Novel Sampling Formulas Associated with Quaternionic Prolate Spheroidal Wave functions

Abstract

The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration problems in quaternion Fourier transform are also investigated.Mathematics Subject Classification (2010). Primary 47B32; Secondary 94A11, 94A20.

Multichannel interpolation of nonuniform samples with application to image recovery

“…To give a simpler approximation for reconstructing a bandlimited function from nonuniform samples, the authors in [10] proposed a new kind of sinc interpolation method and they restricted S n (t) in (1.1) to be of the form sinc[σ(t − tn )], where tn = nT + ζ n and ζ n is a sequence of random variables independent of G(t). This restriction guarantees that the interpolating functions only consist of translation of sinc function, just like most cases of uniform interpolation [11,12,13,14]. However, the restriction strategy simplifies reconstruction problem but introduces error inevitably.…”

Generalized sampling expansions associated with quaternion Fourier transform

“…Lemma 2.1 [16] For every q ∈ H \ R, there is a non-real z ∈ C such that θ(q) ∩ C = {z, z}. Lemma 2.2 [13] If θ(p) ∩ θ(q) = ∅, then θ(p) = θ(q).…”

“…By applying the algorithm in [18] for computing the zeros, we get the zero set of p 3 (λ, s 1 ): Z(s 1 ) = {i} ∪ θ(i √ 3). It was shown in [13] that two eigenvectors of a normal operator are orthogonal if they correspond to two non-similar eigenvalues. So we may set λ 1 = i.…”

“…The theory of quaternion-valued dynamic equations, which is a new direction of mathematical analysis, has received a lot of attention because many physical problems can be described as quaternion models [9,10,11,12]. There are sampling theorems concerning quaternion-valued functions defined on R 2 and the samples in corresponding interpolation formulas are located on uniform grids [13,14]. In this paper the quaternion-valued functions we will study are defined on quaternion skew field and the samples involved in interpolation formulas are non-uniformly spaced.…”

FFT multichannel interpolation and application to image super-resolution

“…Therefore the error analysis of such sampling formulas is of great importance, because the recovered signal does not satisfy the conditions for ideal interpolation in most circumstances. Due to the wide range applications of interpolation, finding new interpolation or sampling formulas with error estimations as well as developing their fast algorithms for implementation have received considerable attentions in recently years, see for instance [3,4,5,6,7].…”