Research Article
Applications of Cesàro Submethod to Trigonometric
Approximation of Signals (Functions) Belonging to Class
Lip(, ) in
-Norm
M. L. Mittal and Mradul Veer Singh
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
Correspondence should be addressed to Mradul Veer Singh; mradul.singh@gmail.com
Received 28 July 2015; Revised 18 November 2015; Accepted 11 February 2016
Academic Editor: Mohsen Tadi
Copyright © 2016 M. L. Mittal and M. V. Singh. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We prove two Teorems on approximation of functions belonging to Lipschitz class Lip(, ) in
-norm using Ces` aro submethod.
Further we discuss few corollaries of our Teorems and compare them with the existing results. We also note that our results give
sharper estimates than the estimates in some of the known results.
Dedicated to Professor S. C. Gupta
1. Introduction
For a given signal (function) ∈
fl
[0, 2], ≥1, let
() fl
(; ) =
0
2
+
∑
=1
(
cos +
sin )
=
∑
=0
(; )
(1)
denote the partial sums, called trigonometric polynomials of
degree (or order) , of the frst ( + 1) terms of the Fourier
series of at a point .
Defne
() =
(; ) =
∑
=0
,
() , ∀ ≥ 0. (2)
Te trigonometric Fourier series of signal is said to be -
summable to , if
() → as →∞.
Troughout ≡ (
,
), a linear operator, will denote an
infnite lower triangular matrix with nonnegative entries and
row sums 1. Such a matrix is said to have monotone rows
if, ∀, {
,
} is either nonincreasing or nondecreasing in ,
0≤≤. A linear operator is said to be regular if it is limit-
preserving over the space of convergent sequences.
Note 1. In (2), ≡ (
,
) behaves as a digital flter if
and
denote the input and output sequences of information,
respectively. By taking diferent values of
,
, we get a variety
of digital flters, For example, if
,
= ( + 1)
−1
, we get a very
popular digital flter known as Ces` aro flter. Psarakis and
Moustakides [1, p. 591] have mentioned that the digital flters
are widely used in various signal processing applications.
During past few decades, various techniques for designing
digital flters have been suggested. It is worth mentioning
that the most common techniques use, as approximation
criterion, the minimization of the
-measure 1≤≤∞.
Tus the designing of digital flters has been recognized as
an approximation problem. Here we are interested in approx-
imations of functions in
1
-space.
A signal (function) in
is approximated by trigono-
metric polynomials
of order (or degree) and the degree
Hindawi Publishing Corporation
Journal of Mathematics
Volume 2016, Article ID 9048671, 7 pages
http://dx.doi.org/10.1155/2016/9048671