Convergence problems of orthogonal series by György Alexits

By György Alexits

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Z71 lC ι ^ = ΛΓ m V i 2 2 L cfc. Since here m is arbitrarily, this implies that 00 Σ ct^3M2. k=it l + Thus we have established that Σά converges and so our statement is proved. 1, as well as its corollary originate also from him. 4 was proved by KHINTCHINE and KOLMOGOROFF [1]. We shall see in the sequel that these theorems are valid even in a wider range. 7). 4 that the Rademacher series Zcnrn(x) can be the expansion in the functions of the completed system of Rademacher functions of an L-integrable function if and only if f(x) is not merely L-, but also L2-integrable.

CkPk(x)· k=n Now let M denote the maximum of σ(χ); then it follows from Schwarz's inequality (6) that 1 1 lç(x)o%x)ql(x)dxijç{x)pl(x)dx^ 4ü -1 1 -I ^ M \ç(x)o(x)ql{x)dx = M -1 and so a(xo)qn(xo) ^Z\ck\ k+n \Mxo)\ = (m+i)YMO(l) = 0(\). Thus we have established our intermediary theorem. Thus all the cases where a, ß are non-integer numbers are settled. In order to consider also integral values ay ß we use the fact (already mentioned) that for the Legendre polynomials (a = 0,ß = 0) the estimate p f °>(x) = 0(1) holds in (—1, 1).

2 , (proved in the following chapter), we may infer that almost everywhere 1 ^ lim -γ- ]£rk(x) = 0. Cantelli's theorem corresponds to the special case λη = η. The relation (22) is also equivalent to a theorem of BOREL, stating that the ratio of the number of the digits 1 to the number of the digits 0 among the first n digits in the dyadic representation of a number x between 0 and 1 tends to 1 for almost every x, when n —> o°. (*)] =0χ{λ) almost everywhere. This means that the arithmetic means of the sequence obtained from the sequence of digits by replacing the digits 0 by —1 tend to zero almost everywhere.

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