The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals – their interconnections

Abstract

It is shown that the Whittaker–Kotel'nikov–Shannon sampling theorem of signal analysis, which plays the central role in this article, as well as (a particular case) of Poisson's summation formula, the general Parseval formula and the reproducing kernel formula, are all equivalent to one another in the case of bandlimited functions. Here equivalent is meant in the sense that each is a corollary of the other. Further, the sampling theorem is equivalent to the Valiron–Tschakaloff sampling formula as well as to the Paley–Wiener theorem of Fourier analysis. An independent proof of the Valiron formula is provided. Many of the equivalences mentioned are new results. Although the above theorems are equivalent amongst themselves, it turns out that not only the sampling theorem but also Poisson's formula are in a certain sense the ‘strongest’ assertions of the six well-known, basic theorems under discussion.

Keywords:

• sampling theorem
• bandlimited signals
• functions of exponential type
• Poisson's summation formula
• reproducing kernel formula
• Paley–Wiener's theorem

AMS Subject Classifications::

• 30D10
• 94A20
• 42C15
• 46E22

Acknowledgements

This article presents the first part of the invited lecture under the title ‘The sampling theorem – a central theorem of analysis’ held by Rudolf Stens at the Workshop ‘Approximation Theory and Signal Analysis’, conducted by Wolfgang zu Castell, Frank Filbir, Rupert Lasser and Jürgen Prestin at Lindau (Lake Constance), March 21–24, 2009. It presents joint work carried out by the five authors during a 3-year period, partly in Chartres, France.

The authors would like to thank Barbara Giese, Lehrstuhl A für Mathematik, RWTH Aachen, for carefully preparing the figures.

The authors are especially grateful to the conductors for inviting and bringing together an international group of participants to the Island City of Lindau. PLB would like to thank the many participants for coming to the Workshop, especially Hubert Berens, Erlangen, his first student and co-author and initiator with him of the research team at Aachen in 1964–1968. The invited lecture was capably chaired by Maurice Dodson, York University, UK, a long-standing friend of the authors.

Notes

Notes

1. Georges (Jean Marie) Valiron, born 7 September 1884 in Lyon, died March 1955 in Paris. He received his Agrégé de mathématiques in 1908, first taught at the lyceum of Besançon (Doubs) while continuing his studies in complex function theory. At Besançon he was the teacher of Georges Bloch (1894–1948) and his brother André. In 1914, he wrote his dissertation ‘Sur les fonctions entières d'ordre nul et d'ordre fini et en particulier sur les fonctions à correspondance régulière’ under É. Borel at the Université de Paris. Thereafter he taught as Professor in Valence (Drôme), and in March–June 1921 he presented a two-hour course on ‘Dirichlet series and factorial series’ at the reorganized University of Strasbourg where he remained until 1931. M. Fréchet was his colleague there from 1921 to 1927. Already in 1923 appeared his well-known ‘Lectures on the general theory of integral functions’ (220 pp; reprinted Chelsea 1949; latest edition, Iyer Press, March 2007) based on lectures he had presented at the University College of Wales (Aberystwyth). In 1931, he received the Chair for analysis at the Faculté des Sciences at Paris. In 1942, appeared his ‘Théorie des fonctions’ (real and complex) followed in 1945 by ‘Équations fonctionnelles et applications’ (reprinted in one volume by Masson 1966, by J. Gabay 1989). His speciality was complex function theory (entire and meromorphic functions). One of his four doctoral students is the Fields Medallist Laurent Schwartz, and according to the Mathematics Genealogy Project (http://genealogy.math.ndsu.nodak.edu/index.php) Valiron has 1860 academic descendants. Schwartz received his doctorate in 1943 at Clermond-Ferrand where the Université Louis Pasteur-Strasbourg was evacuated during WW II. Valiron was the president of the Société Mathématique de France in 1938, and received the Prix Poncelet in 1948. Liubomir Nikolov Tschakaloff (Любомир Николов Чакалов, also transliterated as Tschakalov, Chakalov or similarly) was born on 18 February 1886 into the family of an impoverished tailor of Samokov in Bulgaria, one of eleven children. The young Tschakaloff went to school in Samokov, a small town near Sofia, and then completed his schooling in the town of Plovdiv. By 1904, his attachment to mathematics was so strong that it prompted him to walk from Samokov to Sofia to enrol in the University there. He entered the University in the autumn of that year as a mathematics student. He graduated with honours in 1908, and in 1909 became an assistant at Sofia University. During the period 1910–1912 he pursued advanced studies at the Universities of Leipzig and Göttingen, coming into contact with some of the most famous mathematicians of the age, particularly Hilbert and Klein; Edmund Landau encouraged him to study number theory and analysis. The results of these studies became his Habilitationsschrift Analytical characteristics of the Riemann function ζ(z). In 1922, he became Professor at Sofia University. A second two-year period of study abroad, from 1924 to 1925, found him in Paris, Pisa and finally Naples, where he obtained his doctorate in 1925 with a dissertation on Riccati equations. Tschakaloff is best known for his work in entire and univalent functions, mean value theorems and Gaussian quadrature. His work was characterized by an ability to find original and incisive methods, which were often powerful enough to find wider uses. His scientific creativity was interrupted by two world wars, but apart from these exceptional times he produced a steady stream of scholarly work between 1910 and 1963. He published 112 works during his life, including books on analytic functions and differential equations. He was a member of the Bulgarian Academy of Sciences, as well as several foreign Academies. He died in September 1963. For further details see http://www.math.bas.bg/~serdica/tschakaloff.html. The authors thank Professor Virginia Kiryakova, Sofia, for supplying biographical information.

2. This result was proved in response to a question raised by one of the authors at the conference SampTA 05, Samsun, Turkey, July 2005, of whether Kluvánek's theorem implies an abstract approximate sampling theorem, in analogy with the classical case treated in 30.

3. Note, however, that an equivalence grouping can be of interest even when the logic value of the propositions is unknown. An example is the collection of propositions equivalent to the Riemann hypothesis.

4. In fact, big 𝒪 can be replaced by little o 38, p. 98], but this is much harder to prove and will not be used in the sequel.

5. Let us point out that Riemann's theorem on removable singularities has been used here. The same applies to the function g in connection with Valiron's formula (proof of Theorem 5.1).

6. An elementary proof of this expansion can be found in the appendix. Note that we are not using formula (20) as a side result, since it is a particular case of CSF.

7. This is only used to keep the formulae shorter. We do not apply any results from Hilbert space theory in particular we do not make use of the theory of orthogonal expansions in Hilbert spaces. In every case the inner product notation can be replaced by integrals.

8. The reader will observe that this is Bessel's inequality for the orthonormal system (sinc(· − k)) _k∈ℤ. See also the proof of Theorem 4.10.