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

Olenko, A. Ya.; Pogány, T. K.

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.

Reviewer: A. D. Borisenko (Kyïv)

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

Keywords:

exact upper estimate; interpolation error; Paley-Wiener functions; weak Cramér stochastic processes; rate of convergence