Jacobi theta-functions and systems of integer shifted Gaussian functions. (English. Russian original) Zbl 1219.42023
J. Math. Sci., New York 173, No. 2, 231-241 (2011); translation from Sovrem. Mat. Prilozh. 67, 107-116 (2010).
The authors consider the system of all integer translates \(\varphi(x-k)\) \((k\in \mathbb Z)\) of the Gaussian function
\[
\varphi(x) = \exp (- \frac{x^2}{2 \sigma^2}) \quad (x \in \mathbb R)
\]
with fixed \(\sigma >0\). It is shown that this system is incomplete in \(L_2(\mathbb R)\). Using a Jacobi theta function, the authors prove the existence of a function
\[
f(x) = \sum_{k=-\infty}^{\infty} f_k\, \varphi(x-k)
\]
such that the system \(\{f(x-m);\, m \in \mathbb Z\}\) is orthonormal in \(L_2(\mathbb R)\). Further, Riesz constants of the system \(\{\varphi(x-k);\, k \in \mathbb Z\}\) are determined. Finally, a Lagrange function of the form
\[
g(x) = \sum_{k=-\infty}^{\infty} g_k\, \varphi(x-k)
\]
for cardinal interpolation is constructed.
Reviewer: Manfred Tasche (Rostock)
MSC:
| 42C30 | Completeness of sets of functions in nontrigonometric harmonic analysis |
| 42C15 | General harmonic expansions, frames |
| 33E05 | Elliptic functions and integrals |
| 41A05 | Interpolation in approximation theory |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
integer shifted Gaussian; integer translates; Gaussian function; incomplete system; orthonormal system; Riesz system; Riesz constants; Jacobi theta function; Lagrange function; cardinal interpolationReferences:
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