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Degree of convergence of the functions of trigonometric series in Sobolev spaces and its applications
Journal of Inequalities and Applications volume 2022, Article number: 59 (2022)
Abstract
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.
1 Introduction
Sobolev spaces are vector spaces whose elements are functions defined on domains in an \(\mathbb{N} \)-dimensional Euclidean space \(\mathbb{R}\mathbbm{^{}\mathbb{N}\mathbbm{}}\) and whose partial derivatives satisfy certain integrability conditions. In order to develop and elucidate the properties of these spaces and mappings between them, we require some machinery of general topology and real and functional analysis.
In one of the classical approximation theories, the properties of approximation of orthogonal function systems, polynomials, and trigonometric have been studied in \(L^{q}\)-norm, and mostly in the maximum norm by [3–5, 25, 27, 28, 30, 31].
The \(L^{q}\)-norm for \(q<\infty \) captures the “height” and “width” of a function. In mathematical terms “width” is same as the measure of support of the function. The Sobolev norms capture “height”, “width”, and “oscillations”. The Fourier transform measures oscillation (or frequency or wavelength) by decay of the Fourier transform i.e. the “oscillation” of a function is translated to “decay” of its Fourier transform. Sobolev norms measure “oscillation” via its derivatives (or regularity).
The idea of the best approximation of a function by a polynomial was aggravated by P. L. Chebyshev. This idea chronologically pioneered the discovery of Weierstrass theorem and formed the basis of the modern constructive theory of functions.
The quantity
which gives a measure of the deviation(error) of \(f(t)\) from the polynomial \(P_{\nu}(t)=c_{0}+c_{1}t+\cdots +c_{\nu}t^{\nu}\) corresponding to it, has been given the title of the best approximation of order ν of this function. If the polynomial \(P_{\nu}\) is a trigonometric polynomial \(T_{\nu}\) of degree ν, then the best approximation of a function \(f\in C^{*}\) 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.
2 Notations and preliminaries
In this section, we present notations, definitions, and known results.
2.1 Notations
-
(i)
\(C^{*}\) —\(C[K] \) with the continuous 2π-periodic functions on \(\mathbb{R}\).
-
(ii)
vraisup —The essential upper bound vrai \(\sup f(t)\) is the lower bound of all the numbers M, for which \(f(t)>M\) on a set of measure zero.
2.2 Sobolev spaces
For \(1\leq q<\infty \), the space \(L^{q}[0,2\pi ]\) consists of all measurable functions on \([0,2\pi ]\) such that
and the norm is defined by
When \(q=2\),
The νth order modulus of smoothness of a function \(f:A\rightarrow \mathbb{R}\) is defined by
where
For \(\nu =1\), \(\omega _{1}(f,t)\) is called the modulus of continuity of f [8].
Assume that X is an open subset of \(\mathbb{R}\mathbbm{^{}\mathbb{N}\mathbbm{}}\). The Sobolev space \(W^{\nu ,q}(\mathbf{X})\), \(\nu = 1,2,3,\ldots \) , consists of functions \(f\in L^{q}(\mathbf{X})\) such that, for every multi-index β with \(|\beta |\leq \nu \), the weak derivative \(D^{\beta}f\) exists and \(D^{\beta}f \in L^{q}(\mathbf{X})\).
Thus,
The norm of (2) is defined by
and
The semi-norm of (2) is defined by
and
When \(q=2\), the Sobolev space \(W^{\nu ,2}(\mathbf{X})\) is a Hilbert space with the inner product
where
and
For \(\nu =1\), \(q=2\), the Sobolev space is defined by
and its norm is defined by
Example 2.1
([2])
For \(1\leq q\leq \infty \), the function \(f(t)=|t|\) belongs to \(W^{1,q}(\mathbf{X})\), where \(\mathbf{X}=(-1,+1)\) and
Remark 2.2
Here, we discuss some important properties of the Sobolev space.
-
(i)
For \(1\leq q\leq \infty \) and \(\nu =1,2,\ldots \) , the Sobolev space \(W^{\nu ,q}(\mathbf{X})\) is a Banach space.
-
(ii)
For \(1\leq q< \infty \) and \(\nu =1,2,\ldots \) , the Sobolev space \(W^{\nu ,q}(\mathbf{X})\) is separable.
Remark 2.3
-
(i)
For \(\nu =0\), the Sobolev space reduces in \(L^{q}\) space i.e. \(W^{0,q}(\mathbf{X})=L^{q}(\mathbf{X})\).
