On the ``prediction'' problem
By weight the author means any measurable function \(w\) such that \(0\leq w(t)\leq 1\) for \(t\in\mathbb{T}\). The set of all such functions endowed with an \(L^1\) distance forms a complete metric space \(W\). The following theorem is proved. (i) There exists a weight \(w>0\) a.e. such that any function \(f\in L^2(w,\mathbb{T})\) can be represented as a series of the form \(f= \sum_{n>0} c(n) e^{int}\), which converges in the norm. (ii) For any \(p>2\), the coefficients \(c(n)\) can be chosen in \(\ell_p\) with an arbitrarily small norm. (iii) The set of such weights is residual in the space \(W\). This result is discussed in the context of the classical Szegő-Kolmogorov ``prediction'' theorem.
scientific article; zbMATH DE number 1744345