Frequency domain of weakly translation invariant frame MRAs

Frequency domain of bandlimited frame multiresolution analyses (MRAs) plays a key role when derived framelets are applied into narrow-band signal processing and data analysis. In this study, we give a characterization of frequency domain of weakly translation invariant frame scaling functions [Formula: see text] with frequency domain [Formula: see text]. Based on it, we further study convex and ball-shaped frequency domains. If frequency domain of bandlimited frame scaling function [Formula: see text] is convex and completely symmetric about the origin, then it must be weakly invariant and [Formula: see text]. If [Formula: see text] has a ball-shaped frequency domain, the ball radius must be bounded by [Formula: see text]. These frequency domain characters are owned uniquely by frame scaling functions and not by orthogonal scaling functions.

A note on the applications of one primary function in deep neural networks

By applying fundamental mathematical knowledge, this paper proves that the function [Formula: see text] is an integer no less than [Formula: see text] has the property that the difference between the function value of middle point of arbitrarily two adjacent equidistant distribution nodes on [Formula: see text] and the mean of function values of these two nodes is a constant depending only on the number of nodes if and only if [Formula: see text] By them, we establish an important result about deep neural networks that the function [Formula: see text] can be interpolated by a deep Rectified Linear Unit (ReLU) network with depth [Formula: see text] on the equidistant distribution nodes in interval [Formula: see text] and the error of approximation is [Formula: see text] Then based on the main result that has just been proven and the Chebyshev orthogonal polynomials, we construct a deep network and give the error estimate of approximation to polynomials and continuous functions, respectively. In addition, this paper constructs one deep network with local sparse connections, shared weights and activation function [Formula: see text] and discusses its density and complexity.

Multidimensional periodic discrete wavelets

In this paper, the definition of a periodic discrete multiresolution analysis is provided. The characterization of such systems is obtained in terms of properties of scaling sequences. Wavelet systems generated by such multiresolution analyses are defined and described. Decomposition and reconstruction formulas for the associated discrete wavelet transform are provided.

Uncertainty principles for the windowed offset linear canonical transform

The windowed offset linear canonical transform (WOLCT) can be identified as a generalization of the windowed linear canonical transform (WLCT). In this paper, we generalize several different uncertainty principles for the WOLCT, including Heisenberg uncertainty principle, Hardy’s uncertainty principle, Donoho–Stark’s uncertainty principle and Nazarov’s uncertainty principle. Finally, as application analogues of the Poisson summation formula and sampling formulas are given.

The wave packet transform in the framework of linear canonical transform

In this paper, we introduce the concept of linear canonical wave packet transform (LCWPT) based on the idea of linear canonical transform (LCT) and wave packet transform (WPT). Parseval’s identity and some properties of LCWPT are discussed. The inversion formula of LCWPT is formulated. Moreover, the composition of LCWPTs is defined and some properties are studied related to it. The LCWPTs of Mexican hat wavelet function are obtained.

Optimization in the construction of cardinal and symmetric wavelets on the line

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

Multivariate wavelet leaders Rényi dimension and multifractal formalism in mixed Besov spaces

In this paper, we first establish a general lower bound for the multivariate wavelet leaders Rényi dimension valid for any pair [Formula: see text] of functions on [Formula: see text] where [Formula: see text] belongs to the Besov space [Formula: see text] with [Formula: see text] and [Formula: see text] belongs to [Formula: see text] with [Formula: see text]. We then prove the optimality of this result for quasi all pairs [Formula: see text] in the Baire generic sense. Finally, we compute both iso-mixed and upper-multivariate Hölder spectra for all pairs [Formula: see text] in the same [Formula: see text]-set. This allows to prove (respectively, study) the Baire generic validity of the upper-multivariate (respectively, iso-multivariate) multifractal formalism based on wavelet leaders for such pairs.