-
(ii)
For \(\nu =1,2,3,\ldots \) , \(W^{\nu ,q}(\mathbf{X})=Lip(\nu ,q)\).
-
(iii)
For \(\beta =\nu \), we have \(W^{1,q}(\mathbf{X})=Lip(1,q)\).
-
(iv)
For \(\nu =1\), \(q\rightarrow \infty \), \(Lip(1,q)=Lip(1)\).
2.3 Fourier and derived Fourier series
Let f be a 2π-periodic Lebesgue integrable function defined on \([-\pi ,\pi ]\). The Fourier series of f is given by
The \(\nu ^{th}\) partial sum of (9) is given by
where
and \(D_{\nu}(s)\) (Dirichlet kernel) is defined by
The derived Fourier series of (9) is given by
which is obtained by differentiating (9) term by term.
The \(\nu ^{th}\) partial sum of (13) is given by
where
and
2.4 Conjugate Fourier and conjugate derived Fourier series
The conjugate series of (9) is given by
which is said to be a conjugate Fourier series.
The \(\nu ^{th}\) partial sum of (15) is given by
where the function f̃, the conjugate to a 2π-periodic function f, is given by
where
The derived series of (15) is given by
which is said to be a conjugate derived Fourier series.
The \(\nu ^{th}\) partial sum of (19) is given by
where the function \(\tilde{f}^{\prime }\), the conjugate to a 2π-periodic function f̃, is given by
where
The following result is relevant to our discussion.
Theorem 2.4
([10])
Let \(f\in L^{q}(\mathbb{R})\) with \(1< q\leq \infty \). The following properties are equivalent:
-
(i)
\(f\in W^{1,q}(\mathbb{R}) \);
-
(ii)
∃ a constant C such that for all \(s\in (\mathbb{R})\)
$$ \Vert \tau _{s}f-f \Vert _{L^{q}}(\mathbb{R})\leq C \vert s \vert .$$
Moreover, one can choose \(C= \|f^{\prime }\|_{L^{q}}(\mathbb{R})\) in (ii) and \((\tau _{s}(f))(t)=f(t+s)\).
2.5 Riesz means
Let \(\sum_{\nu =0}^{\infty}u_{\nu}\) be an infinite series such that \(s_{k}= \sum_{\nu =0}^{k}u_{\nu}\). Let \(p_{\nu}\) be a nonnegative, nondecreasing sequence of numbers such that
The sequence-to-sequence transformation defined by
is called Riesz means or \((R,p_{\nu})\) means of the sequence \(\{s_{ \nu}\}\). The series \(\sum_{\nu =0}^{\infty}u_{\nu}\) is said to be summable to the sum s by Riesz method if we can write \(t_{\nu}\rightarrow s \) as \(\nu \rightarrow \infty \).
The necessary and sufficient conditions for the \((R,p_{\nu})\) method to be regular are given by
2.6 Degree of convergence
The degree of convergence of a summation method to a given function f is a measure how fast \(T_{\nu}\) converges to f, which is given by
where \(\lambda _{\nu}\rightarrow \infty \) as \(\nu \rightarrow \infty \).
3 Main results
In this section, we study the following results.
3.1 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 \(0< s\leq \frac{1}{\nu +1}\), \(M_{\nu}(s)=\mathcal{O} (\frac{p_{\nu}(\nu +1)}{P_{\nu}} )\).
Proof
For \(0< s\leq \frac{1}{\nu +1}\), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\) and \(\sin (k+\frac{1}{2})s\leq (k+\frac{1}{2})s\).
Thus,
□
Lemma 3.3
Let \(\{p_{n}\}\) be a nonnegative and nondecreasing sequence, then for \(\frac{1}{\nu +1}< s\leq \pi \), \(M_{\nu}(s)=\mathcal{O} (\frac{p_{\nu}}{s^{2}P_{\nu}} )\).
Proof
For \(\frac{1}{\nu +1}< s\leq \pi \), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\), \(|\sin s|\leq 1\).
Now, using Abel’s transformation, we have
Thus,
□
Proof of Theorem 3.1
Using (10), the Riesz transform of the sequence \(\{s_{\nu}(t)\}\) is given by
Thus,
Using (14), the Riesz transform of the sequence \(\{s_{\nu}^{\prime }(t)\}\) 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
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
□
3.2 Degree of convergence of a function of conjugate Fourier series
Consider a series
We note that (38) is a conjugate series of a Fourier series \(\sum_{\nu =2}^{\infty}\frac{\cos (\nu t)}{\log \nu}\), but it is not a Fourier series that can be easily observed by the following theorem.
