Abstract
The question on affine Riesz basis of Walsh affine systems is considered. An affine Riesz basis is constructed, generated by a continuous periodic function that belongs to the space on the real line, which has a derivative almost everywhere; in connection with the construction of this example, we note that the functions of the classical Walsh system suffer a discontinuity and their derivatives almost vanish everywhere. A method of regularization (improvement of differential properties) of the generating function of Walsh affine system is proposed, and a criterion for an affine Riesz basis for a regularized generating function that can be represented as a sum of a series in the Rademacher system is obtained.
Highlights
Let {0,1}k k 0be the set of all finite multi-indices ( 0,( 0, l 1), denote by: Iraqi Journal of Science, 2021, Vol 62, No 12, pp: 4875-4884( 0, k 1, 0, l 1) W W 0 W k 1( 0, k 1), The first is the operator W k 1 and the last is W 0
Rademacher system was first introduced by the German mathematician Hans Rademacher in 1922
Sarsenbi, and Terekhin [12] studied an affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space
Summary
Rademacher system was first introduced by the German mathematician Hans Rademacher in 1922. Rademacher system is an incomplete orthonormal system in L2[0,1]. Definition (1.2)[2]: Let r be the function defined on [0,1) by : 1, 0 t 1/ 2
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