Absolutely convergence factors of Fourier series

Author: Giorgi Tutberidze
Keywords: Fourier series, Absolutely convergece factors, Fourier coefficient
Annotation:

In 1962, A. Olevsky (see [1]) proved that if f(x)∈L_2 (0,1) and (a_n)∈l_2, then there exist orthonormal systems 〖(φ〗_n (x)), such that c_n (f)=c∙a_n ,n=1,2,..., and c is on absulute constant which does not depend on n. Let f_0 (x)=1, x∈[0,1] and a_n= {█( 0, if n≠2^k@1/(√k log⁡(k+1)),if n=2^k,n=1,2,…)┤ Then Then there exists an orthonormal system 〖(φ〗_n (x)), such that a_n=c∙c_n (f_0 ). It follows that for 1<p<2. So, there exists an orthonormal system 〖(φ〗_n (x)), for which , when p∈(1,2). Our main aim is to get such systems from generalized orthonormal systems set for which the series compiled by furrier coefficients are absolutely convergent in power of p>1 for a proper factor.

Lecture files:

Absolutely convergence factors of Fourier series [en]
აბსოლუტურად კრებადობის მამრავლები ფურიეს მწკრივებისათვის [ka]