Degree of convergence of the functions of trigonometric series in Sobolev spaces and its applications

In this paper, we study the degree of convergence of the functions of Fourier series and conjugate Fourier series in Sobolev spaces using Riesz means. We also study some applications of our main results and observe that our results are much better than earlier results.

The quantity vrai sup a≤t≤b f (t) -P ν (t) [29], which gives a measure of the deviation(error) of f (t) from the polynomial P ν (t) = c 0 + c 1 t + · · · + c ν t ν corresponding to it, has been given the title of the best approximation of order ν of this function. If the polynomial P ν is a trigonometric polynomial T ν of degree ν, then the best approximation of a function f ∈ C * is given by In this paper, we study the degree of convergence of the functions of Fourier series and conjugate Fourier series in Sobolev norms using Riesz means. However, detailed objectives of this paper will be presented in Sect. 3. Organization of the paper is as follows: In Sect. 2, we give important definitions and known results related to our work. In Sect. 3, we mention detailed objectives of the proposed problems and obtain their results. Applications and their numerical results are discussed in Sect. 4, while conclusion is given in Sect. 5.

Notations and preliminaries
In this section, we present notations, definitions, and known results.
Assume that X is an open subset of R N . The Sobolev space W ν,q (X), ν = 1, 2, 3, . . . , consists of functions f ∈ L q (X) such that, for every multi-index β with |β| ≤ ν, the weak derivative D β f exists and D β f ∈ L q (X). Thus, The norm of (2) is defined by and The semi-norm of (2) is defined by and When q = 2, the Sobolev space W ν,2 (X) is a Hilbert space with the inner product where For ν = 1, q = 2, the Sobolev space is defined by and its norm is defined by Remark 2.2 Here, we discuss some important properties of the Sobolev space.

Fourier and derived Fourier series
Let f be a 2π -periodic Lebesgue integrable function defined on [-π, π]. The Fourier series of f is given by (a ν cos νt + b ν sin νt).
The ν th partial sum of (9) is given by where and D ν (s) (Dirichlet kernel) is defined by The derived Fourier series of (9) is given by which is obtained by differentiating (9) term by term.
The ν th partial sum of (13) is given by where and

Conjugate Fourier and conjugate derived Fourier series
The conjugate series of (9) is given bỹ which is said to be a conjugate Fourier series. The ν th partial sum of (15) is given bỹ where the functionf , the conjugate to a 2π -periodic function f , is given bỹ where The derived series of (15) is given bỹ which is said to be a conjugate derived Fourier series. The ν th partial sum of (19) is given bỹ = -2(ν + 1 2 ) π π 0 ρ t (s) sin(ν + 1 2 )s 4 sin s where the functionf , the conjugate to a 2π -periodic functionf , is given bỹ where The following result is relevant to our discussion.
The following properties are equivalent:

Riesz means
Let ∞ ν=0 u ν be an infinite series such that s k = k ν=0 u ν . Let p ν be a nonnegative, nondecreasing sequence of numbers such that The sequence-to-sequence transformation defined by The necessary and sufficient conditions for the (R, p ν ) method to be regular are given by

Degree of convergence
The degree of convergence of a summation method to a given function f is a measure how fast T ν converges to f , which is given by where λ ν → ∞ as ν → ∞.

Main results
In this section, we study the following results.

Degree of convergence of a function of Fourier series
The degree of approximation of a function in function spaces, viz. Lipschitz, Hölder, generalized Hölder, generalized Zygmund, and Besov spaces, using different means of Fourier series, has been studied by the authors [7, 12, 13, 15, 17-19, 21, 22, 24] etc.
Since the degree of approximation of a function of Fourier series in the above mentioned spaces only gives the degree of the polynomial with respect to the function, but the degree of convergence of a function of Fourier series gives the convergence of the polynomial with respect to the function. The degree of convergence of a function of Fourier series in Sobolev spaces gives a much better result than that of the earlier results obtained using the spaces other than Sobolev spaces. Therefore, in this subsection, we study the degree of convergence of a function in Sobolev spaces using the Riesz means of Fourier series and establish the following theorem.

