Error Bound of Periodic Signals in the Hölder Metric

We obtain two theorems to determine the error bound between input periodic signals and processed output signals, whenever signals belong to 𝐻𝜔-space and as a processor we have taken (𝐶,1)(𝐸,1)-mean and generalized an early result of Lal and Yadav in (2001).


Introduction
Chandra 1 was first to extend Pr össdorf's 2 result to find the degree of approximation of a continuous function using the N örlund transform.Later on, Mohapatra and Chandra 3 obtained a number of interesting results on the degree of approximation in the H ölder metric using matrix transforms, which generalize all the previous results based on Cesàro and N örlund transforms.In 1992, Singh 4 introduced H ω -space in place of H α -space and obtained several results on the degree of approximation of functions and deduced many previous results based on H α -spaces.In 1996, Das et al. 5 used H α,p -space in place of H α -space and obtained degree of approximation of functions and generalized the results of Mohapatra and Chandra 3 .In 2000, Mittal and Rhoades 6 also obtained the degree of approximation of functions in a normed space and generalized the results of Singh 4 by removing the hypothesis of monotonicity of the rows of the matrix.Singh and Soni 7 , and Mittal et al. 8 used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals.

Definitions and notations
Let the transforms A:

2.4
Let s t ∈ C 2π be a 2π-periodic analog signal whose Fourier trigonometric expansion be given by and let {s n t } be the sequence of partial sums of 2.5 .
Let the E, 1 and C, 1 transforms for the sequence {s n } be defined by respectively.
The product C, 1 E, 1 -transform is expressed as the C, 1 -transform of E, 1transform of {s n } and is given by sequence-to-sequence transformation see, e.g., 9 : The sequence {s n } is said to be summable C, where is Banach space 2 and the metric induced by the norm • α on H α is said to be H ölder metric.We write 16

2.17
International Journal of Mathematics and Mathematical Sciences

Known result
Lal and Yadav 10 established the following theorem to estimate the error between the input signal s t and the signal obtained after passing through the C, 1 E, 1 -transform.
Theorem A. If a function s : R → R is 2π-periodic and belonging to class Lip α, 0 < α ≤ 1, then the degree of approximation by C, 1 E, 1 means of its Fourier series is given by 3.1

Main result
The object of this paper is to generalize the above result under much more general assumptions.We will measure the error between the input signal s t and the processed output signal t n s; t 4.4

Lemmas
We will use following lemmas.
Lemma 5.1.Let φ t 1 t be defined in 2.16 , then for s ∈ H ω , we have It is easy to verify.Lemma 5.2.Let K n t be defined in 2.17 , then where "C" is an absolute constant, not necessarily the same at each occurrence. Proof.

6.7
Now from 5.1 , Lemmas 5.3 and 5.4, we have 6.8 6.9 Now noting that we have, from 6.6 and 6.8 , and from 6.7 and 6.9 , we have 6.12 Thus, from 2.13 , 6.11 and 6.12 , we have sup

6.15
This completes the proof of Theorem 4.1.
Proof of Theorem 4.2.Follows analogously as the proof of Theorem 4.1 with slight changes, so we omit details.