The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis

Abstract

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 APSF ⇐ ASF ⇒ EMSF ⇔ 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 After this manuscript was accepted for publication the authors learned that special issues were being prepared to honour Heinrich Wefelscheid on the occasion of his 70th anniversary. Since they valued his unstinted service to mathematics as a whole, thus the founding of the present journal and his leading editorship over the years, his publication of the Collected Works, in several volumes, of W. Blaschke and E. Landau, they expressed interest that the present paper, already dedicated to A. Ostrowski, be included in the special issues. 360 P. L. Butzer et al. Results. Math.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.Mathematics Subject Classification (2010). 65B15, 65D32, 94A20.

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

“…Thus the conclusion of the present paper, together with the papers [15,16] and [14], is the full equivalence of the groupings (A) and (B). In other words, the seven theorems for non-bandlimited functions are all equivalent to the corresponding ones for the classical bandlimited functions.…”

“…1. For a proof of ASF ⇔ EMSF the reader is referred to [16]. Thus each of the seven formulae GPDF, ASF, ARKF, PSF, PDPS, FERZ and EMSF is equivalent to each other, in the sense that each is a corollary of each of the others.…”

“…In the present paper, dealing with the grouping (A): GPDF, ASF, ARKF, PSF, PDPS, FERZ, as well as the Euler-Maclaurin summation formula EMSF (handled in [16])-all for non-bandlimited functions-the basic result is that all seven theorems are equivalent amongst themselves. On the other hand, in our joint paper [15], treating the grouping (B), namely the classical sampling theorem CSF, the general Parseval formula GPF (i. e., the GPDF for bandlimited functions), the reproducing kernel formula GPF (i. e., GPDF for bandlimited functions) and the PSF-all for bandlimited functions-it was shown that the latter four theorems are also all equivalent amongst themselves.…”

An alternative to the Euler–Maclaurin summation formula: approximating sums by integrals only

“…The special case of Corollary 5.7 with f (x) = 1 x+1 for x ≥ 0 was, essentially, the basis of Knuth's method in [17] to compute a decimal approximation to Euler's constant; cf. [17, formula (7)]. In that case, one can take m 0 = 1 and F(x) = ln(x + 1) for x ≥ 0.…”

Equivalence of zeta function technique and Abel–Plana formula in regularizing the Casimir energy of hyper-rectangular cavities

“…7,[18][19][20][21][22][23][24][25][26][27][28][29][30] And since they mostly turn out to be in agreement, the connection between the zeta function method and the other regularization methods has attracted some attention. [31][32][33][34] In mathematics, Butzer et al 35 have recently proved the equivalence between any two of the summation formulae of Euler-Maclaurin, Abel-Plana, and Poisson, which play some part in various regularization methods, in their finite and well-defined cases.…”