Theorem 3.1 Let f be a 2π -period and Lebesgue integrable function belonging to Sobolev spaces W 1,2 , then the degree of convergence of a function f of Fourier series using Riesz means is given by
The following lemmas are required for the proof of Theorem 3.1.

Lemma 3.2 Let {p n } be a nonnegative and nondecreasing sequence, then for
Thus,

Lemma 3.3 Let {p n } be a nonnegative and nondecreasing sequence, then for
Thus, Proof of Theorem 3.1 Using (10), the Riesz transform of the sequence {s ν (t)} is given by Thus, Using (14), the Riesz transform of the sequence {s ν (t)} is given by Thus, Now, using the definition of Sobolev norm given in (8), we have Using the definition of L 2 norm, we have Using generalized Minkowski's inequality [6], we have Using Theorem 2.4, we get Now, using Lemma 3.2, we get Now, using Lemma 3.3, we get From (31) and (32), we have Using the definition of L 2 norm, we get Using generalized Minkowski's inequality [6], we get Now, using Lemma 3.2, we get Now, using Lemma 3.3, we get From (35) and (36), we have From (33) and (37), we have

Consider a series
We note that (38) is a conjugate series of a Fourier series ∞ ν=2 cos(νt) log ν , but it is not a Fourier series that can be easily observed by the following theorem.

Theorem 3.4 ([9])
If a ν > 0, a ν ν = ∞, then a ν sin νt is not a Fourier series. Hence, there exists a trigonometric series with coefficients tending to zero which are not Fourier series.
As discussed in Sect. 3.1, the degree of convergence of a function of conjugate Fourier series also gives the convergence of the polynomial with respect to the function. The degree of convergence of a function of conjugate Fourier series in Sobolev spaces gives a much better result than that of the results using the spaces other than Sobolev spaces. Therefore, in this subsection, we study the degree of convergence of conjugate of a function in Sobolev spaces using the Riesz means of conjugate Fourier series and establish a following theorem. 2 1 s 3 ds .

Theorem 3.5 Letf be a 2π -period and Lebesgue integrable function belonging to Sobolev spaces W 1,2 , then the degree of convergence of a functionf of conjugate Fourier series using Riesz means is given by
The following lemmas are required for the proof of Theorem 3.5. Proof For 0 < s ≤ 1 ν+1 , sin( s 2 ) ≥ s π and | cos ks| ≤ 1.

Lemma 3.9 Let {p n } be nonnegative and nondecreasing, then for
Now, using Abel's transformation, we have
Now, using the definition of Sobolev norm given in (8), we have Using the definition of L 2 norm, we have Using generalized Minkowski's inequality [6], we have Using Theorem 2.4, we get Now, using Lemma 3.6, we get Now, using Lemma 3.7, we get From (45) and (46), we have Using the definition of L 2 norm and generalized Minkowski's inequality [6], we get Now, using Lemmas 3.8 and 3.9, we get Now, using Lemmas 3.10 and 3.11, we get From (49) and (50), we have From (47) and (51), we have 2 1 ρ t (s) 2 1 s 3 ds .

Applications
In this section, we study some applications of our main results.

Application on the degree of convergence of a function of Fourier series in Sobolev norm using Riesz means
Consider a function f (t) = t 3 and P -1 = p -1 = 0 and p ν = 1 ∀ ν ≥ 0 and P ν = 1 + ν.

Conclusion
From Table 1 and Figs. 1(a) to 1(f ), we observe that the degree of convergence of Fourier series f (t) = t 3 is much better than that of earlier results, and from Table 2 and Figs. 2(a) to 2(f ), we observe that the degree of convergence of conjugate Fourier seriesf (t) = ∞ ν=2 sin νt log ν for ν ≥ 2 is much better than that of earlier results. Also, from Table 1 and Table 2, we observe that the convergence of Fourier series is faster than the convergence of conjugate Fourier series.