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Example text

This proves that condition (2) is also satisfied. Condition (4) is satisfied if we have chosen X(x) and y(x) possessing bounded derivatives. Finally, for condition (3) we note that (4) cn(k)=. C ,I =mpr ~~6,'. +m'qh Suppose first n = 0. ,P ~ ( ~ ' X , . r, never belongs to A. t If E is of the type H,so is its closure, and a subset of a U-set is also a U-set. the star meaning that ( m I for k -, m . Write T = TI + / m' I # 0. We shall prove that T tends to zero + T2, where TI is extended to the indices I m I 5 N, The Uniqueness of the Expansion in Trigonometric Series 52 + I m' 1 5 N .

This is because we only know that if t-' E S, the Fourier-Stieltjes coefficients of the Lebesgue measure constructed on the set do not tend to zero. , the derived series ynenil is not a Fourier-Stieltjes series). A negative proof of this kind would be rather difficult to establish. In general, to prove that a set E is a set of the type U,one tries to prove that it belongs to a family of sets of which one knows, by certain properties of theirs, that they are U sets. In this connection, we shall make use of the following theorem.

Do there exist other limit points of the numbers 7, and, if so, which ones? 3. It has been shown in Chapter IV that the infinite product is, for 0 < f < 3, the Fourier-Stieltjes transform of a positive measure whose support is a set E ( 0 of the Cantor type and of constant rate of dissection f . t We know that r(u) o(1) for u --r 0 0 , if and only if P1does not belong to the class S. Let - where tl-l and ft-I both belong to the class S, so that neither rl(u) nor rs(u) tends to zero for u = oo. What is the behavior of the product as u -,oo?

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