Some problems arising from prediction theory and a theorem of Kolmogorov (Q793284)

Let \(T=\{e^{2\pi ix},\quad 0\leq x<1\}\) be the unit circle provided with normalized Lebesgue measure m. Denote \(P_ 0=\{g\in P| \hat g(0)=0\}\) where P is the vector space of all trigonometric polynomials and ĝ(k) are the Fourier coefficients of g. Let \(d_ p(\mu)=\inf_{g\in P_ 0}\int | 1+g(x)|^ pd\mu(x).\) For \(1<p<\infty\), \(d_ p(\mu)>0\) iff \(w^{-p'/p}\in L^ 1(T),\) and, more precisely \(d_ p(\mu)=\{\int w(x)^{-p'/p}dm(x)\}^{-p'/p}\) where \(d\mu(x)=w(x)dm(x)+d\mu_ s(x)\) is the Lebesgue decomposition of \(\mu\). Some other related problems are considered.