- Open Access
Euler-Hausdorff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to the generalized Lipschitz class
© Lal and Mishra; licensee Springer 2013
- Received: 20 November 2012
- Accepted: 17 January 2013
- Published: 19 February 2013
In this paper, three new estimates for the degree of approximation of a function , the conjugate of a function f belonging to classes Lipα and , , by summability operator of conjugate series of the Fourier series have been determined.
MSC:42A24, 41A25, 42B05, 42B08.
- trigonometric approximation
- Fourier series
- conjugate series of Fourier series
- Hausdorff matrix summability means ( means)
- means and summability operator
- generalized Lipschitz class
- generalized Minkowski’s inequality
The degree of approximation of the function and , , classes have been determined by several investigators like Alexits , Sahney and Goel , Chandra , Qureshi [4, 5] and Qureshi and Nema . After quite a good amount of work on the degree of approximation of functions by different summability means of its Fourier series, Lal and Singh  established the degree of approximation of conjugates of functions by product means of conjugate series of a Fourier Series in the following form.
where is means of conjugate series of the Fourier series.
where δ is an arbitrary positive number such that , , , conditions (1.1) and (1.2) hold uniformly in x and is means of the Fourier series.
Working in a slightly different direction, in this paper, the degree of approximation of a function , the conjugate of a function f belonging to classes Lipα and , , by the product summability operator of conjugate series of the Fourier series has been established. The results are new, sharper and better than previously known results. Furthermore, some interesting particular estimates have been also derived from the main theorems.
with n th partial sum .
is called ‘conjugate series’ of the Fourier series (2.1) with n th partial sum .
where Δ is the forward difference operator defined by and .
Let be an infinite series with n th partial sum .
If as , is said to be summable to the sum s by the Hausdorff matrix summability method ( means) (Boos and Cass ).
Let , .
If as , is said to be summable to s by the Euler method (Hardy , p.180).
If as , is said to be summable to s by the Euler-Hausdorff matrix summability means, i.e., means.
if , ,
if and , ,
if and .
A function if for (Titchmarsh , p.426).
if for , (def. 5.38 of McFadden ).
if (), where ξ is a modulus of continuity, that is, ξ is a non-negative, non-decreasing, continuous function with the properties , (Khan ).
If , class coincides with the class , and if , then reduces to Lipα class. Thus, it is obvious that .
Let u be a modulus of continuity such that is positive, non-decreasing.
Then . Thus, .
In this paper, three new estimates for the degree of approximation of a function , the conjugate of a function f belonging to the classes Lipα and () by summability operator have been determined in the following form.
For the proof of our theorems, the following lemmas are required.
Lemma 4.1 for .
Lemma 4.2 for .
is known as generalized Minkowski’s inequality where the generalization is simply replacing a finite sum by a definite (Lebesgue) integral (see Chui ).
This completes the proof of Theorem 3.1.
Thus, Theorem 3.3 is completely established.
The following corollaries can be derived from our theorem.
If we take in Theorem 3.2, then it reduces to Corollary 8.4.
The independent proofs of Corollaries 8.3 to 8.6 can be developed along the same lines as the theorems.
Some interesting estimates parallel to the main theorems and Corollary 8.4, Corollary 8.5 and Corollary 8.6 can also be obtained for , , , summability operators.
The degree of approximation of functions of class has been determined without using conditions like (1.1) and (1.2) of Theorem 1.2.
Authors are thankful to DST-CIMS, Banaras Hindu University, Varanasi for encouragement to this work.
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