Discrete-Time Multi-Scale Systems

Abstract

Abstract. We introduce multi-scale filtering by the way of certain double convolution systems. We prove stability theorems for these systems and make connections with function theory in the poly-disc. Finally, we compare the framework developed here with the white noise space framework, within which a similar class of double convolution systems has been defined earlier.

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

“…Other settings include the non-commutative case, [9,23], Riemann compact surfaces [17] functions defined on an homogeneous tree, [18], and others. See for instance [4] for the time-varying case, [10] for the stochastic case and [12] for the case of multi-dimensional systems.…”

Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals

“…Lemma 1 ( [10,17]). Endowed with the composition law •, G θ , forms a group which is a subgroup G. Each γ α ∈ G θ can be written as…”

“…For each n 0, it can be readily seen that Φ (γ) n (z) is the image of z n under T αγ . Being unitary [17], T αγ preserves the scalar product and therefore the family {Φ…”

“…Surprisingly, only the frequency warping effect of the operator has been considered so far while the interpretation of the scale-shift in the time domain did not seem to hold the interest of the authors of [12], [13] (see also [14] and [15]). Now, the group action viewpoint of the scale-shift enlarges the perspective (see [16] and [17]) and leads to an exact and purely harmonic analysis approach of the Mellin transform for discrete-time signals.…”

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis

“…Analogues of Schur functions were introduced by Agler, see [1,2]. Besides the Schur-Agler classes, we mention that counterparts of Schur functions have been studied in the time-varying case, [6], the stochastic case, [7], the Riemann surface case [14], and the multiscale case, [8,9]. A number of other cases also are of importance.…”