Paley-Wiener theorem for the \(q\)-Bessel transform and associated \(q\)-sampling formula. (English) Zbl 1172.33308
For \(0 < q < 1\), \(v > -1\), and \(a = q^n, n \in \mathbb{Z}\), a \(q\)-Bessel Paley-Wiener space \(PW_{q,a}^v\) is defined (see Definition 1). Two characterizations of the functions in this space are obtained (see Theorems 2 and 3), one using the \(q\)-Bessel operator and the other using the \(q\)-Bessel translation operator. Finally, a \(q\)-sampling formula with sampling points \(q^n\), \(n \in \mathbb{Z}\)) is presented (see Theorem 4). To give a flavor of the results, let us quote the \(q\)-sampling formula:
Theorem 4. If \(f \in PW_{q,a}^v\), then for all \(z \in \mathbb{C}\) we have \[ \begin{split} f(z) = \frac{(1-q)^2}{(1-q^{2v+2})}c_{q,v}^2a^{2v+2}\sum_{n \in \mathbb{Z}} q^{n(2v+2)}f(q^n)\\ \times \left[ \frac{q^{2n}j_{v+1}(aq^n,q^2)j_v(aq^{-1}z,q^2) - z^2j_{v+1}(az,q^2)j_v(aq^{-1}q^n,q^2)}{q^{2n}-z^2}\right], \end{split} \] where \(j_v(x,q)\) is the normalized \(q\)-Bessel function of order \(v\).
Theorem 4. If \(f \in PW_{q,a}^v\), then for all \(z \in \mathbb{C}\) we have \[ \begin{split} f(z) = \frac{(1-q)^2}{(1-q^{2v+2})}c_{q,v}^2a^{2v+2}\sum_{n \in \mathbb{Z}} q^{n(2v+2)}f(q^n)\\ \times \left[ \frac{q^{2n}j_{v+1}(aq^n,q^2)j_v(aq^{-1}z,q^2) - z^2j_{v+1}(az,q^2)j_v(aq^{-1}q^n,q^2)}{q^{2n}-z^2}\right], \end{split} \] where \(j_v(x,q)\) is the normalized \(q\)-Bessel function of order \(v\).
Reviewer: Jeremy Lovejoy (Paris)
MSC:
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
| 33D60 | Basic hypergeometric integrals and functions defined by them |
| 33D90 | Applications of basic hypergeometric functions |
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