Uniform convergence of Fourier series with respect to
orthogonal polynomials
Ryszard Szwarc
Institute of Mathematics, Wroclaw University
Abstract
Let sn (f ) denote the nth partial sum of the classical Fourier series of a continuous
2π periodic function f (θ). We know that the quantities
ksn (f )k∞ need not to be
R 2π
uniformly bounded since the Lebesgue numbers 0 |Dn (θ)|dθ behave like constant
multiple of log n. Therefore sn (f ) 6⇒ f for some f ∈ Cper (R). The question arises:
Do there exist a measure space and an orthogonal system such that the partial sums
are uniformly bounded in k · k∞ norm ? We answer this question in the positive.
This is a joint work with Josef Obermaier from Munich.
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