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Degree of approximation of the conjugate of signals (functions) belonging to -class by means of conjugate trigonometric Fourier series
Journal of Inequalities and Applications volume 2012, Article number: 278 (2012)
In this paper, we determine the degree of approximation of the conjugate of 2π-periodic signals (functions) belonging to (, )-class by using Cesàro-Euler means of their conjugate trigonometric Fourier series. Our result generalizes the result of Lal and Singh (Tamkang J. Math. 33(3):269-274, 2002).
Let be a given infinite series with , the sequence of its n th partial sum. The sequence-to-sequence transform
defines the Cesàro means of order one of . The series is said to be summable to s, if . The sequence-to-sequence transform
defines the Euler means of order of . By super imposing the means on means of , we get means of denoted by and defined by
The series is said to be summable to s, if .
For a given 2π-periodic Lebesgue integrable signal (function), let
denote the th partial sum, called trigonometric polynomial of degree n (or order n), of the Fourier series of .
The conjugate of Fourier series of f is defined by
and its n th partial sum is defined as
The conjugate of f denoted by is defined by
where [, p.131].
A function , if
The -norm for is defined by
The -norm is defined by
The degree of approximation of a function by trigonometric polynomials of degree n is given by
This method of approximation is called trigonometric Fourier approximation (tfa). We also write
and , the integral part of .
2 Known result
Various investigators such as Dhakal , Lal and Singh , Mittal et al. [4, 5], Nigam , Qureshi [7, 8] have studied the degree of approximation in various function spaces such as Lipα, , and weighted by using triangular matrix summability and product summability , . Recently, Lal and Singh  have determined the degree of approximation of the conjugate of by means of conjugate Fourier series. Lal and Singh  have proved the following.
Theorem 1 
If is a 2π-periodic and function, then the degree of approximation of its conjugate function by the product means of conjugate series of Fourier series of f satisfies, for ,
is means of series (1.5).
3 Main result
Recently, Nigam and Sharma  have studied the degree of approximation of functions belonging to -class through means of their Fourier series. In this paper, we use the means of conjugate Fourier series of to determine the degree of approximation of the conjugate of f, which in turn generalizes the result of Lal and Singh . More precisely we prove
Theorem 2 Let be a 2π-periodic, Lebesgue integrable function and belong to the -class with and . Then the degree of approximation of , the conjugate of by means of its conjugate Fourier series is given by
where δ is an arbitrary number such that and for .
Remark 1 The authors have used conditions implied by (3.2) and (3.3), but not mentioned in the statement of Theorem 1 [, pp.271-272].
5 Proof of Theorem 2
The integral representation of is given by
Therefore, we have
Now, denoting transform of by , we write
Using Lemma 1, Hölder’s inequality, condition (3.2) and Minkowiski’s inequality, we have
Now, we consider
Using Lemma 2, Hölder’s inequality, condition (3.3) and Minkowiski’s inequality, we have
Combining (5.1)-(5.3), we have
This completes the proof of Theorem 2.
Remark 2 The proof of Theorem 2 for , i.e., , can be written by using sup norm while using Hölder’s inequality.
Corollary 1 When , then means reduces to means.
Hence, Theorem 2 reduces to Theorem 1.
Corollary 2 If is a 2π-periodic, Lebesgue integrable and belonging to the Lipα () class, then the degree of approximation of , the conjugate of , with by means of its Fourier series is given by
Proof If in Theorem 2, then for ,
For , we can write an independent proof by using and in and . □
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The authors would like to thank the referee for his valuable comments and suggestions for the improvement of the manuscript.
The authors declare that they have no competing interests.
SS has identified the problem of this paper and US has suggested the solution and corrected the manuscript written by SS.
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Sonker, S., Singh, U. Degree of approximation of the conjugate of signals (functions) belonging to -class by means of conjugate trigonometric Fourier series. J Inequal Appl 2012, 278 (2012). https://doi.org/10.1186/1029-242X-2012-278
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