T · C1 summability of a sequence of Fourier coefficients

doi:10.1016/j.amc.2008.07.010

Applied Mathematics and Computation

Volume 204, Issue 2, 15 October 2008, Pages 702-706

In memory of Prof. Brian Kuttner, 1908–1992.

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Abstract

Let B_n(x) denote the nth term of the conjugate series of a Fourier series of function f. Mohanty and Nanda [R. Mohanty, M. Nanda, On the behavior of Fourier coefficients, Proc. Am. Math. Soc. 5 (1954) 79–84] were the first to establish a result for C₁-summability of the sequence {nB_n(x)}. Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795] improved it for the product summability H₁ · C₁, which was generalized by various investigators using different summability methods with different set of conditions. In this note, we extend the result of Mittal [M.L. Mittal, On the $‖ T ‖ \cdot C_{1}$ summability of a sequence of Fourier coefficients, Bull. Cal. Math. Soc. 81 (1989) 25–31], which in turn generalizes the results of Prasad [K. Prasad, On the (N, p_n) · C₁ summability of a sequence of Fourier coefficients, Indian J. Pure Appl. Math. 12 (7) (1981) 874–881] and Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795].

Introduction

Let $\sum u_{n}$ be a given infinite series with the sequence of partial sums {s_n}. Let T ≡ (a_n,k) be an infinite lower triangular matrix with real constants. The sequence-to-sequence transformation $t_{n} = \sum_{k = 0}^{n} a_{n, k} s_{k}, n = 0, 1, 2, \dots,$ defines the T-transform of the sequence {s_n}. The series $\sum u_{n}$ is said to be T-summable to s, if lim_n→∞t_n = s. The necessary and sufficient condition for the regularity of T method is lim_n→∞ a_n,k = 0 ∀k.

The transform T reduces to the Nörlund transform N_p, generated by the sequence of coefficients {p_n}, if $a_{n, k} = \{\begin{matrix} p_{n - k} / P_{n}, & 0 ⩽ k ⩽ n, \\ 0, & k > n, \end{matrix}$ where $P_{n} (= \sum_{k = 0}^{n} p_{k} \neq 0)$ and P₋₁ = 0 = p₋₁.

Let f be a 2π-periodic function (or signal) in L₁ [−π, π]. The Fourier series associated with f at a point x is defined by $a_{0} / 2 + \sum_{k = 1}^{\infty} (a_{k} \cos kx + b_{k} \sin kx) = \sum_{k = 0}^{\infty} A_{k} (x) .$ The conjugate series of Eq. (3) is defined by $\sum_{k = 1}^{\infty} (b_{k} \cos kx - a_{k} \sin kx) = \sum_{k = 1}^{\infty} B_{k} (x) .$ We write: $\begin{matrix} ψ (t) = ψ_{x} (t) = f (x + t) - f (x - t) - l, Ψ (t) = \int_{0}^{t} | ψ (u) | d u, \\ Q (n, t) = \sum_{k = 1}^{n} a_{n, k} {\sin kt / {kt}^{2} - \cos kt / t}, A_{n, k} = \sum_{r = k}^{n} a_{n, r}, 0 ⩽ k ⩽ n, \\ Δ_{k} a_{n, k} = a_{n, k} - a_{n, k + 1}, V_{n, k} = (n - k + 1) a_{n, k} / A_{n, k}, \end{matrix}$ and τ = [1/t], the largest integer contained in 1/t; where l is a fixed constant.

In this paper, each matrix T is regular and has non-negative entries with A_n,0 = 1 ∀n ⩾ 0.

Also C₁ and H₁, respectively, denote the Cesáro and the Harmonic summabilities of order one. The product summability T · C₁ is obtained by superimposing T-summability on C₁-summability.

