The degree of approximation of functions belonging to Lipα, , and , -classes through trigonometric Fourier approximation using different summability matrices with monotone rows has been proved by various investigators like Khan [1], Mittal et al. [2, 3], Mittal, Rhoades and Mishra [4], Qureshi [5], Chandra [6], Leindler [7], Rhoades et al. [8]. Recently Lal [9] has proved a theorem on the degree of approximation of a function f belonging to the Lipα () class by summability method of its Fourier series. Lal [9] has assumed monotonicity on the generating sequence . The approximation of a function , conjugate to a 2π periodic function to () using product ()-summability has not been studied so far. In this paper, we obtain a new theorem on the degree of approximation of a function , conjugate to a 2π periodic function () class without monotonicity condition on the generating sequence .
Let be a given infinite series with the sequence of n th partial sums . Let be a non-negative sequence of constants, real (R) or complex, and let us write
The sequence to sequence transformation defines the sequence of Nörlund means of the sequence , generated by the sequence of coefficients . The series is said to be summable to the sum s if exists and is equal to a finite number s. In the special case, in which
the Nörlund summability reduces to the familiar summability.
The product of summability with a summability defines summability. Thus the mean is given by .
If as , then the infinite series or the sequence is said to be summable to the sum s if exists and is equal to s.
Let be a 2π-periodic function and Lebesgue integrable. The Fourier series of is given by
(1.1)
with th partial sum called the trigonometric polynomial of degree (order) n of the first () terms of the Fourier series of f.
The conjugate series of Fourier series (1.1) is given by
(1.2)
A function if
-norm of a function is defined by .
The degree of approximation of a function by the trigonometric polynomial of order n under the sup norm is defined by [10]
and of a function is given by .
The conjugate function is defined for almost every x by
We note that and are also trigonometric polynomials of degree (or order) n.
Abel’s transformation: The formula
(1.3)
where , , if , , which can be verified, is known as Abel’s transformation and will be used extensively in what follows.
If are non-negative and non-increasing, the left-hand side of (1.3) does not exceed in absolute value. In fact,
(1.4)
We write throughout the paper
(1.5)
, where τ denotes the greatest integer not exceeding , , .