A Closed Equation in Time Domain for Band-Limited Extensions of One-Sided Sequences

Abstract

The paper suggests a method of optimal extension of one-sided semi-infinite sequences of a general type by traces of band-limited sequences in deterministic setting, i.e. without probabilistic assumptions. The method requires to solve a closed linear equation in the time domain connecting the past observations of the underlying process with the future values of the band-limited process. Robustness of the solution with respect to the input errors and data truncation is established in the framework of Tikhonov regularization.

Optimal data recovery and forecasting with dummy long-horizon forecasts

“…The setting of this paper targets situations where this condition is satisfied for classes Y of processes with some spectrum degeneracy, as a generalization of the setting from [10,11], where the case of n = 1, V = R, X = ℓ 2 was consider, with Y being a set of band-limited processes, in the notations of the present paper. In [10], finite sets M were considered. In [10], the set D = {t ∈ Z : t ≤ 0} was considered.…”

“…The paper considers data recovery problems for sequences in pathwise setting, i.e. without probabilistic assumptions, using the approach suggested in [10,11] in an extended setting, allowing more general type of approximating processes, more general domains for missing data, and more general sequences with a multidimensional index.…”

“…An estimate of the missing value has to be done based on the intrinsic properties of this sole sequence and the observed values. The paper suggests a method of optimal recovering missing values of sequences (discrete time processes) based on extension of the approach from [10,11]. The optimality criterion is pathwise; it does not involve an expectation on a probability space.…”

“…The optimality criterion is pathwise; it does not involve an expectation on a probability space. In [10] band-limited extensions of onesided sequences with one-dimensional index were considered. In [11], data recovery based on band-limited approximations was considered for sequences with one-dimensional index with finite number of missing values at consequent points.…”

“…We considered approximations by band-limited processes (Section V), in the setting that is close to the setting from [10,11] but with more general type of the domain for the missing data. In this case, the recovery equation ( 5) is in the time domain.…”