In this part, we present a survey of mean-periodicity phenomena which arise in connection with classical questions in complex analysis, partial differential equations, and more generally, convolution equations. A common feature of the problem we shall consider is the fact that their solutions depend on techniques and ideas from complex analysis. One finds in this way a remarkable and fruitful interplay between mean-periodicity and complex analysis. This is exactly what this part will try to explore.

Complex Analysis and Convolution Equations

Linear differential equations of infinite order and theta functions

Leon Ehrenpreis: A Note on His Mathematical Work Leon Ehrenpreis passed away on August 16, 2010, at the age of eighty, after a life enriched by his passions formathematics,music, running, and the scriptures. I first met Leon in 1978, when I was a doctoral student under Carlos Berenstein (himself a student of Leon’s), at a time when Carlos had invited him to lecture at the complex analysis seminar at the University of Maryland. I had just begun to attempt my first (of many) readings of his work on the fundamental principle [11] and I was in awe of his presence. And yet, I discovered a gentle human being, with a passion for mathematics and a genuine desire to help the newcomer (me, in that case) to understand its mysteries. What follows in this section is a quick overview of Ehrenpreis’s work, mostly biased by my own interest and work on the fundamental principle. It is of course impossible to pay full tribute to the complexity of Ehrenpreis’s work in such a short note. The subsequent sections are short remembrances, both mathematical and personal, of Leon’s life, from some of his friends and collaborators. Leon Ehrenpreis’s mathematical career started with his 1953 dissertation [4],which hewrotewhile doing his doctoral work at Columbia University under the guidance of Chevalley. The dissertation, whose main results later appeared in [8], already shows some of Ehrenpreis’s most characteristic

/pdf/remembering-leon-ehrenpreis-1930-2010-2rrgda4o8l.pdf

Remembering Leon Ehrenpreis: 1930-2010

Publisher Summary
Hamburger's theorem, which characterizes the Riemann zeta function by means of its functional equation, has tempted a number of researchers to find an alternative proof and/or a generalization. A characterization of hyper-functions has been proved on Rn whose supports, as well as the supports of their Fourier transforms, are contained in Zn. Claiming the existence of functional equations of a certain family of L-functions with Grosencharacters, a formula is derieved similar to the Poisson summation formula. The chapter explains the method of Ehrenpreis, Kawai, and Yoshimoto to prove the Hamburger theorem for the Epstein zeta function, a typical example of zeta functions associated with prehomogeneous vector spaces.

The Hamburger Theorem for the Epstein Zeta Functions

The paper examines the problem of determining the distribution function for random summing of independent, identically distributed (i.i.d.) ran- dom variables. The aim of this paper is to develop a suitable approach to deter- mine the total claim cost distribution. There are few applications in actuarial field when the insurance companies look for a given period total damage possi- ble structure, according to each claim size and policies frequency function. We try various analytical methods and introduce two numerical approaches which can be used whenever the solution cannot be derived analytically. Examples are given to demonstrate that the numerical Laplace anti-transform procedure is reliable and provides precise approximations of results. Mathematics Subject Classifications (2000). 91B46

Total claim cost distribution: analytical and numerical methods

Poisson's summation formula tells us that the Fourier transform of the sum of Dirac's (J-functions supported by the integral points is again the sum of Dirac's (^-functions supported by the integral points. In this paper we first consider the converse problem, that is, we characterize a distribution which is a sum of distributions supported by the integral points and whose Fourier transform is again of the same form. Using this result we give another proof of the classical theorem of Hamburger on the characterization of the zata function of Riemann. We also show a generalization of the result to the zeta function associated to the imaginary quadratic field Q(^ — 1) -. In what follows, we use the notation 2 (f) or f to denote the Fourier transform of f normalized in the form I f (x) exp(27Tv — 1 ) dx. We denote by Z>£ (resp., xa} (a= (al9 •--, am) eZJ) the differential operator d™/da%* — da%? (resp., the monomial x"l~>x°tf), where Z+ m denotes the set of ra-tuple of non-negative integers and \a\ = XI #/• j=i We denote Dirac's d-f unction supported at x — n by 8(x — 7i) and its derivative Dax8(x-n) by 8(a}(x-n). Now the first result is stated as follows:

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=18&iss=2&rank=12

Poisson's Summation Formula and Hamburger's Theorem

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Leon Ehrenpreis

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Poisson's Summation Formula and Hamburger's Theorem