Theorem 3.4
([9])
If \(a_{\nu}>0\), \(\sum \frac{a_{\nu}}{\nu}=\infty \), then \(\sum a_{\nu} \sin \nu t \) is not a Fourier series. Hence, there exists a trigonometric series with coefficients tending to zero which are not Fourier series.
One can see [9] for more details on conjugate Fourier series.
The degree of approximation of a conjugate function in function spaces, viz. Lipschitz, Hölder, generalized Hölder, generalized Zygmund, and Besov spaces, using different means of conjugate Fourier series, has been studied by the authors [7, 11, 12, 16, 17, 19, 20, 23, 26] etc.
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.
Theorem 3.5
Let f̃ be a 2π-period and Lebesgue integrable function belonging to Sobolev spaces \(W^{1,2}\), then the degree of convergence of a function f̃ of conjugate Fourier series using Riesz means is given by
The following lemmas are required for the proof of Theorem 3.5.
Lemma 3.6
Let \(\{p_{n}\}\) be a nonnegative and nondecreasing sequence, then for \(0< s\leq \frac{1}{\nu +1}\), \(\tilde{M}_{\nu}(s)=\mathcal{O} (\frac{1}{s} )\).
Proof
For \(0< s\leq \frac{1}{\nu +1}\), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\) and \(|\cos ks|\leq 1\).
Thus,
□
Lemma 3.7
Let \(\{p_{n}\}\) be a nonnegative and nondecreasing sequence, then for \(\frac{1}{\nu +1}< s\leq \pi \), \(\tilde{M}_{\nu}(s)=\mathcal{O} (\frac{p_{\nu}}{s^{2}P_{\nu}} )\).
Proof
For \(\frac{1}{\nu +1}< s\leq \pi \), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\).
Now, using Abel’s transformation, we have
Thus,
□
Lemma 3.8
Let \(\{p_{n}\}\) be a nonnegative and nondecreasing sequence, then for \(0< s\leq \frac{1}{\nu +1}\), \(\tilde{M}_{\nu _{1}}^{\prime }(s)=\mathcal{O} ( \frac{(\nu +1)p_{\nu}}{P_{\nu}} )\).
Proof
For \(0< s\leq \frac{1}{\nu +1}\), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\) and \(\sin (k+\frac{1}{2})s\leq (k+\frac{1}{2})s\).
Thus,
□
Lemma 3.9
Let \(\{p_{n}\}\) be nonnegative and nondecreasing, then for \(\frac{1}{\nu +1}< s\leq \pi \), \(\tilde{M}_{\nu _{1}}^{\prime }(s)=\mathcal{O} ( \frac{ p_{\nu}}{s^{2}P_{\nu}} )\).
Proof
For \(\frac{1}{\nu +1}< s\leq \pi \), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\).
Now, using Abel’s transformation, we have
Thus,
□
Lemma 3.10
Let \(\{p_{n}\}\) be a nonnegative and nondecreasing sequence, then for \(0< s\leq \frac{1}{\nu +1}\), \(\tilde{M}_{\nu _{2}}^{\prime }(s)=\mathcal{O} (\frac{1}{s^{2}} )\).
Proof
For \(0< s\leq \frac{1}{\nu +1}\), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\) and \(|\cos ks|\leq 1\).
Thus,
□
Lemma 3.11
Let \(\{p_{n}\}\) be nonnegative and nondecreasing, then for \(\frac{1}{\nu +1}< s\leq \pi \), \(\tilde{M}_{\nu _{2}}^{\prime }(s)=\mathcal{O} ( \frac{p_{\nu}}{s^{3}P_{\nu}} )\).
Proof
For \(\frac{1}{\nu +1}< s\leq \pi \), \(\sin (\frac{s}{2})\geq \frac{s}{\pi}\).
Now, using Abel’s transformation, we have
Thus,
□
Proof of Theorem 3.5
Using (16), the Riesz transform of the sequence \(\{\tilde{s}_{\nu}(t)\}\) is given by
Thus,
Using (20), the Riesz transform of the sequence \(\{\tilde{s}_{\nu}^{\prime }(t)\}\) is given by
Thus,
where
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
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
□
4 Applications
In this section, we study some applications of our main results.
4.1 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 \(p_{\nu}=1\) \(\forall \ \nu \geq 0 \) and \(P_{\nu}=1+\nu \).
