CONTENTS Introduction Story One. Historical notes Story Two. Some methods for deriving the cardinal and allied series Story Three. L 2 and L p theory Story Four. The cardinal series and LCA groups Story Five. Extensions to higher dimensions Conclusion INTRODUCTION

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Five short stories about the cardinal series

A basic property of algebraic polynomials ${{P}_{n}}\left( t \right): = \sum\nolimits_{{k = 0}}^{n} {{{a}_{k}}{{t}^{k}}}$ of degree ≤ n is that any such polynomial is fully and uniquely determined by its values P n (t v ) at an arbitrary given set of n + 1 distinct points t 0, t 1,…, t v,…, t n .The converse question, whether it is always possible to find a suitable polynomial P n (x) which takes on the function (polynomial) values y 0, y 1,…, y n associated with any n + 1 distinct abscissas (interpolation or nodal points) t 0, t 1, …, t n , is answered by the Lagrangian interpolation formula: 

$$ P_n \left( t \right): = \sum\limits_{k = 0}^n {y_k l_k \left( t \right) = \sum\limits_{k = 0}^n {P_n \left( {t_k } \right)l_k \left( t \right)} } $$

(1)

where, for k = 0, 1,…, n, 

$$\begin{array}{*{20}{c}} {{{l}_{k}}\left( t \right): = {{l}_{{k,n}}}\left( t \right) = \frac{{{{\omega }_{{n + 1}}}\left( t \right)}}{{\left( {t - {{t}_{k}}} \right){{{\omega '}}_{{n + 1}}}\left( {{{t}_{k}}} \right)}},} & {{{\omega }_{{n + 1}}}\left( t \right): = \left( {t - {{t}_{0}}} \right) \ldots \left( {t - {{t}_{n}}} \right).} \\ \end{array}$$

(2)

Indeed, l k (t) is the only algebraic polynomial of degree n which possesses the property 

$${{l}_{k}}\left( {{{t}_{\nu }}} \right) = {{\delta }_{{k,\nu }}} = \left\{ {\begin{array}{*{20}{c}} 0 & {for{\mkern 1mu} \nu \ne \kappa } \\ 1 & {for{\mkern 1mu} \nu \ne \kappa .} \\ \end{array} } \right.$$

(3)

Hence the polynomial P n (t) of degree n, thus determined, is unique.

An Introduction to Sampling Analysis

1. Introduction 2. Fundamentals of Fourier Analysis 3. Fourier Analysis in Systems Theory 4. Fourier Transforms in Probability, Random Variables and Stochastic Processes 5. The Sampling Theory 6. Generalizations of the Sampling Theorem 7. Noise and Error Effects 8. Multidimensional Signal Analysis 9. Time-Frequency Representations 10. Signal Recovery 11. Signal and Image Synthesis: Alternating Projections Onto Convex Sets 12. Mathematical Morphology and Fourier Analysis on Time Sales 13. Applications 14. Appendices 15. Reference

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Handbook of Fourier Analysis & Its Applications

http://link.springer.com/content/pdf/10.1007/978-3-0348-8968-1_9.pdf

Sampling Theory of Signal Analysis

This paper is concerned with the two summation formulae of Euler–Maclaurin (EMSF) and Abel–Plana (APSF) of numerical analysis, that of Poisson (PSF) of Fourier analysis, and the approximate sampling formula (ASF) of signal analysis. It is shown that these four fundamental propositions are all equivalent, in the sense that each is a corollary of any of the others. For this purpose ten of the twelve possible implications are established. Four of these, namely the implications of the grouping ${\text{APSF}\Leftarrow\text{ASF}\Rightarrow\text{EMSF}\Leftrightarrow\text{PSF}}$ are shown here for the first time. The proofs of the others, which are already known and were established by three of the above authors, have been adapted to the present setting. In this unified exposition the use of powerful methods of proof has been avoided as far as possible, in order that the implications may stand in a clear light and not be overwhelmed by external factors. Finally, the four propositions of this paper are brought into connection with four propositions of mathematical analysis for bandlimited functions, including the Whittaker–Kotel’nikov–Shannon sampling theorem. In conclusion, all eight propositions are equivalent to another. Finally, the first three summation formulae are interpreted as quadrature formulae.

/pdf/the-summation-formulae-of-euler-maclaurin-abel-plana-poisson-40trvmjt08.pdf

P.L. Butzer

Papers

The Euler-MacLaurin summation formula, the sampling theorem, and approximate integration over the real axis