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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.

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.

Framelets (a.k.a. wavelet frames) are of interest in both theory and applications. Quite often, tight or dual framelets with high vanishing moments are constructed through the popular oblique extension principle (OEP). Though OEP can increase vanishing moments for improved sparsity, it has a serious shortcoming for scalar framelets: the associated discrete framelet transform is often not compact and deconvolution is unavoidable. Here we say that a framelet transform is compact if it can be implemented by convolution using only finitely supported filters. On the other hand, in sharp contrast to the extensively studied scalar framelets, multiframelets (a.k.a. vector framelets) derived through OEP from refinable vector functions are much less studied and are far from well understood. Also, most constructed multiframelets often lack balancing property which reduces sparsity. In this paper, we are particularly interested in quasi-tight multiframelets, which are special dual multiframelets but behave almost identically as tight multiframelets. From any compactly supported \emph{refinable vector function having at least two entries}, we prove that we can always construct through OEP a compactly supported quasi-tight multiframelet such that (1) its associated discrete framelet transform is compact and has the highest possible balancing order; (2) all compactly supported framelet generators have the highest possible order of vanishing moments, matching the approximation/accuracy order of its underlying refinable vector function. This result demonstrates great advantages of OEP for multiframelets (retaining all the desired properties) over scalar framelets., 33 pages, 20 figures

This paper deals with the use of the first two vanishing moments for constructing cubic spline-wavelets orthogonal to polynomials of the first degree. A decrease in the supports of these wavelets is shown in comparison with the classical semiorthogonal wavelets. For splines with homogeneous Dirichlet boundary conditions of the second order, an algorithm of the shifted wavelet transform is obtained in the form of a solution of a tridiagonal system of linear equations with a strict diagonal dominance

Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let ϕ ∈ L 2 ( R d ) be an arbitrary compactly supported real-valued M -refinable function with a general dilation matrix M and ϕ ˆ ( 0 ) = 1 such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary M -refinable function ϕ a directional compactly supported real-valued quasi-tight M -framelet in L 2 ( R d ) associated with a directional quasi-tight M -framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M -framelet becomes a directional tight M -framelet in L 2 ( R d ) . Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M -refinable function ϕ a compactly supported quasi-tight M -framelet in L 2 ( R d ) with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.

As a generalization of orthonormal wavelets in L 2 ( R ) L_2({\mathbb {R}}) , tightframelets (also called tight wavelet frames) are of importance in wavelet analysis and applied sciences due to their many desirable properties in applications such as image processing and numerical algorithms. Tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Consequently, a large family of refinable functions cannot be used to construct tight framelets. This motivates us to introduce the notion of a quasi-tight framelet, which is a dual framelet but behaves almost like a tight framelet. It turns out that the study of quasi-tight framelets is intrinsically linked to the problem of the generalized matrix spectral factorization for matrices of Laurent polynomials. In this paper, we provide a systematic investigation on the generalized matrix spectral factorization problem and compactly supported quasi-tight framelets. As an application of our results on generalized matrix spectral factorization for matrices of Laurent polynomials, we prove in this paper that from any arbitrary compactly supported refinable function in L 2 ( R ) L_2({\mathbb {R}}) , we can always construct a compactly supported one-dimensional quasi-tight framelet having the minimum number of generators and the highest possible order of vanishing moments. Our proofs are constructive and supplemented by step-by-step algorithms. Several examples of quasi-tight framelets will be provided to illustrate the theoretical results and algorithms developed in this paper.

In this paper, we study Boas transforms of wavelets and obtain a sufficient condition under which the Boas transform of a wavelet is the derivative of another wavelet. Also, a characterization of t...

A formula for calculating moments for wavelet packets is derived and a sufficient condition for moments of wavelet packets to be vanishing is obtained. Also, the convolution and cross-correlation theorems for Hilbert transform of wavelets are proved. Finally, using MRA of L2(R), some results on the vanishing moments of the scaling functions, wavelets and their convolution in two dimension are given. Keywords: Wavelet packets, Hilbert transformation, Moments, Wavelet packets, Mathematics Subject Classification: 42C40, 44A15, 44A60, 65T60

In this note we demonstrate an equivalent condition for bi-freeness, inspired by the well-known "vanishing of alternating centred moments" condition from free probability. We show that all products satisfying a centred condition on maximal monochromatic \chi-intervals have vanishing moments if and only if the family of pairs of faces they come from is bi-free, and show that similar characterisations hold for the amalgamated and conditional settings. In addition, we construct a bi-free unitary Brownian motion and show that conjugation by this process asymptotically creates bi-freeness; these considerations lead to another characterisation of bi-free independence., Comment: Updated to accepted manuscript. 15 pages, 4 figures. Comments welcome

