The classical sampling or WKS theorem on reconstructing signals from uniformly spaced samples assumes the signals to be band-limited (i.e., with spectrum in a bounded interval [−W, W ]). This assumption was later weakened to a disjoint translates condition on the spectrum which led to an extension of the sampling theorem to multi-band signals with spectra in a union of finite intervals. In this article the disjoint translates condition is replaced by a more natural 'null intersection' condition on spectral translates. This condition is shown to be equivalent to an analogue of Plancherel's isometric formula when the spectrum has finite measure. Thus to some extent multi-band sampling theory has a logical structure similar to classical Fourier analysis. The relationship between the null intersection condition and the isometric formula is illustrated by considering the consequences when the null intersection condition does not hold. In this case, the sampling representation cannot hold for any function, whereas the isometric formula can still hold for some functions.
I. Kluvánek extended the Whittaker-Kotel'nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the 'band limited' condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an 'almost disjoint' translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse ofKluvánek's theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of / .
Dedicated with thanks to Univ.-Prof. P. L. Butzer, on the occasion of his retirement.ABSTRACT. Functions belonging to various Paley-Wiener spaces have representations in sampling series. When a function does not belong to such a space, the sampling series may converge, not to the object function but to an "alias" of it, and an aliasing error is said to occur. Aliasing error bounds are derived for one-and two-channel sampling series analogous to the WhittakerKotel'nikov-Shannon series, and for the multi-band sampling series, and a "derivative" extension of it, due to Dodson, Beaty, et al. The Poisson summation formula is a basic tool throughout.Aliasing in the one-channel case is shown to arise from a transformation with similarities to a projection. Where possible, the sharpness of the error bounds is discussed.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.