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When is the Fourier transform of an elementary function elementary?
- Publication Year :
- 2000
- Publisher :
- arXiv, 2000.
-
Abstract
- Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the form f(x)/t, where f is a cubic form in many variables (the simplest case after quadratic forms). It turns out that this classification is closely related to Zak's classification of Severi varieties. The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method.<br />Comment: 31 pages, latex; the proof of Theorem 3.10 given in the previous version was found to be incomplete. We have added a remark about this and a reference to a paper by Chaput and Sabatino that gives a complete proof
- Subjects :
- Discrete mathematics
Prehomogeneous vector space
Distribution (number theory)
General Mathematics
010102 general mathematics
Multiplicative function
General Physics and Astronomy
Rational function
Quadratic form (statistics)
01 natural sciences
Combinatorics
symbols.namesake
Mathematics - Algebraic Geometry
Fourier transform
0103 physical sciences
symbols
FOS: Mathematics
Classification theorem
010307 mathematical physics
0101 mathematics
Gamma function
Algebraic Geometry (math.AG)
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....231a7c269b051687e10aca1b0d45517f
- Full Text :
- https://doi.org/10.48550/arxiv.math/0003009