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Orlicz spaces, spline systems, and Brownian motion. (English) Zbl 0814.46022
Summary: There are three results proved in this paper. The first one characterizes the Hölder classes in Orlicz spaces by the coefficients of the orthogonal spline expansions of the Franklin type. The second one gives a sharp estimate for the correlation of two random variables obtained as a composition of two Borel functions with the components of a given two- dimensional Gaussian vector. The third one is obtained with the help of the first two and it states that the Wiener measure is concentrated on the Banach space of Hölder functions with exponent \({1\over 2}\) but in the norm of the Orlicz space \(L^*_ M\) with \(M(t)= \exp(t^ 2)- 1\).

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60J65 Brownian motion
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
60E15 Inequalities; stochastic orderings
41A15 Spline approximation
62J10 Analysis of variance and covariance (ANOVA)
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