Then \(\phi _{t}(s)=0\) and \(dg_{t}(s)=6s^{2}\,ds\).
Therefore, \(M_{\nu}(s)=\mathcal{O}(1)\) for \(0< s\leq \frac{1}{\nu +1}\) and \(M_{\nu}(s)=\mathcal{O} (\frac{1}{s^{2}(\nu +1)} )\) for \(\frac{1}{\nu +1}< s\leq \pi \).
Then, we have
Since \(\|T_{\nu}(t)\|_{2}=0\), the degree of convergence of \(f(t)=t^{3}\) is obtained by
Now, we draw the graphs of \(T_{\nu}(f)\) for different values of ν (see Fig. 1).
Remark 4.1
From Table 1 and Figs. 1(a) to 1(f), we observe that the result obtained in Theorem 3.1 is much better than earlier results.
4.2 Application on the degree of convergence of a function of conjugate Fourier series in Sobolev norm using Riesz means
Consider a conjugate function \(\tilde{f} (t)= \sum_{\nu =2}^{\infty} \frac{\sin \nu t}{\log \nu}\) for \(\nu \geq 2\) and \(P_{-1}=p_{-1}=0 \) and \(p_{\nu}=1 \) \(\forall\ \nu \geq 0 \) and \(P_{\nu}=1+\nu \).
Then \(\varphi _{t}(s)= \sum_{\nu =2}^{\infty} \frac{2\cos \nu t \sin \nu s}{\log \nu}\), \(\|\varphi _{t}(s)\|_{2}= \sum_{\nu =2}^{\infty} \frac{s}{\log \nu}\) and \(\rho _{t}(s)= \sum_{\nu =2}^{\infty} \frac{2\sin \nu t \cos \nu s}{\log \nu}\), \(\|\rho _{t}(s)\|_{2}= \sum_{\nu =2}^{\infty} \frac{1}{\log \nu}\).
Therefore, \(\tilde{M}_{\nu}(s)=\mathcal{O} (\frac{1}{s} )\) for \(0< s\leq \frac{1}{\nu +1}\), \(\tilde{M}_{\nu}(s)= \mathcal{O} (\frac{1}{s^{2}(\nu +1)} )\) for \(\frac{1}{\nu +1}< s\leq \pi \), \(\tilde{M}_{\nu _{1}}(s)=\mathcal{O}(1)\) for \(0< s\leq \frac{1}{\nu +1}\), \(\tilde{M}_{\nu _{1}}(s)=\mathcal{O} (\frac{1}{(\nu +1)s^{2}} )\) for \(\frac{1}{\nu +1}< s\leq \pi \), \(\tilde{M}_{\nu _{2}}(s)=\mathcal{O} (\frac{1}{s^{2}} )\) for \(0< s\leq \frac{1}{\nu +1}\), \(\tilde{M}_{\nu _{2}}(s)=\mathcal{O} (\frac{1}{s^{3}} )\) for \(\frac{1}{\nu +1}< s\leq \pi \).
Then, we have
and
Thus, the degree of convergence of \(\tilde{f}(t)= \sum_{\nu =2}^{\infty} \frac{\sin \nu t}{\log \nu}\) for \(\nu \geq 2\) is obtained by
Now, we draw the graphs of \(\tilde{T}_{\nu}(f)\) for different values of ν (see Fig. 2).
Remark 4.2
From Table 2 and Figs. 2(a) to 2(f), we observe that the result obtained in Theorem 3.5 is much better than earlier results.
Remark 4.3
From Table 1 and Table 2, we also observe that the convergence of Fourier series is faster than the convergence of conjugate Fourier series.
5 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 series \(\tilde{f}(t)= \sum_{\nu =2}^{\infty} \frac{\sin \nu t}{\log \nu}\) for \(\nu \geq 2\) 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.
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Acknowledgements
The authors are thankful to Prof. K.N. Singh, Vice-Chancellor, Central University of South Bihar, Gaya for his encouragement to this work.
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HKN framed the problems. HKN and SY carried out the results and wrote the manuscripts. Both the authors contributed equally to the writing of this paper. All the authors read and approved the final manuscripts.
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Nigam, H.K., Yadav, S. Degree of convergence of the functions of trigonometric series in Sobolev spaces and its applications. J Inequal Appl 2022, 59 (2022). https://doi.org/10.1186/s13660-022-02794-0
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DOI: https://doi.org/10.1186/s13660-022-02794-0