Abstract
We prove that : if f (x) is a measurable function, finite almost everywhere on [0,1] then there is a Schauder series, which converges unconditionally to f almost everywhere on [0,1] and for every e > 0 there is a measurable set E contained in [0,1], with | E |> 1 −e, such that for each f ∈ C ( E ) there is a Schauder series, which unconditionally converges to f in the uniform norm.
Highlights
We recall the definition of the Faber-Schauder system:
We present some results which have an immediate bearing on our investigations
Note the following question, which arises when we investigate the convergence of the greedy algorithm for the new, corrected function e f x and is of independent interest
Summary
Fourier series f f x ¦ An f Mn x n0 with respect to the Faber-Schauder system, which converges to f uniformly on [0,1] . From this follows: there exists a function f0 x C >0,1@ , whose greedy algorithm with respect to the Faber-Shauder system diverges in measure. W. Korner answering a question raised by Carleson and Coifman constructed a continuous function, whose greedy algorithm with respect to the trigonometric systems diverges almost everywhere.
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