means of the Fourier series has been studied by Alexits [8], Sahney and Goel [9], Chandra [10], Qureshi [11], Qureshi and Neha [12], Rhoades [13], etc. Nothing seems to have been done so far in the direction of present work. In present paper, we establish two new theorems on degree of approximation of function f belonging to Lip (a,r) (r ≥ 1) and to weighted class W(Lr, ξ (t))(r ≥ 1) by Kl-means on its Fourier series and two other new theorems on degree of approximation of function f, conjugate of a 2π-periodic function f belonging to Lip (a,r) (r > 1) and to weighted class W(Lr,ξ (t)) (r ≥ 1) by Kl-means on its conjugate Fourier series
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