Degree of approximation of the conjugate of signals (functions) belonging to -class by means of conjugate trigonometric Fourier series
© Sonker and Singh; licensee Springer 2012
Received: 2 April 2012
Accepted: 15 October 2012
Published: 28 November 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).
The series is said to be summable to s, if .
denote the th partial sum, called trigonometric polynomial of degree n (or order n), of the Fourier series of .
where [, p.131].
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 
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
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
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.
For , we can write an independent proof by using and in and . □
The authors would like to thank the referee for his valuable comments and suggestions for the improvement of the manuscript.
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