Sampling and Quasi-Sampling in Rotation Invariant Paley-Wiener Spaces

Abstract

The rotation-invariant Paley-Wiener space PW _B with $B = \overline {K_r (0)} {\text{ }} \subset {\text{ }}\mathbb{R}^n$ is decomposed into a direct orthogonal sum of rotation-invariant subspaces H _k (B), k ∈ ℕ₀. An orthonormal basis $B_k : = \left\{ {v \in \mathbb{N},j = 1,...,a_k } \right\}$ for each of these subspaces, and hence for PWB _B , is constructed. The associated expansion of f ∈ PW _B is given by

$$f(x) = \mathop \sum \limits_{k = 0}^\infty \mathop \sum \limits_{v = 1}^\infty \mathop \sum \limits_{j = 1}^{a_k } c_{k,v} \left\{ {\int {_{s^{n - 1} } f(r_{k,v} u} )\psi _{v,j}^{(k)} (r_{k,v} u)d\sigma _{n - 1} (u)} \right\}\psi _{v,j}^{(k)} (x),$$

where S ^n-1 is the unit sphere in ℝⁿ. Here the numbers r _k,v > 0 and c _k,v ∈ ℝ are independent of f. For such an expansion, which could be called ?quasi-sampling expansion? only information of f on the discrete set of spheres r _k,v S ^n-1 is needed. For n = 1 this is essentially Shannon’s sampling theorem.

By means of the above expansion we derive sampling theorems for the subspaces

$$\mathop \oplus \limits_{k \in J} H_k (B) \subset PW_B {\text{ }}(J \subset \mathbb{N}_0 ,\left| J \right| < \infty )$$

of the form $f\left( x \right) = \mathop \sum \limits_{k \in J} \mathop \sum \limits_{v = 1}^\infty R_v^{\left( k \right)}\left( {\left| x \right|} \right)\mathop \sum \limits_{\mu = 1}^N f\left( {{r_{k,v}}t_\mu ^{k,v}} \right)Y_\mu ^{k,v}\left( {\frac{x}{{\left| x \right|}}} \right)$ with $t_\mu ^{\left( {k,v} \right)}{\text{ }} \in {S^{n - 1}}$, radial functions $R_v^{\left( k \right)}:\left( {0,\infty } \right) \to \mathbb{C}$ and spherical function Y ^(k,v)_μ : S ^n-1 → ℂ. These expansions show the spherical symmetry of the inderlying spaces. For J = {0} one obtains a sampling theorem for the space H₀(B) of radial Paley-Wiener functions, which can also be deduces from Kramer’s lemma.