Abstract
In this paper, we prove an analog of Younis’s result (Younis in Int J Math Math Sci 9(2):301–312, 1986 , Theorem 5.2) on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying the Dini-Lipschitz functions in the space \(\mathbb {L}_{2}^{(\alpha ,\beta )}\).
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1 Introduction
It is well known that many problems for partial differential equations are reduced to a power series expansion of the desired solution in terms of special functions or orthogonal polynomials (such as Laguerre, Hermite, Jacobi, etc., polynomials). In particular, this is associated with the separation of variables as applied to problems in mathematical physics (see, e.g., [1, 2]).
In this article, we obtain an analog of Younis’s Theorem [3, Theorem 5.2] on the description of the image under the discrete Fourier–Jacobi transform of a class of functions satisfying the Dini-Lipschitz condition in weighted function spaces on \([-1,1]\). We now give the exact statement of this theorem.
Suppose that f(x) is a function in the \(L^{2}(\mathbb {R})\) space (all functions below are complex-valued), \(\Vert .\Vert _{L^{2}(\mathbb {R})}\) is the norm of \(L^{2}(\mathbb {R})\), and \(\delta \) is an arbitrary number in the interval (0, 1).
Theorem 1.1
[3, Theorem 5.2] Let \(f\in L^{2}(\mathbb {R})\). Then the following conditions are equivalents:
-
(i)
\(\Vert f(x+h)-f(x)\Vert _{L^{2}(\mathbb {R})}=O\left( h^{\delta } \left( \log \dfrac{1}{h} \right) ^{-\zeta } \right) \), as \(h\rightarrow 0, 0<\delta <1, \zeta >0\),
-
(ii)
\(\displaystyle \int _{|\lambda |\ge r}|\widehat{f}(\lambda )|^{2}d\lambda =O(r^{-2\delta }\left( \log r \right) ^{-2\zeta })\) as \(r\rightarrow \infty \),
where \(\widehat{f}\) stands for the Fourier transform of f.
This theorem has been generalized in the case of noncompact rank 1 Riemannian symmetric spaces [4], and was extented in [5] for the Dunkl transform in the space \(L^{2}(\mathbb {R}^{d}, w_{k}(x)dx)\), where \(w_{k}\) is a weight function invariant under the action of an associated reflection group, using a generalized spherical mean operator. The Younis’s Theorem 1.1 has been generalized recently for a class of functions satisfying the Dini-Lipschitz condition for the Jacobi–Dunkl transform in [6] and also for the Cherednik–Opdam transform in [7].
On the other hand, in [8, Theorem 2.17], Younis characterizes the set of functions in \(L^{2}([-\pi ,\pi ])\) satisfying the Lipschitz condition by means of an asymptotic estimate growth of the norm of their discrete Fourier transform. More precisely, we have:
Theorem 1.2
[8, Theorem 2.17] Let \(f\in L^{2}([-\pi ,\pi ])\). Then the following conditions are equivalents:
-
(i)
\(\Vert f(x+h)-f(x)\Vert _{L^{2}([-\pi ,\pi ])}=O(h^{\delta })\), as \(h\rightarrow 0, \delta \in (0,1)\),
-
(ii)
\(\sum \nolimits _{|n|\ge N}|\widehat{f}(n)|^{2}=O(N^{-2\delta })\) as \(N\rightarrow \infty \),
where \(\widehat{f}(n)\) stands for the nth Fourier transform coefficient of f.
We emphasize that Younis’s Theorem 1.2 has been generalized to the case of compact. Recently, it has also been extended to general compact Lie groups [9], and also for the Dunkl and Fourier–Bessel transform of a class of functions satisfying the Lipschitz condition [10, 11].
In our present paper, we investigate among other things the validity of Theorem 1.2 in case of functions of the wider Dini-Lipschitz class in weighted function spaces on \([-1,1]\). For this purpose, we use a generalized translation operator which was defined by Flensted-Jensen an Koornwinder (see [12]).
