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# ${L}_{r}$-Approximation of signals (functions) belonging to weighted $W({L}_{r},\xi (t))$-class by ${C}^{1}\cdot {N}_{p}$ summability method of conjugate series of its Fourier series

- Vishnu Narayan Mishra
^{1}Email author, - Vaishali Sonavane
^{1}and - Lakshmi Narayan Mishra
^{2}

**2013**:440

https://doi.org/10.1186/1029-242X-2013-440

© Mishra et al.; licensee Springer. 2013

**Received:**23 January 2013**Accepted:**30 July 2013**Published:**18 September 2013

## Abstract

Recently, Lal (Appl. Math. Comput. 209:346-350, 2009) has determined the degree of approximation of a function belonging to Lip*α* and weighted $W({L}_{r},\xi (t))$-classes using product ${C}^{1}\cdot {N}_{p}$ summability with non-increasing weights $\{{p}_{n}\}$. In this paper, we determine the degree of approximation of function $\tilde{f}$, conjugate to a 2*π*-periodic function *f* belonging to weighted $W({L}_{r},\xi (t))$-class by dropping the monotonicity on the generating sequence $\{{p}_{n}\}$ with a new (proper) set of conditions, which in turn generalizes the results of Mishra *et al.* (Bull. Math. Anal. Appl., 2013) on $Lip(\xi (t),r)$-class and rectifies (removes) the errors of Mishra *et al.* (Mat. Vesn., 2013). Few examples and applications are also highlighted in this manuscript.

**MSC:** 42B05, 42B08, 40G05, 41A10.

## Keywords

- generalized Lipschitz $W({L}_{r},\xi (t))$ ($r\ge 1$)-class of functions
- conjugate Fourier series
- degree of approximation
- ${C}^{1}$ means
- ${N}_{p}$ means
- product summability ${C}^{1}\cdot {N}_{p}$ transform

## 1 Introduction

Approximation by trigonometric polynomials is at the heart of approximation theory. The most important trigonometric polynomials used in the approximation theory are obtained by linear summation methods of Fourier series of 2*π*-periodic functions on the real line (*i.e.*, Cesàro means, Nörlund means and Product Cesàro-Nörlund means, *etc.*). Much of the advance in the theory of trigonometric approximation is due to the periodicity of the functions. The method of summability considered here was first introduced by Woronoi [1]. Summability techniques were also applied on some engineering problems like, Chen and Jeng [2] implemented the Cesàro sum of order $(C,1)$ and $(C,2)$, in order to accelerate the convergence rate to deal with the Gibbs phenomenon, for the dynamic response of a finite elastic body subjected to boundary traction. Chen *et al.* [3] applied regularization with Cesàro sum technique for the derivative of the double layer potential. Similarly, Chen and Hong [4] used Cesàro sum regularization technique for hyper singularity of dual integral equation. The degree of approximation of functions belonging to Lip*α*, $Lip(\alpha ,r)$, $Lip(\xi (t),r)$ and $W({L}_{r},\xi (t))$ ($r\ge 1$)-classes by Nörlund (${N}_{p}$) matrices and general summability matrices has been proved by various investigators like Govil [5], Khan [6], Qureshi [7], Mohapatra and Chandra [8], Leindler [9], Rhoades *et al.* [10], Guven and Israfilov [11], Bhardwaj and Gupta [12] and Mishra *et al.* [13–20]. Here, Lal [21] has assumed monotonicity on the generating sequence $\{{p}_{n}\}$ to prove their theorems. The approximation of function $\tilde{f}$, conjugate to a periodic function $f\in W({L}_{r},\xi (t))$ ($r\ge 1$) using product $({C}^{1}\cdot {N}_{p})$-summability has not been studied so far. In this paper, we obtain a new theorem on the degree of approximation of function $\tilde{f}$, conjugate to a periodic function $f\in W({L}_{r},\xi (t))$-class using semi-monotonicity on the generating sequence $\{{p}_{n}\}$ and a proper set of the conditions.

