A study on degree of approximation by Karamata summability method
© Nigam and Sharma; licensee Springer. 2011
Received: 25 January 2011
Accepted: 12 October 2011
Published: 12 October 2011
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.
Keywordsdegree of approximation Lip(α,r) class W(L r ,ξ(t)) class of functions Fourier series conjugate Fourier series Kλ-summability Lebesgue integral
The method K λ was first introduced by Karamata  and Lotosky  reintroduced the special case λ = 1. Only after the study of Agnew , an intensive study of these and similar cases took place. Vuĉkoviĉ  applied this method for summability of Fourier series. Kathal  extended the result of Vuĉkoviĉ . Working in the same direction, Ojha , Tripathi and Lal  have studied Kλ-summability of Fourier series under different conditions. The degree of approximation of a function f ∈ Lip α by Cesàro and Nörlund means of the Fourier series has been studied by Alexits , Sahney and Goel , Chandra , Qureshi , Qureshi and Neha , Rhoades , etc. But nothing seems to have been done so far in the direction of present work. Therefore, in present paper, we establish two new theorems on degree of approximation of function f belonging to Lip (α,r) (r ≥ 1) and to weighted class W(L r , ξ (t))(r ≥ 1) by Kλ-means on its Fourier series and two other new theorems on degree of approximation of function , conjugate of a 2π-periodic function f belonging to Lip (α,r) (r > 1) and to weighted class W(L r ,ξ (t)) (r ≥ 1) by Kλ-means on its conjugate Fourier series.
2 Definitions and notations
The numbers are known as the absolute value of stirling number of first kind
with n th partial sums s n (f;x).
with n th partial sums .
Throughout this paper, we will call (2.5) as conjugate Fourier series of function f.
The degree of approximation of a function f: R → R by a trigonometric polynomial t n of degree n under sup norm || ||∞ is defined by
(definition 5.38 of McFadden ).
If β = 0, our newly defined weighted i.e. W (L r , ξ (t)) reduces to Lip (ξ (t), r), if ξ (t) = t α then Lip (ξ (t), r) coincides with Lip (α, r) and if r → ∞ then Lip (α, r) reduces to Lip α.
3 The main results
3.1 Theorem 1
where s n is Kλ-mean of Fourier series (2.4).
3.2 Theorem 2
where δ is an arbitrary positive number such that s (1 - δ) - 1 > 0, , 1 ≤ r ≤ ∞, conditions (3.4) and (3.5) hold uniformly in x, s n is Kλ-mean of Fourier series (2.4).
3.3 Theorem 3
3.4 Theorem 4
For the proof of our theorems, following lemmas are required.
4.1 Lemma 1
4.2 Lemma 2
4.3 Lemma 3
4.4 Lemma 4
(McFadden ), Lemma 5.40) If f(x) belongs to Lip(α,r) on [0,π], then φ(t) belongs to Lip(α,r) on [0,π].
5 Proof of the theorems
5.1 Proof of Theorem 1
This completes the proof of Theorem 1.
5.2 Proof of Theorem 2
Then f ∈ W (L r ,ξ(t))⇒ φ ∈ W(L r , ξ (t)).
This completes the proof of Theorem 2.
5.3 Proof of Theorem 3
This completes the proof of Theorem 3.