On the seven-diagonals splitting for the cubic spline wavelets with six vanishing moments on an interval

This study uses a zeroing property of the first six moments for constructing a splitting algorithm for the cubic spline wavelets. First, we construct a system of cubic basic spline-wavelets, realizing orthogonal conditions to all polynomials up to any degree. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the orthogonality to all polynomials up to the fifth degree on the closed interval. The originality of the study consists of obtaining implicit finite relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, the resulting transformation matrix has seven diagonals, instead of five as in the previous case with four zero moments. A modification of the system is performed, which ensures a strict diagonal dominance, and, consequently, the stability of the calculations. The comparative results of numerical experiments on approximating and calculating the derivatives of a discrete function are presented.

The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

On the construction of the orthonormal wavelet in the Hardy space H2(ℝ)

We are concerned with the orthonormal wavelet [Formula: see text] in the Hardy space [Formula: see text] which is a closed subspace of [Formula: see text] without negative frequency components. It is well known that there does not exist an [Formula: see text]-wavelet such that [Formula: see text] is continuous on [Formula: see text] and satisfies [Formula: see text] for some [Formula: see text]. The aim of this paper is to find a critical decay rate in the existing [Formula: see text]-wavelet under the condition that [Formula: see text] is continuous on [Formula: see text]. Moreover, we also construct a concrete [Formula: see text]-wavelet having infinite vanishing moments.

Operators on the vanishing moments subspace and Stieltjes classes for M-indeterminate probability distributions

PurposeIn this work the author gathers several methods and techniques to construct systematically Stieltjes classes for densities defined on R+.Design/methodology/approachThe author uses complex integration to obtain integrable functions with vanishing moments sequence, and then the author considers some operators defined on the vanishing moments subspace.FindingsThe author gather several methods and techniques to construct systematically Stieltjes classes for densities defined on R+. The author constructs explicitly Stieltjes classes with center at well-known probability densities. The author gives a lot of examples, including old cases and new ones.Originality/valueThe author computes the Hilbert transform of powers of |ln⁡x| to construct Stieltjes classes by using a recent result connecting the Krein condition and the Hilbert transform.

Non-Destructive Diagnostics of Concrete Beams Strengthened with Steel Plates Using Modal Analysis and Wavelet Transform

Externally bonded reinforcements are commonly and widely used in civil engineering objects made of concrete to increase the structure load capacity or to minimize the negative effects of long-term operation and possible defects. The quality of adhesive bonding between a strengthened structure and steel or composite elements is essential for effective reinforcement; therefore, there is a need for non-destructive diagnostics of adhesive joints. The aim of this paper is the detection of debonding defects in adhesive joints between concrete beams and steel plates using the modal analysis approach. The inspection was based on modal shapes and their further processing with the use of continuous wavelet transform (CWT) for precise debonding localization and imaging. The influence of the number of wavelet vanishing moments and the mode shape interpolation on damage imaging maps was studied. The results showed that the integrated modal analysis and wavelet transform could be successfully applied to determine the exact shape and position of the debonding in the adhesive joints of composite beams.

Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov–Sobolev Spaces

In this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece of our present work are characterizations of these new spaces based on the hyperbolic wavelet transform. Hereby we treat both, the standard approach using wavelet systems equipped with sufficient smoothness, decay, and vanishing moments, but also the very simple and basic hyperbolic Haar system. The second major question we pursue is the relationship between the novel hyperbolic spaces and the classical anisotropic Besov–Lizorkin-Triebel scales. As our results show, in general, both approaches to resolve an anisotropy do not coincide. However, in the Sobolev range this is the case, providing a link to apply the newly obtained hyperbolic wavelet characterizations to the classical setting. In particular, this allows for detecting classical anisotropies via the coefficients of a universal hyperbolic wavelet basis, without the need of adaption of the basis or a-priori knowledge on the anisotropy.

Gaussian and Golden Wavelets: A Comparative Study and their Applications in Structural Health Monitoring

In this work, a comparative analysis between Gaussian and Golden wavelets is presented. These wavelets are generated by the derivative of specific base functions. In this case, the order of the derivative also indicates the number of vanishing moments of the wavelet. Although these wavelets have a similar waveform, they have several distinct characteristics in time and frequency domains. These distinctions are explored here in the scale space. In order to compare the results provided by these wavelets for a real signal, they are used in the decomposition of a signal inserted in the context of structural health monitoring.

A characterization of nonhomogeneous wavelet bi-frames for reducing subspaces of Sobolev spaces

For nonhomogeneous wavelet bi-frames in a pair of dual spaces $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$ ( H s ( R d ) , H − s ( R d ) ) with $s\neq 0$ s ≠ 0 , smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other one for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in $L^{2}(\mathbb{R}^{d})$ L 2 ( R d ) . In this paper, we introduce the reducing subspaces of Sobolev spaces, and characterize the nonhomogeneous wavelet bi-frames under the setting of a general pair of dual reducing subspaces of Sobolev spaces.

Some Properties of Fractional Boas Transforms of Wavelets

In this paper, we introduce fractional Boas transforms and discuss some of their properties. We also introduce the notion of wavelets associated with fractional Boas transforms and give some results related to their vanishing moments. Finally, a comparative study of Hilbert transforms and fractional Boas transforms is done.

Fourier–Boas-Like Wavelets and Their Vanishing Moments

In this paper, we propose Fourier–Boas-Like wavelets and obtain sufficient conditions for their higher vanishing moments. A sufficient condition is given to obtain moment formula for such wavelets. Some properties of Fourier–Boas-Like wavelets associated with Riesz projectors are also given. Finally, we formulate a variation diminishing wavelet associated with a Fourier–Boas-Like wavelet.