Probability measure representation of norms associated with the notion of entropy

Abstract:

One of the applications of Banach spaces introduced by B. Korenblum [1,2] is a new convergence test [2] for Fourier series including both Dirichlet-Jordan and the Dini-Lipschitz tests [3], The norms of the spaces are given in terms of $\kappa$-entropy where $\kappa \left ( s \right ) \geqslant 0$, $0{\text { < }}s \leqslant 1$, is a nondecreasing concave function such that $\kappa \left ( 1 \right ) = 1$. The $\kappa$-norms fill the gap between the uniform and the variation norms. The original proof of the general properties of $\kappa$-norms uses both combinatorial and approximation arguments which are rather complicated. We give a simple proof introducing a probabilistic representation of the norms so that the $\kappa$-norm of a real function $f$ on $T = R / 2\pi Z$ is the expectation of the mean oscillation of $f$ on a subinterval of $T$, chosen in a suitable random process.