International audience; In this paper, we consider the design of wavelet filters based on the Thiran common-factor approach proposed in Selesnick [2001]. This approach aims at building finite impulse response filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complex wavelet. Unfortunately it is not possible to construct wavelets which are both finitely supported and analytic. The wavelet filters constructed using the common-factor approach are then approximately analytic. Thus, it is of interest to control their analyticity. The purpose of this paper is to first provide precise and explicit expressions as well as easily exploitable bounds for quantifying the analytic approximation of this complex wavelet. Then, we prove the existence of such filters enjoying the classical perfect reconstruction conditions, with arbitrarily many vanishing moments.

Let A be an expansive linear map on $${{\mathbb {R}}}^d$$ preserving the integer lattice and with $$| \det A|=2$$ . We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on $${{\mathbb {R}}}^2$$ and to the Quincunx dilation on $${{\mathbb {R}}}^3$$ because we can remove the hypothesis of the existence of the self-affine tile set. Our construction is based on low pass filters by Han in dimension one with the dyadic dilation and multiresolution theory. Finally, for some particular dilation matrices, we realize that unidimensional Daubechies low pass filers can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments.

Let d ≥ 1. For any A ∈ Z d × d such that | det A | = 2 , we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dilations.

In this paper, we study the continuity properties of wavelet transforms in the Gelfand-Shilov spaces with the use of a vanishing moment condition. Moreover, we also compute the Fourier transforms and the wavelet transforms of concrete functions in the Gelfand-Shilov spaces.

We propose an approach to construct a family of two-dimensional compactly supported real-valued symmetric quincunx tight framelets $\{\phi; \psi_1,\psi_2,\psi_3\}$ in $L_2(R^2)$ with arbitrarily high orders of vanishing moments. Such symmetric quincunx tight framelets are associated with quincunx tight framelet filter banks $\{a;b_1,b_2,b_3\}$ having increasing orders of vanishing moments and enjoying the additional double canonical properties: \[ b_1(k_1,k_2)=(-1)^{1+k_1+k_2} a(1-k_1,-k_2), b_3(k_1,k_2)=(-1)^{1+k_1+k_2} b_2(1-k_1,-k_2). \] For a low-pass filter $a$ which is not a quincunx orthonormal wavelet filter, we show that a quincunx tight framelet filter bank $\{a;b_1,\ldots,b_L\}$ with $b_1$ taking the above canonical form must have $L\ge 3$ high-pass filters. Thus, our family of symmetric double canonical quincunx tight framelets has the minimum number of generators. Numerical calculation indicates that this family of symmetric double canonical quincunx tight framelets can be arbitrarily smooth. Using one-dimensional filters having linear-phase moments, in this paper we also provide a second approach to construct multiple canonical quincunx tight framelets with symmetry. In particular, the second approach yields a family of $6$-multiple canonical real-valued quincunx tight framelets in $L_2(R^2)$ and a family of double canonical complex-valued quincunx tight framelets in $L_2(R^2)$ such that both of them have symmetry and arbitrarily increasing orders of smoothness and vanishing moments. Several examples are provided to illustrate our general construction and theoretical results on canonical quincunx tight framelets in $L_2(R^2)$ with symmetry, high vanishing moments, and smoothness. Symmetric quincunx tight framelets constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing.

Motivated by the need of short FIR filters for perfect-reconstruction multirate systems, the main objective of this paper is to derive the shortest filters for such filter banks with M channels, for any integer M ⩾ 2 , based on the M -dilated refinement sequence p m of the mth order cardinal B-spline. By imposing the additional constraint of lth order sum rule on the M -dual low-pass sequence a m , the smoothness property of the M -dual scaling function, along with its corresponding analysis wavelets, is studied, and consequently yielding the lth order vanishing moment for each of the M − 1 synthesis (spline) wavelets. Several illustrative examples and tables of the filter systems are also included in this paper.

In this paper, we establish a boundedness criterion of a class of inhomogeneous Journe’s type of multi-parameter singular integral operators on multi-parameter local Hardy spaces h p ( R n 1 × ⋯ × R n k ) recently introduced in Ding et al. (2019). The consideration of such inhomogeneous Journe’s type of multi-parameter singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces. The lack of the vanishing moments of both the corresponding functions in the multi-parameter local Calderon reproducing formula and the multi-parameter local Littlewood–Paley–Stein functions, as well as the lack of vanishing moments of atoms in the multi-parameter local Hardy spaces h p ( R n 1 × ⋯ × R n k ) with large size all create substantial difficulties in our proofs.

