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Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal and image processing. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for 2 × 2 matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry. [ABSTRACT FROM AUTHOR]

As a generalization of orthonormal wavelets in L2(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 L2(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. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets {φ; ψ1, ψ2, ψ3} in L2(ℝ2) with symmetry property and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks {a; b1, b2, b3} having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties: b1(k1, k2) = (-1)1+k1+k2a(1 - k1,-k2), b3(k1, k2) = (-1)1+k1+k2 b2(1 - k1,-k2), ∀ k1, k2 ∈ ℤ. Moreover, the supports of all the high-pass filters b1, b2, b3 are no larger than that of the low-pass filter a. For a low-pass filter a which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank {a; b1, …, bL} with b1 taking the above canonical form must have L ≥ 3 highpass filters. Thus, our family of double canonical quincunx tight framelets with symmetry property has the minimum number of generators. Numerical calculation indicates that this family of double canonical quincunx tight framelets with symmetry property 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 property. In particular, the second approach yields a family of 6-multiple canonical real-valued quincunx tight framelets in L2(ℝ2) and a family of double canonical complex-valued quincunx tight framelets in L2(ℝ2) such that both of them have symmetry property 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 L2(ℝ2) with symmetry property, high vanishing moments, and smoothness. Quincunx tight framelets with symmetry property constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing due to their polynomial preserving property, full symmetry property, short support, and high smoothness and vanishing moments. [ABSTRACT FROM AUTHOR]

We consider the coorbit theory associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G = R d ⋊ H . The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups , containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found. As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved. [ABSTRACT FROM AUTHOR]

A new method for the construction of compactly supported Parseval wavelet frames in L 2 ( R d ) with any preassigned approximation order n for arbitrary matrix dilation M is proposed. The number of wavelet functions generating a frame constructed in this way is less or equal to ( d + 1 ) | det M | − d . The method is algorithmic, and the algorithm is simple to use. The number of generating wavelet functions can be reduced to | det M | for a large class of matrices M . [ABSTRACT FROM AUTHOR]

We construct several families of tight wavelet frames in L 2 ( R 2 ) 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 R 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. [ABSTRACT FROM AUTHOR]

This paper proposes a performant method for R-wave detection. It is based on the analysis of ECG by the multiscale product (MP) using continuous wavelets with one and two vanishing moments. Adaptive thresholding and maxima detection allowed the localization of the QRS. The method can be applied for heart rate monitoring or being a part of a fetal ECG extraction process. Tests on real signals including abdominal ECG from pregnant women have shown that the proposed method can be effective with a preference to the one vanishing moment wavelet namely the first derivative of the Gaussian. [ABSTRACT FROM AUTHOR]

We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L(ℝ), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(ℝ), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gröchenig, and they can be informally described as follows: Given a function ψ ∈ L(ℝ) satisfying fairly mild decay, smoothness and vanishing moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation. [ABSTRACT FROM AUTHOR]

Dual wavelet frames and their associated dual framelet filter banks are often constructed using the oblique extension principle. In comparison with the construction of tight wavelet frames and tight framelet filter banks, it is indeed quite easy to obtain some particular examples of dual framelet filter banks with or without symmetry from any given pair of low-pass filters. However, such constructed dual framelet filter banks are often too particular to have some desirable properties such as balanced filter supports between primal and dual filters. From the point of view of both theory and application, it is important and interesting to have an algorithm which is capable of finding all possible dual framelet filter banks with symmetry and with the shortest possible filter supports from any given pair of low-pass filters with symmetry. However, to our best knowledge, this issue has not been resolved yet in the literature and one often has to solve systems of nonlinear equations to obtain nontrivial dual framelet filter banks. Given the fact that the construction of dual framelet filter banks is widely believed to be very flexible, the lack of a systematic algorithm for constructing all dual framelet filter banks in the literature is a little bit surprising to us. In this paper, by solving only small systems of linear equations, we shall completely settle this problem by introducing a step-by-step efficient algorithm to construct all possible dual framelet filter banks with or without symmetry and with the shortest possible filter supports. As a byproduct, our algorithm leads to a simple algorithm for constructing all symmetric tight framelet filter banks with two high-pass filters from a given low-pass filter with symmetry. Examples will be provided to illustrate our algorithm. To explain and to understand better our algorithm and dual framelet filter banks, we shall also discuss some properties of our algorithms and dual framelet filter banks in this paper. [ABSTRACT FROM AUTHOR]

