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Degree of approximation of the conjugate of signals (functions) belonging to Lip(α,r)-class by (C,1)(E,q) means of conjugate trigonometric Fourier series

Abstract

In this paper, we determine the degree of approximation of the conjugate of 2π-periodic signals (functions) belonging to Lip(α,r) (0<α1, r1)-class by using Cesàro-Euler (C,1)(E,q) means of their conjugate trigonometric Fourier series. Our result generalizes the result of Lal and Singh (Tamkang J. Math. 33(3):269-274, 2002).

MSC:41A10.

1 Introduction

Let n = 0 u n be a given infinite series with { s n }, the sequence of its n th partial sum. The sequence-to-sequence transform

C n 1 = 1 n + 1 k = 0 n s k ,n=0,1,2,,
(1.1)

defines the Cesàro means of order one of { s n }. The series n = 0 u n is said to be (C,1) summable to s, if lim n C n 1 =s. The sequence-to-sequence transform

E n q = 1 ( 1 + q ) n k = 0 n ( n k ) q n k s k ,q>0,n=0,1,2,,
(1.2)

defines the Euler means of order q>0 of { s n }. By super imposing the (C,1) means on (E,q) means of { s n }, we get (C,1)(E,q) means of { s n } denoted by C n 1 E n q and defined by

C n 1 E n q = 1 n + 1 k = 0 n E k q = 1 n + 1 k = 0 n ( q + 1 ) k v = 0 k ( k v ) q k v s v .
(1.3)

The series n = 0 u n is said to be (C,1)(E,q) summable to s, if lim n C n 1 E n q =s.

For a given 2π-periodic Lebesgue integrable signal (function), let

s n (f;x)= a 0 2 + k = 1 n ( a k coskx+ b k sinkx)
(1.4)

denote the (n+1)th partial sum, called trigonometric polynomial of degree n (or order n), of the Fourier series of f L 1 [π,π].

The conjugate of Fourier series of f is defined by

k = 1 ( b k coskx a k sinkx),
(1.5)

and its n th partial sum is defined as

s n ˜ (f;x)= k = 1 n ( b k coskx a k sinkx)
(1.6)

The conjugate of f denoted by f ˜ is defined by

2π f ˜ (x)= lim ϵ 0 ϵ π ψ(t)cot(t/2)dt,

where ψ(t)=f(x+t)f(xt) [[1], p.131].

A function fLipα, if

| f ( x + t ) f ( x ) | =O ( t α ) ,

and fLip(α,r) if

( 0 2 π | f ( x + t ) f ( x ) | r d x ) 1 / r =O ( t α ) ,0<α1,r1.

The L r -norm for f L r [π,π] is defined by

f r = ( 0 2 π | f ( x ) | r d x ) 1 / r ,r1.

The L -norm is defined by

f =sup { | f ( x ) | : x R } .

The degree of approximation E n (f) of a function fLip(α,r) by trigonometric polynomials T n (x) of degree n is given by

E n (f)= min T n f T n r .

This method of approximation is called trigonometric Fourier approximation (tfa). We also write

K n (t)= 1 n + 1 k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v cos ( v + 1 / 2 ) t sin ( t / 2 )

and τ=[1/t], the integral part of 1/t.

2 Known result

Various investigators such as Dhakal [2], Lal and Singh [3], Mittal et al. [4, 5], Nigam [6], Qureshi [7, 8] have studied the degree of approximation in various function spaces such as Lipα, Lip(α,r), Lip(ξ(t),r) and weighted ( L r ,ξ(t)) by using triangular matrix summability and product summability (C,1)(E,1), (N, p n )(E,1). Recently, Lal and Singh [3] have determined the degree of approximation of the conjugate of fLip(α,r) by (C,1)(E,1) means of conjugate Fourier series. Lal and Singh [3] have proved the following.

Theorem 1 [3]

If f:RR is a 2π-periodic and Lip(α,r) function, then the degree of approximation of its conjugate function f ˜ (x) by the (C,1)(E,1) product means of conjugate series of Fourier series of f satisfies, for n=0,1,2, ,

M n ( f ˜ )=Min ( C E ) n 1 f ˜ r =O ( n α + 1 / r ) ,
(2.1)

where

( C E ) n 1 = 1 n + 1 k = 0 n ( 1 2 k i = 0 k ( k i ) S i ) ,

is (C,1)(E,1) means of series (1.5).

