( 1) f~~~~~~(x) Y. anOn (X), n=0 ' b where an f (x)'n (x) dE (x). While the local convergence properties of

The mean convergence of orthogonal series.

CONTENTS ??1. Introduction ??2. Fourier coefficients ??3. Convergence ??4. Convergence to 0 and to +? ??5. Divergent series ??6. Summability ??7. The problem of the representation of functions ??8. Absolute convergence and summability ??9. Unconditional convergence ??10. Fourier series and integrals ??11. Basic problemsReferences

http://iopscience.iop.org/article/10.1070/RM1964v019n01ABEH004136/pdf

Solved and unsolved problems in the theory of trigonometric and orthogonal series

of every function/(x) in Lp(a, b) converges to/(x) in the pth mean. A classical theorem of Marcel Riesz [l](2) asserts that the functions (2/ir)lli cos nx, « = 0, 1, • • • , form a basis for Lp(0, it) for all p>\\. An examination of the literature discloses only three further sets for which the property has been studied (3): the Haar functions [2], the Fourier transforms of the Laguerre polynomials [3, 4], and the Walsh functions [14]. These sets are bases for Lp(0, 1), Lp( — oo, oo ), and Lp(0, 1) respectively, for all values of p>l (in the case of the Haar functions for p = l also). It is the object of this study to investigate the extent to which the property of p-mean convergence is shared by other sets of functions. The methods of functional analysis supply useful necessary and sufficient conditions, but there seems to be no general method for determining whether a given set actually fulfills them, and each must be examined on its own merits. It is natural to begin with a set of functions as nearly like {cos rax}as possible. The particular choice made here is the set {cos anx\\ arising from a simple Sturm-Liouville problem

/pdf/the-mean-convergence-of-orthogonal-series-i-4hef58np01.pdf

The mean convergence of orthogonal series. I

Series with respect to the Haar system

The methodology of fractal sets generates new procedures for the analysis of functions whose graphs have a complex geometric structure. In the present paper, a method for the definition of fractal functions is described. The new mappings are perturbed versions of classical bases as Legendre polynomials, etc. The new elements are non-differentiable and may serve as models for pseudo-random behaviour. The proposed fractal functions have good algebraic properties and good approximation properties as well. In the present paper it is proved that they constitute bases for the most important functional spaces as, for instance, the Lebesgue spaces (1 ≤ p < ∞), where I is a compact interval in the reals.

Juljusz Schauder

Papers

Eine Eigenschaft des Haarschen Orthogonalsystems