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By Raphael Salem

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Extra resources for Algebraic numbers and Fourier analysis (Heath mathematical monographs)

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This proves that Co = O. r , we find its constant term to be Ck. r is identically 0, and that E is a set of the type U. L First application: Sets of the type H. A linear set E C (0, 27r) is said to be "of the type H" if there exists an interval (a, (3) contained in (0, 27r) and an infinite sequence of integers {nd 7 such that, for whatever x E E none of the points of abscissa nkX (reduced modulo 27r) belongs to (a, (3). For example, the points of Cantor's ternary set constructed on (0, 27r): X El = 27r [ -+ -E2 + ...

If, at a given point x, F(x) has a second generalized derivative equal to s, we shall say that the series (1) is summableRiemann (or summable-R) and that its sum is s. THEOREM H. If the series (1) (an, bn ~ 0) converges to s at the point x, it is also summable-R to s at this point. THEOREM HA. If the series (1) with coefficients tending to zero is summable-R to zero for all the points of an interval, it converges to zero in this interval (consequence of the principle of "localization"). THEOREM HI.

Since l/w is actually a pole for 21 Therefore w is a number of the class S, · Ps 1lm Qs' because Jl -:;t. O. ) EXERCISE Let a be a natural positive integer > 2. Then a is a limit point for the numbers of the class S. (Considering the equation zn(z - a) - 1 = 0, the result for a > 2 is a straightforward application of Rouche's theorem. ) Chapter III APPLICATIONS TO THE THEORY OF POWER SERIES; ANOTHER CLASS OF ALGEBRAIC INTEGERS 1. A generalization of the preceding results Theorem A' of Chapter 11 can be extended, and thus restated in the following way.