2 Preliminaries
Throughout the paper, \(\alpha \) and \(\beta \) are arbitrary real numbers with \(\alpha \ge \beta \ge -1/2\) and \(\alpha \ne -1/2\). We put \(w(x)=(1-x)^{\alpha }(1+x)^{\beta }\) and consider problems of the approximation of functions in the Hilbert spaces \(L_{2}([-1,1],w(x)dx)\). Let \(P_{n}^{(\alpha ,\beta )}(x)\) be the Jacobi orthogonal polynomials , \(n\in \mathbb {N}_{0}:=\{0,1,2,\ldots \}\) (see [13] or [14]). The polynomials \(P_{n}^{(\alpha ,\beta )}(x), n\in \mathbb {N}_{0},\) form a complete orthogonal system in the Hilbert space \(L_{2}([-1,1],w(x)dx)\). It is known (see [14], Ch. IV) that
The polynomials
are called normalized Jacobi polynomials.
In what follows it is convenient to change the variable by the formula \(x=\cos t, t\in I:=[0,\pi ]\). We use the notation
Let \(\mathbb {L}_{2}^{(\alpha ,\beta )}\) denote the space of square integrable functions f(t) on the closed interval I with the weight function \(\rho (t)\) and the norm
The Jacobi differential operator is defined as
The function \(\varphi _{n}(t)\) satisfies the differential equation
with the initial conditions \(\varphi _{n}(0)=1\) and \(\varphi _{n}^{'}(0)=0\).
Lemma 2.1
The following inequalities are valid for Jacobi functions \(\varphi _{n}(t)\)
-
(i)
For \(t\in (0,\pi /2]\) we have
$$\begin{aligned} | \varphi _{n}(t)|< 1. \end{aligned}$$ -
(ii)
For \(t\in [0,\pi /2]\) we have
$$\begin{aligned} 1-\varphi _{n}(t)\le c_{1}\lambda _{n} t^{2}. \end{aligned}$$ -
(iii)
For \(t\in [0,1]\) and \(ts\le 2\) we have
$$\begin{aligned} 1-\varphi _{n}(t)\ge c_{2}\lambda _{n} t^{2}. \end{aligned}$$
Proof
See [15, Proposition 3.5. and Lemma 3.1]
Recall from [15], the Fourier–Jacobi series of a function \(f\in \mathbb {L}_{2}^{(\alpha ,\beta )} \) is defined by
where
\(\square \)
The sequence \(\left\{ a_{n}(f), n\in \mathbb {N}_{0}\right\} \) is called the discrete Fourier–Jacobi transform of f.
Let
be a partial sums of series (1), and let
denote the best approximation of \(f\in \mathbb {L}_{2}^{(\alpha ,\beta )}\) by polynomials of the form
It is well known that
The Jacobi generalized translation is defined by the formula
where \(K(t,s,\theta )\) is a certain function (see [16]).
Below are some properties (see [15]):
-
(i)
\(T_{h}:\mathbb {L}_{2}^{(\alpha ,\beta )}\rightarrow \mathbb {L}_{2}^{(\alpha ,\beta )}\) is a continuous linear operator,
-
(ii)
\(\Vert T_{h}f\Vert \le \Vert f\Vert \),
-
(iii)
\(T_{h}(\varphi _{n}(t))=\varphi _{n}(h)\varphi _{n}(t)\),
-
(iv)
\(a_{n}(T_{h}f)=\varphi _{n}(h)a_{n}(f)\),
-
(v)
\(\Vert T_{h}f-f\Vert \rightarrow 0, \quad h\rightarrow 0\),
-
(vi)
\(\mathcal {B}(T_{h}f)=T_{h}(\mathcal {B}f)\).
The following lemma will be needed in due course.