A bidiagonal matrix is a matrix with non-zero entries along the main diagonal and *either* the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix.

When the diagonal above the main diagonal has the non-zero entries, the matrix is upper bidiagonal. When the diagonal below the main diagonal has the non-zero entries, the matrix is lower bidiagonal.

*n*th partial sums $\{{s}_{n}\}$. Let $\{{p}_{n}\}$ be a non-negative sequence of constants, real or complex, and let us write

*s*if ${lim}_{n\to \mathrm{\infty}}{t}_{n}^{N}$ exists and is equal to

*s*. In the special case in which

The Nörlund summability ${N}_{p}$ reduces to the familiar ${C}^{\alpha}$ summability.

*s*if ${lim}_{n\to \mathrm{\infty}}{t}_{n}^{CN}$ exists and is equal to

*s*.

*π*-periodic function and Lebesgue integrable. The Fourier series of $f(x)$ is given by

with *n* th partial sum ${s}_{n}(f;x)$.

where $\xi (t)$ is positive increasing function of *t*.

If $\beta =0$, then $W({L}_{r},\xi (t))$ reduces to the class $Lip(\xi (t),r)$, if $\xi (t)={t}^{\alpha}$ ($0<\alpha \le 1$), then $Lip(\xi (t),r)$ class coincides with the class $Lip(\alpha ,r)$, and if $r\to \mathrm{\infty}$, then $Lip(\alpha ,r)$ reduces to the class Lip*α*.

${L}_{\mathrm{\infty}}$-norm of a function $f:R\to R$ is defined by ${\parallel f\parallel}_{\mathrm{\infty}}=sup\{|f(x)|:x\in R\}$.

${L}_{r}$-norm of *f* is defined by ${\parallel f\parallel}_{r}={({\int}_{0}^{2\pi}|f(x){|}^{r}\phantom{\rule{0.2em}{0ex}}dx)}^{1/r}$, $r\ge 1$.

*n*under sup norm ${\parallel \parallel}_{\mathrm{\infty}}$ is defined by [22]

and ${E}_{n}(f)$ of a function $f\in {L}_{r}$ is given by ${E}_{n}(f)={min}_{n}{\parallel {t}_{n}-f\parallel}_{r}$.

*x*by

We note that ${t}_{n}^{N}$ and ${t}_{n}^{CN}$ are also trigonometric polynomials of degree (or order) *n*.

where $0\le m\le n$, ${U}_{k}={u}_{0}+{u}_{1}+{u}_{2}+\cdots +{u}_{k}$, if $k\ge 0$, ${U}_{-1}=0$, which can be verified, is known as Abel’s transformation and will be used extensively in what follows.

$\tau =[1/t]$, where *τ* denotes the greatest integer not exceeding $1/t$. Furthermore, *C* denotes an absolute positive constant, not necessarily the same at each occurrence.

We note that the series, conjugate to a Fourier series, is not necessarily a Fourier series. Hence a separate study of conjugate series is desirable and attracted the attention of researchers.

## 2 Known theorem

Lal [21] has obtained the degree of approximation of the functions belonging to $W({L}_{r},\xi (t))$-class by ${C}^{1}\cdot {N}_{p}$ means with monotonicity on the generating sequence $\{{p}_{n}\}$. He proved the following.

**Theorem 2.1**

*If*$f(x)$

*is a*2

*π*-

*periodic function and Lebesgue integrable on*$[0,2\pi ]$

*and is belonging to*$W({L}^{r},\xi (t))$-

*class*,

*then its degree of approximation by*${C}^{1}\cdot {N}_{p}$

*means of its Fourier series*(1.1)

*is given by*

*provided*$\xi (t)$

*satisfies the following conditions*:

*where* *δ* *is an arbitrary number such that* $s(1-\delta )-1>0$, ${r}^{-1}+{s}^{-1}=1$, $1\le r\le \mathrm{\infty}$, *conditions* (2.3) *and* (2.4) *hold uniformly in* *x*.

