By Raphael Salem

ISBN-10: 053498049X

ISBN-13: 9780534980498

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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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