Lemma 2.2
[9, Lemma 4.1] Suppose \(a\in \mathbb {R}, b_{n}\ge 0\) and \(0<c<d\). Then
3 Main results
For every function \(f\in \mathbb {L}_{2}^{(\alpha ,\beta )}\) we define the differences \(\Delta _{h}^{k}f\) of order, \(k\in \mathbb {N}=\{1,2,3,\ldots \}\), with step h, \(0<h<\pi \), by the formulae
where I is the identity operator in \(\mathbb {L}_{2}^{(\alpha ,\beta )}\).
where
Let \(W_{2}^{k}\) be the Sobolev space constructed by the Jacobi operator \(\mathcal {B}\). that is:
where \(\mathcal {B}^{0}f=f, \mathcal {B}^{j}f=\mathcal {B}(\mathcal {B}^{j-1}f), j=1,2,\ldots ,k.\)
Lemma 3.1
If \(f\in W_{2}^{k}\), then
where \(r=0,1,\ldots ,k.\)
Proof
Since \(\mathcal {B}\) is self-adjoint (see [15]), we have
This completes the proof of lemma \(\square \)
Lemma 3.2
If
then
Here, the convergence of the series on the right-hand side is understood in the sense of \(\mathbb {L}_{2}^{(\alpha ,\beta )}\).
Proof
By the definition of the operator \(T_{h}\),
Therefore, for any polynomial
Since \(T_{h}\) is linear, we have
Since \(T_{h}\) is a linear bounded operator in \(\mathbb {L}_{2}^{(\alpha ,\beta )}\) and the set of all polynomials \(Q_{N}(t)\) is everywhere dense in \(\mathbb {L}_{2}^{(\alpha ,\beta )}\), passage to the limit in (2) give the required equality.
\(\square \)
Remark
Since
the Parseval’s identity gives
If \(f\in W^{k}_{2}\), from Lemma 3.1, we have
where \(r=0,1,\ldots ,k.\)
Definition 3.1
Let \(\zeta \in \mathbb {R}\) and \(\delta \in (0,1)\). A function \(f\in W_{2}^{k}\) is said to be in the \((\delta ,\zeta ,2)\)-Jacobi Dini-Lipschitz class, denoted by \(Lip(\delta ,\zeta ,2)\), if
where \(r=0,1,\ldots ,k.\)
Theorem 3.1
Let \(f\in W_{2}^{k}\). The following two conditions are equivalent:
-
(a)
\(f\in Lip(\delta ,\zeta ,2),\)
-
(b)
\(\sum \nolimits _{\begin{array}{c} n\ge N \end{array}}\lambda _{n}^{4r}|a_{n}(f)|^{2}=O(N^{-2\delta }(\log N)^{2\zeta }), \quad as \quad N\rightarrow \infty \).
Proof
\((\textit{a})\Rightarrow (\textit{b})\) Let \(f\in Lip(\delta ,\zeta ,2)\). Then we have
It follows from (3) that
If \(0\le n\le \frac{1}{h}\), then \( nh\le 2\), and from the third inequality of lemma 2.1, we obtain
Therefore,
and, by \(\lambda _{n}\ge n^{2}\),
Putting \(N=\frac{1}{h}\), we may write this inequality in the following form:
From Lemma 2.2, we have
Thus, the first implication is proved.
\((\textit{b})\Rightarrow (\textit{a})\). Suppose now that
It follows from Lemma 2.2 that
According (3), we write
Note that
It follows from (4) and the second inequality in Lemma 2.1 that
On the other hand, it follows from the first inequality of lemma 2.1 that
Consequently,
and this ends the proof of the Theorem. \(\square \)
We conclude this work by the following immediate consequence.
Corollary 3.1
Let \(f\in W_{2}^{k}\), and let
Then
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The authors would like to thank the referee for his valuable comments and suggestions.
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El Ouadih, S., Daher, R. Best trigonometric approximation and Dini-Lipschitz classes. J. Pseudo-Differ. Oper. Appl. 9, 903–912 (2018). https://doi.org/10.1007/s11868-017-0223-y
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DOI: https://doi.org/10.1007/s11868-017-0223-y