ACTA issues

A new proof of the Rademacher--Menshov Theorem

Adam Paszkiewicz

Acta Sci. Math. (Szeged) 71:3-4(2005), 631-642

Abstract. We present a new proof of the following theorem. There exists an orthonormal system $(\Phi_n)_{n\geq1}$ in ${\msbm L}_2(0,1)$, such that corresponding to each sequence $(w_n)_{n\geq1}$ of positive numbers, $w_n =o(\log_2^2n)$ as $n\to\infty $, there is a series $$\sum_{n\geq1} a_n\Phi_n(x)$$ that diverges a.e. and whose coefficients satisfy $$\sum_{n\geq1} a_n^2 w_n < \infty.$$ Our proof {\it does not} depend on the properties of the Hilbert matrix $({1\over i-j})_{i,j\geq1, i\not=j}.$ Possible simplifications of the proof of Tandori's theorem are also discussed. More precisely, we give a new proof of the famous lemma of Menshov being a starting point of a theory of {\it divergence} of orthogonal series.

AMS Subject Classification (1991): 42C15

Keyword(s): orthogonal series, Rademacher-Menshov theorem, Menshov lemma

Received January 19, 2004, and in revised form August 2, 2005. (Registered under 5891/2009.)