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Integrability behaviour of the maximal partial sum of orthogonal series. (English) Zbl 0931.42015

Summary: The Rademacher-Menshov theorem is well known in the theory of orthogonal series. That is, if a sequence \(\{a_k\}\) of real numbers satisfies \(\sum^\infty_{k=1} a^2_k\log^2k<\infty\) then the orthogonal series \(\sum^\infty_{k=1}a_k\phi_k(x)\) converges a.e. Moreover, the maximal partial sum \[ S_*(x):=\sup_{n\geq 1}\left|\sum^n_{k=1}a_k\phi_k(x)\right| \] belongs to \(L^2\). Our main result states that in the case \(\sum^\infty_{k=1}a^2_k\log^2k=\infty\) the maximal sum \(S_*\) does not belong to \(L^2\) in general.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42C15 General harmonic expansions, frames
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