Algebraic Numbers and Fourier Analysis by Raphael Salem

Posted by

By Raphael Salem

Show description

Read Online or Download Algebraic Numbers and Fourier Analysis PDF

Similar mathematical analysis books

Understanding the fast Fourier transform: applications

This can be a educational at the FFT set of rules (fast Fourier remodel) together with an creation to the DFT (discrete Fourier transform). it's written for the non-specialist during this box. It concentrates at the genuine software program (programs written in simple) in order that readers may be in a position to use this know-how once they have comprehensive.

Acta Numerica 1995: Volume 4 (v. 4)

Acta Numerica has proven itself because the best discussion board for the presentation of definitive studies of numerical research themes. Highlights of this year's factor contain articles on sequential quadratic programming, mesh adaption, unfastened boundary difficulties, and particle tools in continuum computations.

Extra info for Algebraic Numbers and Fourier Analysis

Sample text

Then the elements of R are rational integers or integers of an imaginary quadratic field (see [9]). 5. There exist only a finite number of algebraic integers of given degree n, which lie with all their conjugates in a bounded domain of the complex plane (= ~91). the integral being taken in the p-dimensional unit torus P, H. Weyl's criterion becomes lim elri(HVI) + . + e2ri(HV,) n - 0 where (HV,) is the scalar product 6. Let P(x) be a polynomial in a field k. Let K be an extension of k such that, in K, P(x) can be factored into linear factors.

Every set E of the type H(") is a set of uniqueness. PROOF. We can again suppose that E is closed, and we shall take n = 2, the two-dimensional case being typical. Suppose that the family of vectors + and = X(n& = m y,em*&. The sequence of functions (Xk(x)) satisfy the conditions (I), (2), (3), (4) of Theorem I. @) = 0 if nk,/'n, we see that the conditions (3) are satisfied, with is normal. We can assume that A consists of the points (xl, x2)such that the intervals (al, 8,) and (a9, &) being contained in (0,2 ~ ) .

We proceed now to prove that the preceding conditions are sufficient in order that E be a U-set. The points of this set E' are given by where the ej are either 0 or 1. We can, as in the case of symmetrical perfact sets, d&ne a measure carried by this set and prove that its Fourier-Stieltjes transform is fi cos 3. The homogeneous set E whose points are given by (1). where I / f = 8 is an algebraic integer of the class S and the numbers 'll, .... v d are algebraic belonging to the jeld of 8, is a set of the type H'") (n being the degree of 8), and thus a set of uniqueness.

Download PDF sample

Rated 4.51 of 5 – based on 11 votes