Musielak-Orlicz spaces and prediction problems

Volume 64 / 2004

K. Urbanik Banach Center Publications 64 (2004), 207-219 MSC: 60G25, 60G57, 46E30. DOI: 10.4064/bc64-0-16


By a harmonizable sequence of random variables we mean the sequence of Fourier coefficients of a random measure $M$: $$ X_n(M)=\int_0^1e^{2\pi nis}M(ds)\quad (n=0,\pm 1,\ldots) $$ The paper deals with prediction problems for sequences $\{X_n(M)\}$ for isotropic and atomless random measures $M$. The crucial result asserts that the space of all complex-valued $M$-integrable functions on the unit interval is a Musielak-Orlicz space. Hence it follows that the problem for $\{X_n(M)\}$ $(n=0,\pm 1,\ldots)$ to be deterministic is in fact an extremal problem of Szegö's type for Musielak-Orlicz spaces in question. This leads to a characterization of deterministic sequences $\{X_n(M)\} \ (n=0, \pm 1,\ldots)$ in terms of random measures $M$.


  • K. UrbanikInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8, P.O. Box 21
    00-956 Warszawa 10, Poland

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