Applications of the Generalized Kramer-Weiss WKSK Sampling Theorem with its Necessary Compatible Analysis: A Detailed Tutorial
by Abdul J. Jerri
Professor Emeritus
Department of Mathematics, Clarkson University
Potsdam, NY 13676, USA
Dedication: This paper is dedicated to the memory of my dear friend, a pioneer of sampling theory, Professor John Rowland Higgins.
Abstract. This paper, first, deals with the establishing of compatible tools necessary for an applicable generalized Kramer-Weiss sampling theorem. As this theorem involves non-exponential kernels, the first hurdle arises from missing the periodicity property of the Fourier series. The next difficulty is dealing with finding a new approach paralleling the Fourier convolution product and theorem which is an important, easy to use tool. The Fourier convolution product and theorem benefit from their familiar exponential kernel property: eixt · e−ixτ = eix(t−τ).
In contrast, general non-exponential kernels have not been found to have anything that resembles such property. This will be clearly illustrated for the case of the Hankel transform and other familiar integral transforms.
The applications include associating the generalized sampling function with modeling for time-varying systems, helping form a solution to boundary value problems in chemical engineering, for plug flow, and a nonlinear chemical concentration in a cylindrical pellet. Further applications include an optics problem for scattering in a circular aperture and an acoustic problem that awaits our already established aliasing error bound for the Bessel sampling series.
The necessary tools to solve these problems involve deriving a generalization to the Poisson sum formula illustrated for the Hankel transform. This new result was essential for establishing the first aliasing error bound for a generalized sampling series, namely, the Bessel sampling series. It also can be extended to other non-exponential kernel integral transforms. A necessary bound of a Bessel function for deriving a truncation error bound for the associated Jn−sampling series has been established.
In that regard, Bessel-type Hill functions ψR+1 (x) were established. Presently, their transforms to a high power are being used for self-truncating their associated sampling series. A Lemma was developed for inserting such a factor.
For signal analysis and the associated problems, the introduction of the generalized convolution product and its associated theorem in the 1970’s is very important. Now, we would like to replace the “generalized convolution” with the Convolution-Parallel composition, or, in brief, “CP composition”. This is done because convolution means bending together and there is no “bending together” in the case of non-exponential kernel integral transforms. In this paper, we shall use the CP composition! This subject is also the topic of a detailed paper: Submitted [3].
Key words and phrases : Compatible mathematical tools for an applicable WKSK generalized sampling theorem, the sampling strong relation to the orthogonal expansion, the difficulty in moving to non-exponential kernels integral transforms, their complicated convolution product-parallel (CP- composition) and its theorem, applications done in mathematics, chemical engineering and those awaiting the new tools in optics for the scattering in a circular aperture, and in acoustics: needing our already established Jn−type aliasing error bound.
2010 AMS Mathematics Subject Classification — 94A20, 94A11, 62P35,42C10, 44A15, 41A10, 41A05.
A NEW MODIFICATION OF THE FOURIER SERIES FOR ITS CONVERGENCE AND THAT OF ITS CORRESPONDING SHANNON SAMPLING SERIES
The Fourier Series, and in general the Fourier Analysis, is an essential tool for almost all applications of mathematics from physics to engineering and in particular, communications theory and its internet revolution. For the latter, the Fourier Series, is used for establishing the Shannon Sampling Series, as the Shannon tool of the main contribution of the modern communications theory.
Such sampling series has its sampling function varies as 1 over n, which is barely good for the convergence of the series. Since the late 1960’s we have been working with this subject and the Hill Function (B-Splines) of high order m + 1. The Shannon Sampling Series uses just the Hill Function of order one for the orthogonality property of the used tool of Fourier Series.
Our new results use Hill Functions of order m + 1, for the orthogonality used for the Fourier Series, and its corresponding sampling series. The final result shows in raising the decay of the sampling function to power m + 1.
i.e., it varies like 1 divided by n raised to the power m + 1, instead of the regular case of just 1 divided by n, thus a better convergence, which means it is sufficient to use much fewer terms of the series, for a great savings.
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