Representation of Functions in the Weighted Space L^1_μ by Trigonometric and Walsh Series

Authors: Grigorian M.G.¹; Episkoposian S.A.²

Source: Analysis Mathematica, Volume 27, Number 4, 2001 , pp. 261-277(17)

Publisher: Springer

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Abstract:

We prove that there exists a series of the form (*) $$\sum\limits_{k = 1}^\infty {c_{n_k } \varphi _{n_k } \left( x \right)}$$, where { phiv _n(x)} is a subsystem of either the trigonometric or the Walsh system such that $$|\sum\limits_{k = 1}^\infty {c_{n_k } \varphi _{n_k } \left( x \right)} |\mathop < \limits_ - \lambda _{m,} m = 1,2,...;x \in \left[ {0,1} \right]$$; where lambda _n uarr infin as n rarr infin and for each epsiv > 0 a weight function (x) can be constructed such that 0 < (x) 1, $$|\left\{ {x \in \left[ {0,1} \right]:\mu \left( x \right) \ne 1} \right\}| < \varepsilon$$, and series (*) is universal in the space L^1_μ with respect simultaneously to rearrangements as well as to subseries.