3 Main result

Recently, Nigam and Sharma [9] have studied the degree of approximation of functions belonging to Lip(ξ(t),r)-class through (C,1)(E,q) means of their Fourier series. In this paper, we use the (C,1)(E,q) means of conjugate Fourier series of fLip(α,r) to determine the degree of approximation of the conjugate of f, which in turn generalizes the result of Lal and Singh [3]. More precisely we prove

Theorem 2 Let f(x) be a 2π-periodic, Lebesgue integrable function and belong to the Lip(α,r)-class with r1 and αr1. Then the degree of approximation of f ˜ (x), the conjugate of f(x) by (C,1)(E,q) means of its conjugate Fourier series is given by

C n 1 E n q f ˜ r =O ( n α + 1 / r ) ,n=0,1,2,,
(3.1)

provided

(3.2)
(3.3)

where δ is an arbitrary number such that (α+δ)s+1<0 and 1/r+1/s=1 for r>1.

Remark 1 The authors have used conditions ( 0 π / ( n + 1 ) | t ψ ( t ) t α | r d t ) 1 / r =O(1) implied by (3.2) and (3.3), but not mentioned in the statement of Theorem 1 [[3], pp.271-272].

4 Lemmas

We need the following lemmas for the proof of our theorem.

Lemma 1 | K n (t)|=O(1/t)+O((n+1)t) for 0tπ/(n+1)π/(v+1).

Proof

| K n ( t ) | 1 2 π ( n + 1 ) | k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v cos ( v + 1 / 2 ) t sin ( t / 2 ) | = 1 2 π ( n + 1 ) | k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v cos ( v + 1 1 / 2 ) t sin ( t / 2 ) | 1 ( n + 1 ) k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v | cos ( v + 1 ) t cos ( t / 2 ) + sin ( v + 1 ) t sin ( t / 2 ) sin ( t / 2 ) | = 1 ( n + 1 ) k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v [ O ( 1 / t ) + O ( sin ( v + 1 ) t ) ] = O [ 1 ( n + 1 ) t k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v ] + O [ 1 ( n + 1 ) k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v ( v + 1 ) t ] = O [ 1 ( n + 1 ) t ( n + 1 ) ] + O [ 1 ( n + 1 ) ( n + 1 ) ( n + 1 ) t ] = O ( 1 / t ) + O ( ( n + 1 ) t ) ,

in view of sin(v+1)t(v+1)t for 0t<π/(v+1) and ( sin ( t / 2 ) ) 1 <π/t for 0<tπ [[10], p.247]. □

Lemma 2 | K n (t)|=O(1/t)+O(1) for π/(v+1)tπ.

Proof

| K n ( t ) | 1 2 π ( n + 1 ) | k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v cos ( v + 1 / 2 ) t sin ( t / 2 ) | = 1 2 π ( n + 1 ) | k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v cos ( v + 1 1 / 2 ) t sin ( t / 2 ) | 1 ( n + 1 ) k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v | cos ( v + 1 ) t cos ( t / 2 ) + sin ( v + 1 ) t sin ( t / 2 ) sin ( t / 2 ) | = 1 ( n + 1 ) k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v [ O ( 1 / t ) + O ( 1 ) ] = O [ 1 ( n + 1 ) t k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v ] + O [ 1 ( n + 1 ) k = 0 n 1 ( 1 + q ) k v = 0 k ( k v ) q k v ] = O [ 1 ( n + 1 ) t ( n + 1 ) ] + O [ 1 ( n + 1 ) ( n + 1 ) ] = O ( 1 / t ) + O ( 1 ) ,

in view of |sin(v+1)t|1 and ( sin ( t / 2 ) ) 1 π/t for 0<tπ [[10], p.247]. □

5 Proof of Theorem 2

The integral representation of s n ˜ (f;x) is given by

s n ˜ (f;x)= 1 π 0 π ψ(t) cos ( t / 2 ) cos ( n + 1 / 2 ) t 2 sin ( t / 2 ) dt.

Therefore, we have

s ˜ n (f;x) f ˜ (x)= 1 2 π 0 π ψ(t) cos ( n + 1 / 2 ) t sin ( t / 2 ) dt.

