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

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Theorem IV remains valid if one supposes that the generalized second derivative exists and is zero except at the points of a denumerable set E, provided that at these points G is smooth. THEOREM V. , has a derived set of finite or transfinite order which is empty. It was extended much later by Young to the general case where E is supposed only to be denumerable. 44 The Uniqueness of the Expansion in Trigonometric Series From Theorem V we deduce finally: VI. If the series (1) converges to 0 at all points of (0, 27r) except perhaps when x belongs to a denumerable set E, the series vanishes identically.

The same argument leads then to Cn (k) ~ 0 for k = 00, n ¥= O. This concludes the proof that all conditions of the general theorem are satisfied and hence that the set E is a set of uniqueness. " Chapter VI SYMMETRICAL PERFECT SETS WITH CONSTANT RATIO OF DISSECTION; THEIR CLASSIFICATION INTO M-SETS AND U-SETS In this chapter and in the following one we shall make use of the fundamental theorem of Minkowski on linear forms. For the proof we refer the reader to the classical literature. ) MINKOWSKI'S THEOREM.

The problem of the behavior at infinity It is well known in the elementary theory of trigonometric series that if f is absolutely continuous, the Fourier-Stieltjes transform "f(u) = (27r)-1 f,21r euix df tends to zero as I u I ~ 00, because in this case "f(u) is nothing but the ordinary Fourier transform of a function of the class L. The situation is quite different iff is continuous, but singular. In this case "f(u) need not tend to zero, although there do exist singular functions for which "f(u) ~ 0 ([17J, and other examples in this chapter).

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Algebraic numbers and Fourier analysis (Heath mathematical monographs) by Raphael Salem

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