**Remark 2.2** The condition $1/{sin}^{\beta}(t)=\mathrm{O}(1/{t}^{\beta})$, $1/(n+1)\le t\le \pi $ used by Lal [[21], pp.349-350] in writing the proof of Theorem 2.1 is not valid since $sint\to 0$ as $t\to \pi $.

**Remark 2.3**There is a fatal error in the proof of Theorem 2.1 of Lal [[21], p.349], in calculating

note that $-\beta s-s+1<0$. Therefore, one has $\frac{1}{\beta s+s-1}[\frac{1}{{\u03f5}^{\beta s+s-1}}-{(n+1)}^{\beta s+s-1}]$, which need not be $\mathrm{O}({(n+1)}^{\beta s+s-1})$, since *ϵ* might be $\mathrm{O}(1/{n}^{\gamma})$ for some $\gamma >1$.

## 3 Main theorem

It is well known that the theory of approximations, *i.e.*, TFA, which originated from a well-known theorem of Weierstrass, has become an exciting interdisciplinary field of study for the last 130 years. These approximations have assumed important new dimensions due to their wide applications in signal analysis [23], in general and in digital signal processing [24] in particular, in view of the classical Shannon sampling theorem. Mittal *et al.* [25, 26] have obtained many interesting results on TFA using summability methods without monotonicity on the rows of the matrix *T*: a digital filter. Broadly speaking, signals are treated as functions of one variable, and images are represented by functions of two variables. Very recently, Mishra *et al.* [13, 18] obtained the degree of approximation of conjugate of a function using ${C}^{1}\cdot {N}_{p}$ product summability method of its conjugate series of its Fourier series in Lipschitz and weighted spaces, respectively. Therefore, the purpose of the present paper is to generalize the results of Mishra *et al.* [13] on the degree of approximation of a function $\tilde{f}(x)$, conjugate to a 2*π*-periodic function *f* belonging to weighted $W({L}_{r},\xi (t))$-class by ${C}^{1}\cdot {N}_{p}$ means of conjugate series of its Fourier series by dropping the monotonicity condition on the generating sequence $\{{p}_{n}\}$ with the help of a new (proper) set of conditions to rectify the errors of Mishra *et al.* [18]. More precisely, we state our main theorem as follows.

**Theorem 3.1**

*If*$\tilde{f}(x)$,

*conjugate to a*2

*π*-

*periodic function*

*f*

*belonging to*$W({L}_{r},\xi (t))$-

*class*,

*then its degree of approximation by*${C}^{1}\cdot {N}_{p}$

*means of conjugate series of Fourier series*(1.2)

*is given by*

*provided*$\{{p}_{n}\}$

*satisfies*

*and*$\xi (t)$

*satisfies the following conditions*:

*where* *δ* *is an arbitrary number such that* $s(\beta -\delta )-1>0$, ${r}^{-1}+{s}^{-1}=1$, $1\le r\le \mathrm{\infty}$, *conditions* (3.4) *and* (3.5) *hold uniformly in* *x*.

**Note 3.2** $\xi (\frac{\pi}{\sqrt{n}})\le \pi \xi (\frac{1}{\sqrt{n}})$, for $(\frac{\pi}{\sqrt{n}})\ge (\frac{1}{\sqrt{n}})$.

**Note 3.3** Condition ${V}_{n}<C\Rightarrow n{p}_{n}<C{P}_{n}$, [27].

**Note 3.4** The product transform ${C}^{1}\cdot {N}_{p}$ plays an important role in signal theory as a double digital filter [14] and theory of machines in mechanical engineering [14].

## 4 Lemmas

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

**Lemma 4.1** $|{\tilde{M}}_{n}(t)|=\mathrm{O}[1/t]$ *for* $0<t\le \pi /\sqrt{n}$.

*Proof*For $0<t\le \pi /\sqrt{n}$, $sin(t/2)\ge (t/\pi )$ and $|cosnt|\le 1$.

