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Summary: Mean - square and almost sure reconstruction is considered
for spectrally band - limited and non - band - limited stochastic signals
(weakly stationary stochastic processes, homogeneous random fields, random
distributions). The reconstruction error upper bounds are derived in both
of, above mentioned, approaches for stochastic signals. Necessary and
sufficient conditions are derived for the existence of the Fourier -
integral form spectral representation of the sampling cardinal series
expansion of non -band - limited stochastic signals. In the case of
multidimensional signals with correlated coordinates, the mean square (and
the almost sure as well) sampling approximation error upper bounds are
generalized to the so - called truncation cross - error upper bounds for
the sampling cardinal series.
Keywords: Sampling theorems, Band - limited signals, Non - band - limited signals, Weakly stationary stochastic processes, Homogeneous random fields, Random distributions, Sampling cardinal series expansion, Truncation error, Mean square convergence, Almost sure convergence, Fourier - integral representation, Spectral representation, Multidimensional stochastic signals, Correlated signals, Chebyshev inequality, Borel - Cantelli lemma.
Research goals: In the whole investigations just non - band -
limited stochastic signals will be considered, but the principal results
involve many times certain before derived results for band - limited
signals. MAIN GOALS: 1. Mean - square and almost sure sampling
reconstruction of: 1.1. weakly stationary stochastic processes, 1.2.
homogeneous random fields, 1.3. random distributions. 1.4. Approximation
error upper bound evaluations for 1.1.-1.3.. 2. Fourier - integral form
represenation for the sampling cardinal series expansion of non - band -
limited stochastic signals: 2.1.scalar and vectorial processes, 2.2.
homogeneous random fields, 2.3. random distributions. 3. Multidimensional
analogons truncation error upper bound evaluations if the initial signal
possesses correlated coordinates. EXPECTED RESULTS: The realization of
above exposed research program contains these expaected results. Namely,
some recent best results due to the researcher (see e.g. 1.3. and the
pharagraph 2.). Certain improvements of these evaluations and obtaining
the exact connections between the existence of the integral
representations 2. and the continuity of the spectral distribution
function of the considered signals leads us to expected results. Other information about the project.