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Using the Matrix Summability Method to Approximate the Lip (ξ(t), p) Class Functions

Binod Prasad Dhakal

Central Department of Education (Mathematics), Tribhuvan University, Nepal

Abstract: Most of the summability methods are derived from the matrix means. In this paper, author has been determined the degree of approximation of certain trigonometric functions belonging to the Lip (ξ(t), p) class by matrix method.

Keywords: matrix means, degree of approximation, generalized Lipschitz class functions

Received October 18, 2014; Revised November 18, 2014; Accepted November 23, 2014

Citation: Binod Prasad Dhakal, Using the Matrix Summability Method to Approximate the Lip (ξ(t), p) Class Functions. Turkish Journal of Analysis and Number Theory, 2 (6). 198-201. January 2014. doi:10.12691/tjant-2-6-2

1. Introduction

The Fourier series associate with f at point x of 2π periodic function in is given by

[picture] | (1.1)

A function f ∈Lip if

, if

for 0 < α ≤ 1 and an integer .

if

provided is positive increasing function.

If then Lip coincide with Lip and if in Lip than Lip reduce to Lip α.

We observed that Lip for.

We define norm by

[picture] | (1.2)

The degree of approximation En (f) is given by

[picture] | (1.3)

where is a trigonometric polynomial of degree n.

Let be an infinite lower triangular matrix satisfying the condition(see, [4]) of regularity. Let infinite series such that whose partial sum s.

The sequence-to-sequence transformation defines the sequence of lower triangular matrix means of the sequence {sn} generated by the sequence of coefficients (an,k).

The series is said to be summable to the sum s by lower triangular matrix method (see, [1]) if .

In this paper, we use following notations.

[picture] | (1.4)

[picture] | (1.5)

where is the greatest integer not greater than (1/t) and

[picture] | (1.6)

2. Main Theorem

Chandra[5] proved a theorem on the approximation of function belonging to Lip class by Nörlund and Riesz means. Mittal et. al. [3] extended the result of Chandra [5] by using the matrix means on same Lip class function. In [6], Lal & Dhakal proved a theorem on approximation of a Lip class function by matrix means.

Aim of this paper is to extend the theorems of Chandra [5], Mittal et.al. [3] and Lal & Dhakal [6] by using matrix means on Lip class functions as following way:

Theorem. Let T=(an.k) be an infinite regular lower triangular matrix such that the element (an.k) be non-negative, non-decreasing with . If a function is 2π-periodic, Lebesgue integrable on [-π, π] and belonging to Lip class then the degree of approximation of f by lower triangular matrix means of its Fourier series (1.1) satisfies, for n= 0,1,2,3…,

[picture] | (2.1)

provided ξ (t) satisfies the following conditions:

[picture] | (2.2)

[picture] | (2.3)

where is an arbitrary number such that , q the conjugate index of p and conditions (2.2) & (2.3) hold uniformly in x.

3. Lemmas

For the proof of the theorem following lemmas are required.

Lemma 1. Mn (t) = O (n +1), if .

Proof. For , ,

Lemma 2. If (an,k) is non- negative and non- decreasing with , then,

, uniformly in .

Proof: Let Then

by Abel’s lemma,

and

therefore

Also, .

Thus

Lemma 3. , if

Proof: For , sin(t/2),we have

4. Proof of the Theorem

Following [2], we have

then,

or

[picture] | (4.1)

For, I1,

Applying Holder’s inequality and fact that , we have

by condition (2.2) & Lemma 1.

Applying Second Mean Value Theorem for Integrals, we have

[picture] | (4.2)

Applying Holder’s inequality, condition (2.3) and Lemma 2.

taking

[picture] | (4.3)

Combining the conditions (4.1) – (4.3), we have

Now,

[picture] | (4.4)

This completes the proof of the theorem.

5. Corollaries

Following corollaries can be derived from the theorem.

Corollary 1. If then degree of approximation of a function class is given by

for

Corollary 2. If with , then degree of approximation of a function class is given by

References

[1]   | A. Zygmund, Trigonometric series, Cambridge University Press, 1959.
[2]   | E. C. Titchmarsh, Theory of functions, Oxford University Press, 1939.
[3]   | M. L Mittal, B. E. Rhoades, V. N. Mishra and U. Shing, Using infinite matrices to functions of class Lip (α,p) using trigonometric polynomials, J. Math. Anal. Appl, 326(2007), 667-676.
[4]   | O.Töeplitz, Über allgemeine lineare Mittelbildungen, Prace mat. - fiz., 22(1913), 113-119.
[5]   | P. Chanrda, Trigonometric approximation of function in Lp-norm, J. Math. Anal. Appl, 275(2002), 13-676.
[6]   | S. Lal and B. P. Dhakal, On Approximation of functions belonging to Lipschitz class by triangular matrix method of Fourier series, Int. Journal of Math. Analysis, 4(21), 2010, 1041-1047.