A precise upper bound for the error of interpolation of stochastic processes.(Ukrainian, English)Zbl 1097.60023

Teor. Jmovirn. Mat. Stat. 71, 133-144 (2004); translation in Theory Probab. Math. Stat. 71, 151-163 (2005).
Let us denote by $$PW_{\pi}^2$$ the class of Paley-Wiener functions. For all $$f\in PW_{\pi}^2$$ we have $$f(x)=\sum_{n=-\infty}^{\infty} {\text{sinc}}\,(x-n)f(n)$$, where ${\text{sinc}}\,(t)=\begin{cases} {\sin(\pi t)\over \pi t}, & t\not=0,\\ 1, & t=0.\end{cases}$ Let $Y_{N}(f;x)=\sum_{| x-n|\leq N}\text{sinc}\,(x-n)f(n), \quad T_{N}(f,x)=f(x)-Y_{N}(f;x),$
$\| f\|_{\infty}= \inf\{a>0:\;\forall x\in R,\;| f(x)|\leq a\}.$ The main result for the deterministic case is the following. Let $$f\in PW_{\pi}^2$$, then $\| T_{N}(f,\cdot)\|_{\infty}\leq \left(1-{8\over \pi^2}\sum_{n=1}^{N}{1\over (2n-1)^2}\right)^{1/2}\| f\|_2.$ Equality in this relation gives the extremal function $f_{N}^{*}(x)=\sum_{| n-1/2|>N}{\sin(\pi x)\over \pi^{2}(n-1/2)(n-x)}.$ For the weak Cramér stochastic processes the analysis of interpolation error is performed. The rate of the mean square convergence of the interpolation errors is investigated.

MSC:

 60G12 General second-order stochastic processes 94A20 Sampling theory in information and communication theory 26D15 Inequalities for sums, series and integrals 30D15 Special classes of entire functions of one complex variable and growth estimates
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