Multiple Rademacher series in symmetric spaces
Let \(r_1,r_2,\dots\) denote the Rademacher system on the interval \([0,1]\). The author announces several results concerning representation of functions as series of the form \[ f(x)= \sum_{i\neq j} a_{i,j} r_k(x) r_j(x). \] It seems that in rearrangement invariant spaces \(X\) (e.g., \(L^p\) spaces, Orlicz spaces and Marcinkiewicz spaces), the norm of such an \(f\) is equivalent to the \(\ell^2\) norm of the coefficients \(\{a_{i,j}\}\) if and only if \(X\) contains the closure of \(L^\infty\) in the Orlicz space \(L_M\), where \(M(x)= e^x- 1\). On the other hand, when \(1\leq p<2\), there are no rearrangement invariant spaces in which the norm of \(f\) is equivalent to the \(\ell^p\) norm of the coefficients \(\{a_{i,j}\}\).
scientific article; zbMATH DE number 1407629
scientific article