Direct Construction of Superoscillations

IEEE Trans. Signal Process.

2018

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.

Section: Introductionmentioning

confidence: 97%

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

IEEE Trans. Signal Process.

2018

“…For ω = ω 0 , it gives Let us show that the second inequality in (13) holds. Suppose that we use another estimator x(s) = F x| Z\Ms , where F : ℓ 2 (Z \ M s ) → C is some mapping.…”

mentioning

confidence: 99%

On exact and optimal recovering of missing values for sequences

2017

Signal Processing

The paper studies recoverability of missing values for sequences in a pathwise setting without probabilistic assumptions. This setting is oriented on a situation where the underlying sequence is considered as a sole sequence rather than a member of an ensemble with known statistical properties. Sufficient conditions of recoverability are obtained; it is shown that sequences are recoverable if there is a certain degree of degeneracy of the Z-transforms. We found that, in some cases, this degree can be measured as the number of the derivatives of Z-transform vanishing at a point. For processes with non-degenerate Ztransform, an optimal recovering based on the projection on a set of recoverable sequences is suggested. Some robustness of the solution with respect to noise contamination and truncation is established.

“…It can be noted that the setting in [18,21] considers exact matching of the band-limited process and the underlying process in certain points, which is different from our setting.…”

mentioning

confidence: 92%

“…For the solution, we use non-singularity of special sinc matrices obtained in [21] for the solution of the so-called superoscillations problem for continuous time processes; see the references in [18,21].…”

mentioning

confidence: 99%

On causal extrapolation of sequences with applications to forecasting

Applied Mathematics and Computation

2018

The paper suggests a method of extrapolation of notion of one-sided semi-infinite sequences representing traces of two-sided band-limited sequences; this features ensure uniqueness of this extrapolation and possibility to use this for forecasting. This lead to a forecasting method for more general sequences without this feature based on minimization of the mean square error between the observed path and a predicable sequence. These procedure involves calculation of this predictable path; the procedure can be interpreted as causal smoothing. The corresponding smoothed sequences allow unique extrapolations to future times that can be interpreted as optimal forecasts.