We consider an extension of the continuous shearlet transform which additionally uses higher order shears. This extension, called the Taylorlet transform, allows for a detection of the position, the orientation, the curvature and other higher order geometric information of singularities. Employing the novel vanishing moment conditions of higher order, $\int_\mathbb{R} g(t^k)t^m dt=0$, on the analyzing function, we can show that the Taylorlet transform exhibits different decay rates for decreasing scales depending on the choice of the higher order shearing variables. This enables a more robust detection of the geometric information of singularities. Furthermore, we present a construction that yields analyzing functions which fulfill vanishing moment conditions of different orders simultaneously., 21 pages, 3 figures

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i - j &gt, 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump&ndash, diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank&ndash, Nicolson method with Richardson extrapolation combined with the wavelet&ndash, Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.

The paper is concerned with the construction of a cubic spline wavelet basis on the unit interval and an adaptation of this basis to the first-order homogeneous Dirichlet boundary conditions. The wavelets have four vanishing moments and they have the shortest possible support among all cubic spline wavelets with four vanishing moments corresponding to B-spline scaling functions. We provide a rigorous proof of the stability of the basis in the space [Formula: see text] or its subspace incorporating boundary conditions. To illustrate the applicability of the constructed bases, we apply the wavelet-Galerkin method to option pricing under the double exponential jump-diffusion model and we compare the results with other cubic spline wavelet bases and with other methods.

In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation 4 such that their generating wavelet functions have symmetry and the shortest possible supports with respect to their increasing orders of vanishing moments.

An interesting method called oblique extension principle (OEP) has been proposed in the literature for constructing compactly supported MRA tight and dual wavelet frames with high vanishing moments and high frame approximation orders. Many compactly supported MRA wavelet frames have been recently constructed from scalar refinable functions via OEP. Despite the great flexibility and popularity of OEP for constructing compactly supported MRA wavelet frames in the literature, however, the associated fast frame transform is generally not compact and a deconvolution appears in the frame transform. Here we say that a frame transform is compact if it can be implemented by convolutions, coupled with upsampling and downsampling, using only finite-impulse-response (FIR) filters. In this paper we shall address several fundamental issues on MRA dual wavelet frames and fast frame transforms. Basically, we present two complementary results on dual wavelet frames which are obtained via OEP from scalar refinable functions (= refinable function vectors with multiplicity one) and from truly refinable function vectors with multiplicity greater than one. On the one hand, by a nontrivial argument, we show that from any pair of compactly supported refinable spline functions ϕ and ϕ ˜ (not necessarily having stable integer shifts) with finitely supported masks, if we require that the associated fast frame transform be compact, then any compactly supported dual wavelet frames derived via OEP from ϕ and ϕ ˜ can have vanishing moments at most one and the frame approximation order at most two. On the other hand, we prove in a constructive way that from any pair of compactly supported refinable function vectors ϕ and ϕ ˜ with multiplicity at least two and with finitely supported masks, then we can always build a pair of compactly supported dual multiwavelet frames in L 2 ( R ) with the following properties: (i) The associated fast frame transform is compact; therefore, no deconvolution appears in the fast frame transform. (ii) All the frame generators achieve the highest possible order of vanishing moments; the pair of dual multiwavelet frames has the highest possible frame approximation order. (iii) The pair of dual multiwavelet frames and its fast frame transform have the highest possible balancing order; therefore, the difficulty of approximation inefficiency facing most multiwavelet transforms does not appear here in the associated fast frame transform. In short, the two desirable properties (i) and (ii) of dual wavelet frames obtained via OEP from scalar refinable functions are generally mutually conflicting, while they coexist very well for truly refinable function vectors and multiwavelets with the additional property of high balancing orders in (iii). In this paper, we shall present a comprehensive study of the frame approximation order, balancing order and the fast frame transform associated with the dual multiwavelet frames derived from refinable function vectors via OEP. One of the key ingredients in our study of MRA multiwavelet frames is an interesting canonical form of a matrix mask which greatly facilitates the investigation of refinable function vectors and multiwavelets. An algorithm is given in this paper for constructing pairs of dual wavelet frames with compact fast frame transforms and high balancing orders. An example of balanced spline dual wavelet frames with compact fast frame transforms is provided to illustrate the algorithm and results of this paper.

We construct several families of tight wavelet frames in L2(R2)L2(R2) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on RR with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even. The first author was partially supported by MEC/MICINN grant #MTM2011-27998 (Spain).