Now, denoting (C,1)(E,q) transform of s n ˜ (f;x) by C n 1 E n q , we write

C n 1 E n q f ˜ = 1 2 π ( n + 1 ) [ k = 0 n 1 ( 1 + q ) k 0 π ψ ( t ) sin ( t / 2 ) v = 0 k ( k v ) q k v cos ( v + 1 / 2 ) t d t ] = [ 0 π / ( n + 1 ) + π / ( n + 1 ) π ] ψ ( t ) K n ( t ) d t = I 1 + I 2 , say .
(5.1)

Using Lemma 1, Hölder’s inequality, condition (3.2) and Minkowiski’s inequality, we have

| I 1 | = 0 π / ( n + 1 ) | ψ ( t ) | | K n ( t ) | d t [ 0 π / ( n + 1 ) ( | ψ ( t ) | / t α ) r d t ] 1 / r [ lim ϵ 0 ϵ π / ( n + 1 ) ( t α | K n ( t ) | ) s d t ] 1 / s = O ( ( n + 1 ) 1 ) [ lim ϵ 0 ϵ π / ( n + 1 ) ( t α 1 + ( n + 1 ) t α + 1 ) s d t ] 1 / s = O ( ( n + 1 ) 1 ) [ ( lim ϵ 0 ϵ π / ( n + 1 ) t ( α 1 ) s d t ) 1 / s + ( lim ϵ 0 ϵ π / ( n + 1 ) ( n + 1 ) t ( α + 1 ) s d t ) 1 / s ] = O ( ( n + 1 ) 1 ) [ ( n + 1 ) α + 1 1 / s + ( n + 1 ) ( n + 1 ) α 1 1 / s ] = O ( ( n + 1 ) 1 ) [ ( n + 1 ) α + 1 / r + ( n + 1 ) ( n + 1 ) α 1 1 + 1 / r ] = O [ ( n + 1 ) α + 1 / r 1 + ( n + 1 ) α 2 + 1 / r ] = O ( ( n + 1 ) α 1 + 1 / r ) .
(5.2)

Now, we consider

| I 2 | π / ( n + 1 ) π | ψ ( t ) | | K n ( t ) | dt.

Using Lemma 2, Hölder’s inequality, condition (3.3) and Minkowiski’s inequality, we have

| I 2 | [ π / ( n + 1 ) π ( t δ | ψ ( t ) | t α ) r d t ] 1 / r [ π / ( n + 1 ) π ( t α | K n ( t ) | t δ ) s d t ] 1 / s = O ( ( n + 1 ) δ ) [ π / ( n + 1 ) π ( t α t δ ( O ( 1 / t ) + O ( 1 ) ) ) s d t ] 1 / s = O ( ( n + 1 ) δ ) [ π / ( n + 1 ) π ( t α + δ 1 + t α + δ ) s d t ] 1 / s = O ( ( n + 1 ) δ ) [ ( π / ( n + 1 ) π t ( α + δ 1 ) s d t ) 1 / s + ( π / ( n + 1 ) π t ( α + δ ) s d t ) 1 / s ] = O ( ( n + 1 ) δ ) [ ( n + 1 ) ( α δ + 1 ) 1 / s + ( n + 1 ) ( α δ ) 1 / s ] ( ( α + δ ) s + 1 < 0 ) = O [ ( n + 1 ) α + 1 1 / s + ( n + 1 ) α 1 / s ] = O [ ( n + 1 ) α + 1 / r + ( n + 1 ) α 1 + 1 / r ] = O [ ( n + 1 ) α + 1 / r ( 1 + ( n + 1 ) 1 ) ] = O ( ( n + 1 ) α + 1 / r ) .
(5.3)

Combining (5.1)-(5.3), we have

| C n 1 E n q f ˜ | =O ( ( n + 1 ) α + 1 / r ) .

Hence,

C n 1 E n q f ˜ r = ( 0 2 π | C n 1 E n q f ˜ ( x ) | r d x ) 1 / r =O ( n α + 1 / r ) .

This completes the proof of Theorem 2.

Remark 2 The proof of Theorem 2 for r=1, i.e., s=, can be written by using sup norm while using Hölder’s inequality.

6 Corollaries

Corollary 1 When q=1, then (C,1)(E,q) means reduces to (C,1)(E,1) means.

Hence, Theorem  2 reduces to Theorem  1.

Corollary 2 If f:RR is a 2π-periodic, Lebesgue integrable and belonging to the Lipα (0<α1) class, then the degree of approximation of f ˜ (x), the conjugate of f(x)Lipα, with 0<α1 by (C,1)(E,q) means of its Fourier series is given by

C n 1 E n q f ˜ =O ( n α ) for n=0,1,2,.

Proof If r in Theorem 2, then for 0<α<1,

C n 1 E n q f ˜ ( x ) =O ( n α ) .

For α=1, we can write an independent proof by using α=1 and ψ(t)=O(t) in I 1 and I 2 . □

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The authors would like to thank the referee for his valuable comments and suggestions for the improvement of the manuscript.

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Correspondence to Uaday Singh.

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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 Lip(α,r)-class by (C,1)(E,q) 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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Keywords

  • conjugate Fourier series
  • Lip(α,r)-class
  • (C,1)(E,q) means