Algebraic Numbers and Fourier Analysis - download pdf or read online

By Raphael Salem

ISBN-10: 053498049X

ISBN-13: 9780534980498

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Extra resources for Algebraic Numbers and Fourier Analysis

Sample text

We had occasion to cite Minkowski's theorem on linear forms in Chapters I, 111, and V1. We restate it here as follows. Ler . be n linearforms of the n variables X I , . , x, where the cwficients are real and the determinant D of the forms is not zero. There exists a point x with integral coordinates not all zero, x,, . , x, such that provided that 61 - . 6, 2 1 D 1. The result holds if the coefficients aqp arc complex, provided that complex forms figure in conjugate pairs, and that the two 6,'s corresponding to a conjugate pair are equal.

Let A be an interval contiguous to E. The series converges to a linear function in A. Hence, the Fourier series where the star means that there is no constant term, represents in A a function -c,,x a, the constant a = a(A) depending on A. nieniz and gives The integral is equal to since X and A' are zero for x E E. ) Integrating by parts, The Uniqueness of the Expansion in Trigonometric Series 49 Remark. The hypothesis that X'(x) is bounded could be relaxed (which would lead also to a relaxation of the hypothesis (4) of the theorem), but this is of no interest for our applications.

Let R be a ring of real or complex numbers such that 0 is not a limit point of numbers of R. ) 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.

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Algebraic Numbers and Fourier Analysis by Raphael Salem


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