Reproducing Formulas and Double Orthogonality in Bargmann and Bergman Spaces

Abstract

It is observed that the reproducing kernels of the Bargmann spaces in ${\bf C}^n $ act reproducingly over any polyball (apart, of course, from positive constants depending on the radii of the polyball). It is noticed likewise that the reproducing kernels of the Bergman spaces over the unit ball in ${\bf C}^n $ act reproducingly over any ball (that is, ball in the Bergman metric). From these observations the eigenvalues and eigenfunctions of certain concentration operators are found. These eigenfunctions can be viewed as analogues to the prolate spheroidal wave functions in the Paley–Wiener space, but they are simpler and have nice properties which the prolate spheroidal wave functions do not have. Such expansions are exploited to yield analogues to results on sampling of bandlimited signals: necessary density conditions for sampling and interpolation, and jittered sampling. These results can be interpreted as results on irregular discrete representations of particular short-time Fourier and wavelet transf...