Plancherel--Polya-type inequalities for entire functions of exponential type in \(L_{p}(\mathbb R^{d})\)
A classical theorem of Plancherel-Polya states that if a function \(f \in L_p(R), p>0\), is an entire function of exponential type \(\omega\) then for any real sequence \(\{ x_j \} \subset \mathbb{R}\) such that \( \inf_{j \neq m } | x_j - x_m | >0\), the following inequality holds true: \[ \Bigl( \sum_j | f(x_j) |^p \Bigr)^{1/p} \leq C \| f \|_{L_p(R)}. \] Another well-known result of Plancherel and Polya states that if \(x_j = j \in \mathbb{Z}\) is a sequence of integers and \(f \in L_p(\mathbb{R}), 1 \leq p \leq \infty\), is a entire function of exponential type less than \(\pi\), then the following inequality also holds true: \[ \| f \|_{L_p(\mathbb{R})} \leq C \Bigl( \sum_j | f(x_j) |^p \Bigr)^{1/p}, \quad 1 \leq p \leq \infty . \] The sequence of real numbers \(\{ x_j \}\) for which both inequalities hold true \[ c \Bigl( \sum_j | f(x_j) |^p \Bigr)^{1/p} \leq \| f \|_{L_p(\mathbb{R})} \leq C \Bigl( \sum_j | f(x_j) |^p \Bigr)^{1/p} \] are known as sampling sequences. The corresponding inequalities are known as Plancherel-Polya inequalities or as the frame inequalities. The author considers these results for spaces \(E_p^{\omega}(\mathbb{R}^n)\), of entire functions of exponential type \(\leq \omega\) which belongs to \(L_p(\mathbb{R}^n)\). The author proves the following: Suppose that compactly supported distributions \(\{ \Phi_j \}\) satisfy suitable conditions (a typical example is a set of delta functionals \(\Phi_j = \delta_{x_j}\)). For any \(f \in E_p^{\omega}(\mathbb{R}^n), \;1 \leq p \leq \infty\), \[ c \sum_j | \Phi_j (f) |^p \leq \int_{\mathbb{R}^n} | f(x) |^p \,dx \leq C \sum_j | \Phi_j (f) |^p . \] The author also obtains a reconstruction theorem which is a generalization of the classical reslut of \textit{R. J. Duffin} and \textit{A. C. Schaeffer} [Trans. Am. Math. Soc. 72, 341--366 (1952; Zbl 0049.32401)].
scientific article; zbMATH DE number 5149520
scientific article