Algebraic Numbers and Fourier Analysis by Raphael Salem

By Raphael Salem

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Hence, for all x, PROOF. an cos nx + bn sin nx 2:1 - - - - - -n:2- - - - - - = 00 I 2 - aoX - A 4 x- B and the periodicity of the series implies ao = A = 0; next, the series being uniformly convergent, B = 0 and an = b n = 0 for all n. We shall now prove the theorem on page 42: THEOREM. Every set ofpositive measure is a set M. Let EC (0, 27r) and I El> O. It will be enough to prove that there exists a trigonometric series (not vanishing identically) and converging to zero in the complementary set of E, that is, CE.

For this purpose, one uses the well-known fact [9J that the integers of imaginary quadratic fields share with the rational integers the property of not having zero as a limit point. Theorem A" shows, in particular, that if where the an are rational integers, is regular in the neighborhood of z = 0, has only a finite number of poles in I z I < 1, and is uniformly bounded in the neighborhood of the circumference I z I = 1, then fez) is a rational function. This result suggests the following extension.

28 Applications to Power Series,. Another Class of Algebraic Integers The determinant 1 u Ll = • •• U k- 1 k-l 1 PI . PI 1 ••• Pk-l Pho-l k-l being not zero, we can, by Minkowski's theorem (as given at the beginning of Chapter VI and Appendix, 9), find rational integers AI, ... , Ak, such that the number o= A1uk - 1 + ... ••, (3k-1 all less than 1 in absolute value. In other words, o is a number of the class S belonging to the field of u. Its conjugates are all real. Take now J,L = 02h and l'j = (3lh, h being a positive integer such that 1'1 + + ...

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