Section snippets

Known theorems

Varshney [17] improved the result of Mohanty and Nanda [8] on C₁-summability of the sequence {nB_n(x)} by extending it to the product summability H₁ · C₁. He has proved:

Theorem A

[17]. If $Ψ (t) = o (t / \log (1 / t)), as t \to + 0,$ then the sequence {nB_n(x)} is summable H₁ · C₁ to the value l/π.

Various investigators such as Sharma [15], [16], Rao and Sahney [12], Rhoades [13], [14], Pandey [9], Rai [11], Dwivedi [1], Lal [3], Mittal and Kumar [5], Mittal and Prasad [6], Prasad [10] and Mittal [4] have generalized Theorem A by

Main theorem

In the present paper, we extend Theorem C by dropping monotonicity on the sequence {a_n,k}. We prove:

Theorem 1

Let T ≡ (a_n,k) be an infinite lower triangular regular matrix such that $(i) \sum_{k = r}^{n} | Δ_{k} a_{n, n - k} | = O (a_{n, n - r}), (ii) V_{n, r} = O (1) \forall 0 ⩽ r ⩽ n .$ If α(t) be a positive and increasing function of t such that $Ψ (t) = o (t / α (1 / t)), as t \to + 0,$ $and α (n) \to \infty, as n \to \infty,$ then a sufficient condition that the sequence {nB_n(x)} be summable T · C₁ to the value l/π is $\int_{1 / δ}^{n} \frac{A (n, n - [u]}{u α (u)} d u = O (1) .$

Remark 1

(i) If a_n,k ⩽ a_n,k+1 ∀0 ⩽ k < n, as used in Theorem C, then both the

Lemmas

For the proof of Theorem 1, we need the following lemmas:

Lemma 1

[4]. For 0 ⩽ t ⩽ 1/n, under the conditions of regularity of matrix T ≡ (a_n,k), $Q (n, t) = O (n) .$

Lemma 2

[2]. For all values of n and t, $|\sum_{k = 0}^{n} \sin (k + 1) t / (k + 1)| ⩽ 1 + \frac{π}{2} .$

Lemma 3

Let the regular matrix T ≡ a_n,k satisfy conditions (11), then for 1/n ⩽ t ⩽ δ $Q (n, t) = O (t^{- 1} A_{n, n - τ}) + O (t^{- 2} a_{n, 1}) .$

Proof

We have $Q (n, t) = \sum_{k = 0}^{n - 1} a_{n, n - k} \{\frac{\sin (n - k) t}{(n - k) t^{2}} - \frac{\cos (n - k) t}{t}\} = Q_{1} (n, t) + Q_{2} (n, t), say .$ Using Abel’s transformation, Lemma 2, and condition (11), we have $Q_{1} (n, t) = \sum_{k = 0}^{n - 1} a_{n, n - k} t^{- 2} \sin (n - k) t / (n - k) = t^{- 1} \sum_{k = 0}^{τ - 1} a_{n, n - k} \sin ($

Proof of Theorem 1

The C₁ transform of the sequence {nB_n(x)}, denoted by C_n(x), is defined by $C_{n} (x) = (1 / n) \sum_{k = 1}^{n} {kB}_{k} (x) .$ The T · C₁ transform of the sequence {nB_n(x)} denoted by t_n(x) is given by $t_{n} (x) = \sum_{k = 1}^{n} a_{n, k} C_{k} (x) = \sum_{k = 1}^{n} a_{n, k} k^{- 1} \sum_{r = 1}^{k} {rB}_{r} (x) .$ Therefore, following Mohanty and Nanda [8], we have $t_{n} (x) - l / π = \sum_{k = 1}^{n} a_{n, k} (k^{- 1} \sum_{r = 1}^{k} {rB}_{r} (x) - l / π) + o (1) = \frac{1}{π} \int_{0}^{π} ψ (t) \sum_{k = 1}^{n} a_{n, k} (\frac{\sin kt}{{kt}^{2}} - \frac{\cos kt}{t}) d t + o (1) = \frac{1}{π} (\int_{0}^{1 / n} + \int_{1 / n}^{δ} + \int_{δ}^{π}) ψ (t) Q (n, t) d t + o (1) = π^{- 1} \sum_{k = 1}^{3} I_{k} + o (1), say,$ where $I_{1} = \int_{0}^{1 / n} ψ (t) Q (n, t) d t = O (n) \int_{0}^{1 / n} |ψ (t)| d t = O (n) Ψ (1 / n) = O (n) o (1 / n α (n)) = o (1 / α (n)) = o (1), as n \to \infty,$ in

