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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
Journal of Inequalities and Applications volume 2013, Article number: 440 (2013)
Abstract
Recently, Lal (Appl. Math. Comput. 209:346350, 2009) has determined the degree of approximation of a function belonging to Lipα and weighted W({L}_{r},\xi (t))classes using product {C}^{1}\cdot {N}_{p} summability with nonincreasing weights \{{p}_{n}\}. In this paper, we determine the degree of approximation of function \tilde{f}, conjugate to a 2πperiodic 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.
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 2πperiodic functions on the real line (i.e., Cesàro means, Nörlund means and Product CesàroNörlund 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 Lipα, 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 semimonotonicity on the generating sequence \{{p}_{n}\} and a proper set of the conditions.
A bidiagonal matrix is a matrix with nonzero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two nonzero diagonals in the matrix.
When the diagonal above the main diagonal has the nonzero entries, the matrix is upper bidiagonal. When the diagonal below the main diagonal has the nonzero entries, the matrix is lower bidiagonal.
For example, the following matrix is upper bidiagonal:
and the following matrix is lower bidiagonal:
Let {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n} be a given infinite series with the sequence of n th partial sums \{{s}_{n}\}. Let \{{p}_{n}\} be a nonnegative sequence of constants, real or complex, and let us write
The sequence to sequence transformation {t}_{n}^{N}={\sum}_{\nu =0}^{n}{p}_{n\nu}{s}_{\nu}/{P}_{n} defines the sequence \{{t}_{n}^{N}\} of Nörlund means of the sequence \{{s}_{n}\}, generated by the sequence of coefficients \{{p}_{n}\}. The series {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n} is said to be summable {N}_{p} to the sum 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.
The product of {C}^{1} summability with an {N}_{p} summability defines {C}^{1}\cdot {N}_{p} summability. Thus, the {C}^{1}\cdot {N}_{p} mean is given by {t}_{n}^{CN}=\frac{1}{n+1}{\sum}_{k=0}^{n}{P}_{k}^{1}{\sum}_{\nu =0}^{k}{p}_{k\nu}{s}_{\nu}. If {t}_{n}^{CN}\to s as n\to \mathrm{\infty}, then the infinite series {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n} or the sequence \{{s}_{n}\} is said to be summable {C}^{1}\cdot {N}_{p} to the sum s if {lim}_{n\to \mathrm{\infty}}{t}_{n}^{CN} exists and is equal to s.
Let f(x) be a 2πperiodic function and Lebesgue integrable. The Fourier series of f(x) is given by
with n th partial sum {s}_{n}(f;x).
The conjugate series of Fourier series (1.1) is given by
A function f(x)\in Lip\alpha if
and f(x)\in Lip(\alpha ,r), [6] for 0\le x\le 2\pi, if
f(x)\in Lip(\xi (t),r) if
f\in W({L}_{r},\xi (t)), [18, 20] if
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 Lipα.
{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.
The degree of approximation of a function f:R\to R by trigonometric polynomial {t}_{n} of order 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}.
The conjugate function \tilde{f}(x) is defined for almost every x by
We note that {t}_{n}^{N} and {t}_{n}^{CN} are also trigonometric polynomials of degree (or order) n.
Abel’s transformation: The formula
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.
If {v}_{m},{v}_{m+1},\dots ,{v}_{n} are nonnegative and nonincreasing, the lefthand side of (1.3) does not exceed 2{v}_{m}{max}_{m1\le k\le n}{U}_{k} in absolute value. In fact,
We write throughout
\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 2πperiodic 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.349350] 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 ss+1<0. Therefore, one has \frac{1}{\beta s+s1}[\frac{1}{{\u03f5}^{\beta s+s1}}{(n+1)}^{\beta s+s1}], which need not be \mathrm{O}({(n+1)}^{\beta s+s1}), 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 wellknown 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 2πperiodic 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 2πperiodic 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 nonnegative sequence and satisfy (3.2), then
Proof We have
where
and using Abel’s transformation and sin(t/2)\ge (t/\pi ), for 0<t\le \pi, we get
by virtue of the fact that {\sum}_{k=\lambda}^{\mu}exp(ikt)=\mathrm{O}({t}^{1}), 0\le \lambda \le k\le \mu.
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
Let {\tilde{s}}_{n}(f;x) denote the partial sum of series (1.2), we have
Denoting {C}^{1}\cdot {N}_{p} means of {\tilde{s}}_{n}(f;x) by {\tilde{t}}_{n}^{CN}, we write
Clearly, \psi (x+t)\psi (t)\le f(u+x+t)f(u+x)+f(uxt)f(ux).
Hence, by Minkowski’s inequality,
Then f\in W({L}_{r},\xi (t))\Rightarrow \psi \in W({L}_{r},\xi (t)).
Using Hölder’s inequality, \psi (t)\in W({L}_{r},\xi (t)), condition (3.4), sin(t/2)\ge (t/\pi ), for 0<t\le \pi, Lemma 4.1, Note 3.2 and second mean value theorem for integrals, we have
Using Lemma 4.2, we have
Using Hölder’s inequality, conditions (3.3) and (3.5), sint\le 1, sin(t/2)\ge (t/\pi ), for 0<t\le \pi, Note 3.2 and second mean value theorem for integrals, we have
Similarly, as conditions (3.2), (3.3) and (3.5) above, sint\le 1, sin(t/2)\ge (t/\pi ), for 0<t\le \pi, Note 3.2 and second mean value theorem for integrals, we have
Collecting (5.1)(5.4), we have
Now, using the {L}_{r}norm of a function, we get
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 Lpspace 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 matrixvalued functions and enable to investigate perturbations of matrixvalued 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 2πperiodic 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
Thus, we get
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 2πperiodic 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
or,
or,
since otherwise, the righthand side of the equation above will not be \mathrm{O}(1).
Hence
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 2πperiodic function f belonging to the class Lipα (0<\alpha <1) is given by
Proof For r\to \mathrm{\infty} in Corollary 2, we get
Thus, we have
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).
Therefore, there is an easy way to find a sequence that is (C,1) summable, but not convergent. One such sequence is
Here {s}_{1}=1, {s}_{2}=0, {s}_{3}=1, {s}_{4}=0, … .
For (C,1) summability,
Case (i) If n is even, i.e., n=2m, then
Case (ii) If 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.
For
Since it is summable (C,1), also, and (N,p) method is stronger than (C,1) method. Therefore, it is then summable by the product (C,1)(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 nonnegative generating sequence).
The statement CN is stronger than C is equivalent to showing that the matrix CN{C}^{1} is regular.
Since {C}^{1} is a bidiagonal matrix {c}_{nn}^{1}=n+1 and {c}_{n+1}^{1}=(n1), for k<n,
Also,
Therefore,
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.
Let \{{e}^{k}\} denote the column vector with one in position 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, Silchar788010, DistrictCachar, Assam, and he is also a very good active researcher in approximation theory and operator analysis.
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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.
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Authors’ contributions
LNM computed lemmas and established the main theorem with a proper (a new) set of conditions to remove the errors in this direction. LNM studied examples (i) and (ii) in its depth. LNM and VNM conceived of the study and participated in its design and coordination. LNM, VNM contributed equally and significantly in writing this paper. All the authors, i.e., VNM, VS and LNM drafted the manuscript, read and approved the final manuscript.
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Mishra, V.N., Sonavane, V. & Mishra, L.N. {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. J Inequal Appl 2013, 440 (2013). https://doi.org/10.1186/1029242X2013440
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DOI: https://doi.org/10.1186/1029242X2013440
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