Compactly supported Riesz wavelets are of interest in several applications such as image processing, computer graphics and numerical algorithms. In this paper, we shall investigate compactly supported MRA Riesz multiwavelet bases in L 2 ( R ) . An algorithm is presented to derive Riesz multiwavelet bases from refinable function vectors. To illustrate our algorithm and results in this paper, we present several examples of Riesz multiwavelet bases with short support in L 2 ( R ) .

Zero crossings or extrema of a wavelet transform constitute important signatures for signal analysis with the advantage of great simplicity. In this paper, we introduce a fast frequency-estimation method based on zero-crossing counting in the transform domain of a family of differential spline wavelets. The resolution and order of the vanishing moments of the chosen wavelets have a close relation with the frequency components of a signal. Theoretical results on estimating the highest and the lowest frequency components are derived, which are particularly useful for frequency estimation of harmonic signals. The results are illustrated with the help of several numerical examples. Finally, we discuss the connection of this approach with other frequency estimation methods, with the high-order level-crossing analysis in statistics, and with the scaling theorem in computer vision.

Let ϕ be a compactly supported symmetric real-valued refinable function in L 2 ( R ) with a finitely supported symmetric real-valued mask on Z . Under the assumption that the shifts of ϕ are stable, in this paper we prove that one can always construct three wavelet functions ψ 1 , ψ 2 , and ψ 3 such that (i) All the wavelet functions ψ 1 , ψ 2 , and ψ 3 are compactly supported, real-valued and finite linear combinations of the functions ϕ ( 2 ⋅ − k ) , k ∈ Z ; (ii) Each of the wavelet functions ψ 1 , ψ 2 , and ψ 3 is either symmetric or antisymmetric; (iii) { ψ 1 , ψ 2 , ψ 3 } generates a tight wavelet frame in L 2 ( R ) , that is, ‖ f ‖ 2 = ∑ l = 1 3 ∑ j ∈ Z ∑ k ∈ Z | 〈 f , ψ j , k l 〉 | 2 ∀ f ∈ L 2 ( R ) , where ψ j , k l : = 2 j / 2 ψ l ( 2 j ⋅ − k ) , l = 1 , 2 , 3 and j , k ∈ Z ; (iv) Each of the wavelet functions ψ 1 , ψ 2 , and ψ 3 has the highest possible order of vanishing moments, that is, its order of vanishing moments matches the order of the approximation order provided by the refinable function ϕ. We shall give an example to demonstrate that the assumption on stability of the refinable function ϕ cannot be dropped. Some examples of symmetric tight wavelet frames with three compactly supported real-valued symmetric/antisymmetric generators will be given to illustrate the results and construction in this paper.

It is shown that the conjugate quadrature filters that fail to generate an orthonormal multiresolution are symmetrical if their length is at most 6 and have at most N-2 vanishing moments if their length is 2N/spl ges/6. >

A class of r-regular multiwavelets, depending on the smoothness of the multiwavelet functions, is introduced with the appropriate notation and definitions. Oscillation properties of orthonormal systems are obtained in Lemma 1 and Corollary 1 without assuming any vanishing moments for the scaling functions, and in Theorem 1 the existence of r-regular multiwavelets in L2(Rn) is established. In Theorem 2, a particular r-regular multiresolution analysis for multiwavelets is obtained from an r-regular multiresolution analysis for uniwavelets. In Theorem 3, an r-regular multiresolution analysis of split-type multiwavelets, which are perhaps the simplest multiwavelets, is easily obtained by using an r-regular multiresolution analysis for uniwavelets and a (2n − 1)-fold regular multiresolution analysis for uniwavelets. For some split-type multiwavelets, the support or width of the wavelets is shorter than the support or width of the scaling functions without loss of regularity nor of vanishing moments. Examples of split-type multiwavelets in L2(R) are constructed and illustrated by means of figures. Symmetry and antisymmetry are preserved in the case of infinite support.

A systematic study on tight periodic wavelet frames and their approximation orders is conducted. We identify a necessary and sufficient condition, in terms of refinement masks, for applying the unitary extension principle for periodic functions to construct tight wavelet frames. Then a theory on the approximation orders of truncated tight frame series is established, which facilitates the construction of tight periodic wavelet frames with desirable approximation orders. Finally, a notion of vanishing moments for periodic wavelets, which is missing in the current literature, is introduced and related to frame approximation orders and sparse representations of locally smooth functions. As illustrations, the results are applied to two classes of examples: one is band-limited and the other is time-localized.