Abstract

The goal of this paper is to approach a new result on the degree of Fourier series approximation of a function g ∈ W(L^r, ξ(t))(r ≥ 1) class by product summability. Signals are handled as one-variable functions, while images are represented as two-variable functions. The concept’s research is directly tied to the rapidly developing field of information technology. The approximation theory is a trigonometric polynomial approximation. Many academics have researched in similar lines, but our research also proves some fresh findings.

Keywords: Degree of approximation, -means, weighted generalized lipschitz class, Fourier series, Lebesgue integral

26.1 Introduction

Let L = L(0, 2π) denote the space of 2π-periodic and Lebesgue integrable functions on the (0, 2π) Fourier series at point x:

Let ∑a_n be an nth partial sum sequence {s_n} in an infinite series [9].

26.1.1 Definition 1

The transformation from sequence to sequence [5]:

(26.1)

The mean of the sequence {s_n} generalized by the series is defined by the sequence .

26.1.2 Definition 2

Riesz means can also be expressed as:

(26.2)

where R_n = p₀ q₀ + p₁q₁ + ⋯ + p_nq_n(≠ 0), p₍₋₁₎ = q₍₋₁₎ = R₍₋₁₎ = 0.

The series is and is then summable to s.

We have the following regularity conditions for Riesz summability [10]:

1. , For each integer k ≥ 0, as n → ∞.
2. where C can be any positive integer irrespective of n.

26.1.3 Definition 3

The (C,2) summability nth partial sum of the (C,2) transform is given [3]-.

(26.3)

By using the (C,2) method, the infinite series can be summable to the definite as the number.

26.1.4 Definition 4

of {s_n} is a composite transformation over (C,2) as

(26.4)

If is n → ∞, then ∑a_n can be summable to s using means.

26.1.5 Remark 1

If q_n = 1, then in (26.2) the , summability method reduces with summability and it reduces to , summability.

Also, the degree to which of the function g: R → R is approximated by a trigonometric polynomial t_n of degree n:

(26.5)

A function f(x) ∈ Lipα, if

(26.6)

and g(x) ∈ Lip(α, r), for 0 ≤ x ≤ 2π, if

(26.7)

This is also applicable to any function that increases in a positive way ξ(t), g(x) ∈ Lip(ξ(t), r), if

(26.8)

This is also applicable to any function that increases in a positive way ξ(t), an integer r ≥ 1, f ∈ W(L^r, ξ(t))

(26.9)

We will use the notations below as a guide:

(26.10)

Then, the weighted W(L^r, ξ(t)) class is a subset of the Lipα, Lip(α, r), and Lip(ξ(t), r) classes. As a result, the following additions have been made:

for all 0< α ≤ 1 and r≥1.

26.2 Known Result

In 2011, Nigam [1] proved a theorem on (C,1)(E,q) means of the Fourier series. Proceeding the work in 2014, Mishra et al. [4], the product (E,s) (N,p_n, q_n) - summability mean of the Fourier series showed a theorem on the degree of approximation. Pradhan [5] proposed the following theorem in 2016.

26.2.1 Theorem 1

If f ∈ W(L_r, ξ(t)) class is a 2π-periodic function that is integrable in the Lebesgue sense in [0,2π], then the degree of approximation is given by

where is the (E,s) transform of {s_n}, provided ξ(t) has the following requirements: is a decreasing sequence,

26.3 Main Theorem

Various mathematicians including Nigam [2], Deger [6], and Mishra et al. [7, 8] studied the degree of approximation using various summability approaches. By using the Riesz-Cesaro product summability approach, we obtain a novel result on the degree of approximation of function g ∈ W(L^r, ξ(t)) class.

The following theorem is established.

26.3.1 Theorem 2

If g ∈ W(L^r, ξ(t)) with , the degree of approximation by Riesz-Cesaro product mean of the Fourier series satisfies for n = 0,1,2…

(26.11)

where is the transform of s_n, supposing that function ξ(t) satisfies the required criteria,

is a function that does not increase and

(26.12)

(26.13)

where δ is an arbitrarily defined s.t. q^s(β − δ) – 1 > 0

r⁻¹ + s⁻¹ = 1, r ≥ 1 and (26.12) and (26.13) uniformly hold in x.

26.4 Some Auxiliary Results

Uniformly holding the following lemmas is necessary to prove the above theorem.

26.4.1 Lemma-1

For

we get P_n(t) = O(n + 2)

26.4.2 Lemma-2

For (n + 2)⁻¹ < t < π

we get

Using Jordan’s Lemma and sinkt ≤, we have

26.5 Theorem’s Proof

Using Riemann-Lebesgue theorem, we have

The Riesz Cesaro transform of the sequence is given by

(26.14)

where

Applying Minikowski’s inequality, since

and the fact that

and Lemma 1, we have

Using the second mean value theorem for integrals, ξ(t) is a decreasing function and we get

(26.15)

Now,

Using Hölder’s inequality as a starting point,

we put t =1/x, dt = -(1/x²)dx

Because ξ(t) > 0 and are both rising functions, the second mean value theorem is used and

(26.16)

from (26.14) and (26.15), (26.16) becomes

26.6 Applications

From our main theorem, we can derive the following corollaries.

26.6.1 Cor. 1

If ξ(t) = t^α, then weighted class W(L^r, ξ(t)), 1 ≤ r < ∞ reduces to the group of class Lip(α, r) and then a function f is a degree of approximation belonging to the class Lip(α, r), r⁻¹ ≤ α ≤ 1, given by

Proof: Setting produces, the result is β = 0 in (26.5).

26.6.2 Cor. 2

For r → ∞ in Cor. 1 for the class function, g ∈ Lip(α, r) reduces to the class Lip α and a function g belonging to the class Lipα, 0 < α < 1 whose degree of the approximation is given by

26.7 Conclusion

The previous finding on function degree of approximation demonstrated that the result is frequently true in nature and can be simplified to a few special cases. As a result, the current research can be used for a series of issues in the fields of analysis, technology, and engineering. Figure 26.1 shows some unique behaviour in the Fourier series approximations [11].

Acknowledgement

The sincere gratitude goes to the anonymous referees for their careful reading, remarks, and valuable comments, as well as many other useful suggestions for improved presentation. The author wishes to express his appreciation to the members of the editorial board.

References

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