Banach Gelfand Triples for Applications in Physics and Engineering

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The principle of extension is widespread within mathematics. Starting from simple objects one constructs more sophisticated ones, with a kind of natural embedding from the set of old objects to the new, enlarged set. Usually a set of operations on the old set can still be carried out, but maybe also some new ones. Done properly one obtains more completed objects of a similar kind, with additional useful properties. Let us give a simple example: While multiplication and addition can be done exactly and perfectly in the setting of , the rational numbers, the field of real numbers has the advantage of being complete (Cauchy sequences have a limit…) and hence allowing for numbers like or . Finally the even “more complicated” field of complex numbers allows to find solutions to equations like z² = −1. The chain of inclusions of fields, is a good motivating example in the domain of “numbers.”The main subject of the present survey-type article is a new theory of Banach Gelfand triples (BGTs), providing a similar setting in the context of (generalized) functions. Test functions are the simple objects, elements of the Hilbert space L²(^d) are well suited in order to describe concepts of orthogonality, and they can be approximated to any given precision (in the ||·||₂-norm) by test functions. Finally one needs an even larger (Banach) space of generalized functions resp. distributions, containing among others pure frequencies and Dirac measures in order to describe various mappings between such Banach Gelfand triples in terms of the most important “elementary building blocks,” in a clear analogy to the finite/discrete setting (where Dirac measures correspond to unit vectors).Our concrete Banach Gelfand triple is based on the Segal algebra S₀(^d), which coincides with the modulation space M¹(^d) = M(^d), and plays a very important and natural role for time-frequency analysis. We will point out that it provides the appropriate setting for a description of many problems in engineering or physics, including the classical Fourier transform or the Kohn-Nirenberg or Weyl calculus for pseudo-differential operators. Particular emphasis will be given to the concept of w*-convergence and w*-continuity of operators which allows to prove conceptual uniqueness results, and to give a correct interpretation to certain formal expressions coming up in various versions of the Dirac formalism. ©2009 American Institute of Physics