This completes the proof of Lemma 4.1. □

**Lemma 4.2**

*Let*$\{{p}_{n}\}$

*be a non*-

*negative sequence and satisfy*(3.2),

*then*

*Proof*We have

in view of Note 3.3. Combining (4.2), (4.3) and (4.4) yields (4.1).

This completes the proof of Lemma 4.2. □

## 5 Proof of theorem

Clearly, $|\psi (x+t)-\psi (t)|\le |f(u+x+t)-f(u+x)|+|f(u-x-t)-f(u-x)|$.

Then $f\in W({L}_{r},\xi (t))\Rightarrow \psi \in W({L}_{r},\xi (t))$.

This completes the proof of Theorem 3.1.

## 6 Applications

The theory of approximation is a very extensive field, which has various applications. As mentioned in [24], the *Lp*-space in general and ${L}_{2}$ and ${L}_{\mathrm{\infty}}$ in particular play an important role in the theory of signals and filters. From the point of view of the applications, the Sharper estimates of infinite matrices are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix-valued functions and enable to investigate perturbations of matrix-valued functions and compare them. Some interesting applications of the Cesàro summability can be seen in [2–4]. The following corollaries can be derived from Theorem 3.1.

**Corollary 1**

*If*$\beta =0$,

*then the generalized weighted Lipschitz*$W({L}^{r},\xi (t))$ ($r\ge 1$)-

*class reduces to the class*$Lip(\xi (t),r)$,

*and the degree of approximation of a function*$\tilde{f}(x)$,

*conjugate to a*2

*π*-

*periodic function*

*f*

*belonging to the class*$Lip(\xi (t),r)$,

*is given by*

*Proof*The result follows by setting $\beta =0$ in Theorem 3.1, we have

This completes the proof of Corollary 1. □

**Corollary 2**

*If*$\beta =0$, $\xi (t)={t}^{\alpha}$, $0<\alpha \le 1$,

*then the generalized weighted Lipschitz*$W({L}^{r},\xi (t))$ ($r\ge 1$)-

*class reduces to the class*$Lip(\alpha ,r)$, $(1/r)<\alpha <1$

*and the degree of approximation of a function*$\tilde{f}(x)$,

*conjugate to a*2

*π*-

*periodic function*

*f*

*belonging to the class*$Lip(\alpha ,r)$,

*is given by*

*Proof*Putting $\beta =0$, $\xi (t)={t}^{\alpha}$, $0<\alpha \le 1$ in Theorem 3.1, we have

since otherwise, the right-hand side of the equation above will not be $\mathrm{O}(1)$.

This completes the proof of Corollary 2. □

**Corollary 3**

*If*$\beta =0$, $\xi (t)={t}^{\alpha}$

*for*$0<\alpha <1$

*and*$r\to \mathrm{\infty}$

*in*(3.1),

*then*$f\in Lip\alpha $.

*In this case*,

*the degree of approximation of a function*$\tilde{f}(x)$,

*conjugate to a*2

*π*-

*periodic function*

*f*

*belonging to the class*Lip

*α*($0<\alpha <1$)

*is given by*

*Proof*For $r\to \mathrm{\infty}$ in Corollary 2, we get

This completes the proof of Corollary 3. □

**Examples** (i) From Theorem 20 of Hardy’s ‘Divergent Series,’ if a Nörlund method $(N,p)$ has increasing weights $\{{p}_{n}\}$, then it is stronger than $(C,1)$.

Here ${s}_{1}=1$, ${s}_{2}=0$, ${s}_{3}=1$, ${s}_{4}=0$, … .

*n*is even,

*i.e.*, $n=2m$, then

*n*is odd,

*i.e.*, $n=2m+1$, then

Thus, number $\frac{1}{2}$ is assigned to the infinite series $\sum {(-1)}^{n}$.