As one of the major directions in applied and computational harmonic analysis, the classical theory of wavelets and framelets has been extensively investigated in the function setting, in particular, in the function space L2(ℝd). A discrete wavelet transform is often regarded as a byproduct in wavelet analysis by decomposing and reconstructing functions in L2(ℝd) via nested subspaces of L2(ℝd) in a multiresolution analysis. However, since the input/output data and all filters in a discrete wavelet transform are of discrete nature, to understand better the performance of wavelets and framelets in applications, it is more natural and fundamental to directly study a discrete framelet/wavelet transform and its key properties. The main topic of this paper is to study various properties of a discrete framelet transform purely in the discrete/digital setting without involving the function space L2(ℝd). We shall develop a comprehensive theory of discrete framelets and wavelets using an algorithmic approach by directly studying a discrete framelet transform. The connections between our algorithmic approach and the classical theory of wavelets and framelets in the function setting will be addressed. Using tensor product of univariate complex-valued tight framelets, we shall also present an example of directional tight framelets in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we present the lifting factorization and construction of wavelet bi-frames with arbitrary generators and scaling. We show that an arbitrary polyphase matrix of a wavelet bi-frame can be factorized into a series of lifting steps. Based on the proposed factorization, we present a general construction of bi-frames. Especially, we give an explicit formula to construct the bi-frames of two scaling and two generators with symmetry and one vanishing moment. This paper does not involve many theories of wavelet frames in mathematics, but focuses on the algebraic issues related to Laurent polynomials, which are efficient expressions in redundant filter banks associated with wavelet frames. As an extension of the classical two-channel filter bank, the redundant filter bank is more complicated but also more flexible. Furthermore, we present an algorithm to increase the number of vanishing moments to arbitrary order by lifting, which is iterated and is straightforward in implementation. [ABSTRACT FROM PUBLISHER]

Abstract: 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. [Copyright &y& Elsevier]

This paper deals with the construction of spline wavelets on the interval [0, 1], which have zero boundary values and vanishing moments. One begins with some primal scaling functions and their biorthogonal duals. Then desired biorthogonal wavelets are given by the method of Dahmen, Kunoth and Urban. Since the structure of those wavelets looks complicated, one tries to construct those types of wavelets only in primal side finally, without using any dual informations. Some numerical experiments show good effects, although the uniform stability remains to be proved theoretically. [ABSTRACT FROM AUTHOR]

A wavelet is a localized function having a prescribed number of vanishing moments. In this correspondence, we provide precise arguments as to why the Hilbert transform of a wavelet is again a wavelet. In particular, we provide sharp estimates of the localization, vanishing moments, and smoothness of the transformed wavelet. We work in the general setting of non-compactly supported wavelets. Our main result is that, in the presence of some minimal smoothness and decay, the Hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet, whereas its localization is controlled by the number of vanishing moments of the original wavelet. We motivate our results using concrete examples. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new set of higher density wavelet frames with symmetric low-pass and band-pass wavelet filters. Based on the maximally flat low-pass linear-phase FIR filter and spectral factorization, two types of design approaches are proposed, which can respectively obtain odd-length FIR filters and even-length FIR filters. The two compact support wavelets respectively have a specified order of vanishing moments and a high degree of regularity. Several design examples are given. The denoising experiments show that the proposed wavelet frames have better shift invariance and denoising performance than the wavelet frames constructed by I.W. Selesnick. In engineering application, the proposed wavelet frame is applied to extract the fault feature of a roller bearing with out-race fault. [Copyright &y& Elsevier]

We propose structural multidimensional multichannel filter banks with desirable numbers of vanishing moments for the analysis and synthesis banks. For a two-channel filter bank, we use a three-step lifting scheme as opposed to the conventional two-step lifting method in order to provide more symmetry between the analysis and synthesis filters. We show that the resulting filters have more regularity, lower frame bounds ratio, and better frequency selectivity. We also extend our design to a general multichannel framework. Three steps of lifting scheme provides us a degree of freedom which we can benefit from toward a more flexible design. [ABSTRACT FROM AUTHOR]