Particular cases

1.
If we take a_n,k = 1/(n − k + 1) log n for 0 ⩽ k ⩽ n;a_n,k = 0 for k > n and α(t) = log t, then T · C₁ summability reduces to the H₁ · C₁ summability and condition (12) reduces to (5), while condition (11) holds true in view of the Remark 1(i), as in this case 1/(n − k + 1)log n is non-decreasing in k. Condition (13) is obvious since log n → ∞, as n → ∞ and condition (14) is satisfied as given below: $\int_{1 / δ}^{n} \frac{A (n, n - [u])}{u α (u)} d u = O (\frac{1}{\log n}) \int_{1 / δ}^{n} \frac{\log u}{u \log u} d u = O (\frac{1}{\log n}) (\log n - \log (1 / δ)) = O (1), as n \to \infty .$ Hence, Theorem A is a particular case of Theorem 1.
2.

References (17)

R.M. Sharma
On a sequence of Fourier coefficients
Bull. Cal. Math. Soc.
(1969)
G.K. Dwivedi
On a sequence of Fourier coefficients
Annal. Soc. Math. Polon Series—I, Commen. Math.
(1971)
T.H. Gronwall
Über die Gibbssche Erscheinung und die trigonometrische summen sin x + (1/2)sin 2x + (1/3)sin 3x + ⋯ + (1/n)sin nx
Math. Ann.
(1912)
S.N. Lal
On the Nörlund summability of Fourier series and the behavior of Fourier coefficients
Indian J. Math.
(1971)
M.L. Mittal
On the $‖ T ‖ \cdot C_{1}$ summability of a sequence of Fourier coefficients
Bull. Cal. Math. Soc.
(1989)
M.L. Mittal et al.
On a sequence of Fourier coefficients
Bull. Cal. Math. Soc.
(1994)
M.L. Mittal et al.
On a sequence of Fourier coefficients
Indian J. Pure Appl. Math.
(1992)
M.L. Mittal et al.
Approximation of functions (signals) belonging to Lip (ξ(t), p)-class by means of conjugate Fourier series using linear operators
Indian J. Math.
(2005)

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T · C1 summability of a sequence of Fourier coefficients

Abstract

Introduction

Section snippets

Known theorems

Main theorem

Lemmas

Proof of Theorem 1

Particular cases

Bull. Cal. Math. Soc.

On a sequence of Fourier coefficients

Annal. Soc. Math. Polon Series—I, Commen. Math.

Über die Gibbssche Erscheinung und die trigonometrische summen sin x + (1/2)sin 2x + (1/3)sin 3x + ⋯ + (1/n)sin nx

Math. Ann.

On the Nörlund summability of Fourier series and the behavior of Fourier coefficients

Indian J. Math.

On the ‖T‖·C1 summability of a sequence of Fourier coefficients

Bull. Cal. Math. Soc.

On a sequence of Fourier coefficients

Bull. Cal. Math. Soc.

On a sequence of Fourier coefficients

Indian J. Pure Appl. Math.

Approximation of functions (signals) belonging to Lip (ξ(t), p)-class by means of conjugate Fourier series using linear operators

Indian J. Math.

T · C₁ summability of a sequence of Fourier coefficients

On the $‖ T ‖ \cdot C_{1}$ summability of a sequence of Fourier coefficients