This sequence is summable $(C,1)$ to $1/2$, but is clearly not convergent.

Now, take $(N,p)$ to be the Nörlund matrix generated by ${p}_{n}=n+1$. Then this sequence is summable by $(N,p)$ method.

- (ii)
In the first example, we discuss the case in which the Nörlund method is stronger than $(C,1)$. The product ${C}_{1}N$ sums a divergent sequence, where

*N*is the Nörlund method.

We know already that $(C,1)$ sums the sequence ${s}_{n}={\sum}_{k=0}^{n}{(-1)}^{k}$. We will now show that *CN* is stronger than *C*, for any regular Nörlund matrix (with non-negative generating sequence).

The statement *CN* is stronger than *C* is equivalent to showing that the matrix $CN{C}^{-1}$ is regular.

where *e* denotes the sequence of all ones. ${C}^{-1}(e)=e$, so that $CN(e)=C(N(e))=Ce=e$. Thus, not only does $CN{C}^{-1}$ have a finite norm, but it also has row sums equal to one.

*k*and zeros elsewhere. Then

Therefore, $\{{b}_{n}\}$ is a null sequence. Since *N* is a regular Nörlund matrix, with $b:=\{{b}_{n}\}$, $t=N(b)$ is a null sequence. Since *C* is also regular. *Ct* is a null sequence, and $CN{C}^{-1}$ has null columns. Therefore, it is regular, and *CN* is stronger than *C*.

## 7 Conclusion

Various results concerning the degree of approximation of periodic signals (functions) belonging to the generalized weighted class by matrix operator have been reviewed, and the condition of monotonicity on the weights $\{{p}_{n}\}$ has been relaxed (*i.e.*, weakening the conditions on the filter, we improve the quality of digital filter). Further, a proper (correct) set of conditions has been discussed to rectify the errors pointed out in Remarks 2.2 and 2.3. Some interesting application of the operator $({C}^{1}\cdot {N}_{p})$ used in this paper is pointed out in Note 3.4. The theorem of this paper is an attempt to formulate the problem of approximation of the function $\tilde{f}$, conjugate to a periodic function $f\in W({L}_{r},\xi (t))$ ($r\ge 1$)-class through trigonometric polynomials generated by the product summability $({C}^{1}\cdot {N}_{p})$-transform of conjugate series of Fourier series of *f* in a simpler manner by dropping monotonicity on the generating sequence $\{{p}_{n}\}$ with the help of proper (new) set of conditions. Few applications and examples on product summability $({C}^{1}\cdot {N}_{p})$-transform of signals (functions) are also discussed in this manuscript.

## Authors’ information

Dr. VNM is currently working as an Assistant Professor of Mathematics at SVNIT, Surat, Gujarat, India, and he is a very active researcher in various fields of mathematics. A Ph.D. in Mathematics, he is a double gold medalist, ranking first in the order of merit in both B.Sc. and M.Sc. Examinations from the Dr. Ram Manohar Lohia Avadh University, Faizabad.

Dr. VNM has undergone rigorous training from IIT, Roorkee, Mumbai in computer oriented mathematical methods and has experience of teaching post graduate, graduate and engineering students. The second author VS is a research scholar (R/S) in Applied Mathematics and Humanities Department at the Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat (Gujarat), India under the supervision of Dr. VNM. Recently, the third author LNM has joined as a full time research scholar in the Department of Mathematics, National Institute of Technology, Silchar-788010, District-Cachar, Assam, and he is also a very good active researcher in approximation theory and operator analysis.

## Declarations

### Acknowledgements

The authors would like to thank the anonymous learned referees for several useful interesting comments and suggestions about the paper. Special thanks are due to our great master and friend academician Prof. Ravi P. Agarwal, Texas A&M University, Kingsville, TX, USA, for kind cooperation, smooth behavior during communication and for his efforts to send the reports of the manuscript timely.

## Authors’ Affiliations

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