On Fields with Finite Information Density

The existence of a natural ultraviolet cutoff at the Planck scale is widely expected. In a previous Letter, it has been proposed to model this cutoff as an information density bound by utilizing suitably generalized methods from the mathematical theory of communication. Here, we prove the mathematical conjectures that were made in this Letter.Comment: 31 pages, to appear in Phys.Rev.

The Zero-Removing Property and Lagrange-Type Interpolation Series

The classical Kramer sampling theorem, which provides a method for obtaining orthogonal sampling formulas, can be formulated in a more general nonorthogonal setting. In this setting, a challenging problem is to characterize the situations when the obtained nonorthogonal sampling formulas can be expressed as Lagrange-type interpolation series. In this article a necessary and sufficient condition is given in terms of the zero removing property. Roughly speaking, this property concerns the stability of the sampled functions on removing a finite number of their zeros

A New Role of Green′s Function in Interpolation and Sampling Theory

AbstractOne of the most fundamental theorems in interpolation and sampling theory is the Whittaker-Shannon-Kotel′nikov sampling theorem, which gives an algorithm for reconstructing functions that are Fourier transforms of functions with compact supports, from their sampled values at a discrete set of points. Functions that are integral transforms of types other than the Fourier one, may also be reconstructed from their sampled values at a discrete set of points by using a generalization of the Whittaker-Shannon-Kotel′nikov sampling theorem, known as Kramer′s sampling theorem. The main assumption in Kramer′s sampling theorem is that the kernel of the integral transform to be reconstructed has to arise from a self-adjoint boundary-value problem whose eigenvalues are all simple and the eigenfunctions are all generated by one single function. In this paper, we derive a Kramer-type sampling theorem for a larger class of integral transforms than that considered by Kramer′s theorem, but under less stringent assumptions. For example, the kernel of any of these integral transforms to be reconstructed from its sampled values may arise from a non-self-adjoint boundary-value problem with repeated eigenvalues and eigenfunctions that are not necessarily generated by one single function. We only require that the eigenvalues be simple poles of the associated Green′s function. Our technique is based on the use of Green′s functions of non-self-adjoint boundary-value problems to reconstruct the interpolating functions

A Sampling Theorem for Signals Bandlimited to a General Domain in Several Dimensions

AbstractSampling series expansions for functions (signals) that are bandlimited to N-dimensional rectangles (N ≥ 1) have been studied extensively; however, if a function is bandlimited to a general region in RN, not much is known about its sampling series expansion. In this paper, we derive a sampling theorem for functions that are bandlimited (in the sense of Kramer) to finite regions with smooth boundaries in RN. The sampling series expansions obtained for these functions are Lagrange-type interpolation series. Our technique utilizes Green′s function of the region involved. As an application of our sampling theorem, we obtain a new method for summing infinite series in several variables

Wavelet Transform of Periodic Generalized Functions

AbstractThe aim of this paper is to define the wavelet transform for spaces of periodic functions, then extend this definition to spaces of generalized functions larger than the space of periodic Schwartz distributions, such as spaces of periodic Beurling ultradistributions and hyperfunctions on the unit circle. It is shown that the wavelet transforms of such generalized functions are nice and smooth functions defined on an infinite cylinder, provided that the analyzing wavelet is also nice and smooth. For example, it is shown that the growth rate of the derivatives of the wavelet transform is almost as good as that of the analyzing wavelet. More precisely, if the mother wavelet g satisfies Supx ∈ R|xkg(q)(x)| ≤ CAkBqkkβqqα (k, q = 0, 1, 2, …), then the wavelet transform Wg(ƒ) of a periodic Beurling ultradistribution ƒ satisfies sup(r,θ) ∈ Yϵ |rk ∂pθ ∂qrWg(ƒ)(r, θ)| ≤ DAkkαkBpCqppαqq(α + β); k, p, q ≥ 0, where Yϵ = {(r, θ): r ≥ ϵ > 0, θ ∈ T}