In this paper, we present the lifting scheme of wavelet bi-frames along with theory analysis, structure, and algorithm. We show how any wavelet bi-frame can be decomposed into a finite sequence of simple filtering steps. This decomposition corresponds to a factorization of a polyphase matrix of a wavelet bi-frame. Based on this concept, we present a new idea for constructing wavelet bi-frames. For the construction of symmetric bi-frames, we use generalized Bernstein basis functions, which enable us to design symmetric prediction and update filters. The construction allows more efficient implementation and provides tools for custom design of wavelet bi-frames. By combining the different designed filters for the prediction and update steps, we can devise practically unlimited forms of wavelet bi-frames. Moreover, we present an algorithm of increasing the number of vanishing moments of bi-framelets to arbitrary order via the presented lifting scheme, which adopts an iterative algorithm and ensures the shortest lifting scheme. Several construction examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate compactly supported symmetric orthonormal dyadic complex wavelets such that the symmetric orthonormal refinable functions have high linear-phase moments and the antisymmetric wavelets have high vanishing moments. Such wavelets naturally lead to real-valued symmetric tight wavelet frames with some desirable moment properties, and are related to coiflets which are real-valued and are of interest in numerical algorithms. For any positive integer m, employing only the Riesz lemma without solving any nonlinear equations, we obtain a 2 π-periodic trigonometric polynomial $\hat a$ with complex coefficients such that Define $\hat \phi(\xi):=\prod_{j=1}^\infty \hat a(2^{-j}\xi)$ and $\hat \psi(2\xi)=e^{-{{\mathrm{i}}} \xi} {\overline{\hat a(\xi+\pi)}}\hat \phi(\xi)$. Then ψ is a compactly supported antisymmetric orthonormal wavelet with m + 1 − odd m vanishing moments, and ϕ is a compactly supported symmetric orthonormal refinable function with the special linear-phase moments: $\int_{{{\mathbb R}}} \phi(x)dx=1$ and $\int_{{{\mathbb R}}} (x-1/2)^j \phi(x) dx=0$ for all j = 1,..., m + odd m − 1. Both functions ϕ and ψ are supported on [2 − 2 m,2 m − 1]. The mask of a coiflet has real coefficients and satisfies (i), (ii), and (iii), often with a general phase c and the additional condition that the order of the linear-phase moments is equal (or close) to the order of the sum rules. On the one hand, as Daubechies showed in [, ] that except the Haar wavelet, any compactly supported dyadic orthonormal real-valued wavelets including coiflets cannot have symmetry. On the other hand, solving nonlinear equations, [, ] constructed many interesting real-valued dyadic coiflets without symmetry. But it remains open whether there is a family of real-valued orthonormal wavelets such as coiflets whose masks can have arbitrarily high linear-phase moments. This partially motivates this paper to study the complex wavelet case with symmetry property. Though symmetry can be achieved by considering complex wavelets, the symmetric Daubechies complex orthogonal masks in [] generally have no more than 2 linear-phase moments. In this paper, we shall study and construct orthonormal dyadic complex wavelets and masks with symmetry, linear-phase moments, and sum rules. Examples and two general construction procedures for symmetric orthogonal masks with high linear-phase moments and sum rules are given to illustrate the results in this paper. We also answer an open question on construction of symmetric Daubechies complex orthogonal masks in the literature. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

This letter extends the formulation developed by Sherlock and Monro for parameterization of orthonormal wavelet filter banks. Additional constraints are proposed to ensure that the resulting wavelets have at least two vanishing moments. The remaining degrees of freedom are re-parameterized in a convenient form, which results in a convex set of feasible parameter values. An example involving wavelet adaptation in the context of signal compression is presented for illustration. In this case, the additional constraints improve the outcome of the optimization algorithm employed to adjust the filter bank. [ABSTRACT FROM AUTHOR]

Abstract: 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 (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 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 with multiplicity at least two and with finitely supported masks, then we can always build a pair of compactly supported dual multiwavelet frames in 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. [Copyright &y& Elsevier]

Higher order Coiflets for which both the scaling and wavelet functions have a high number of vanishing moments are constructed by the modified Daubechies method. In contrast with Daubechies' filters of lower order Coiflets, our results are more accurate. The properties of phase and vanishing moment of the scaling and wavelet functions and the corresponding filter coefficients are calculated and verified. The results show that the figures of higher order Coiflets are similar to that of lower order ones. The wavelet filter is superior to the scaling filter in linear phase. The phase of the scaling filter is non-linear, while the phase of the wavelet filter is almost linear. With the increase of the order of Coiflet, the linear phase of the wavelet filter changes sharply. The phase of the scaling function is almost zero in the low frequency band. A filter coefficient table of 5 higher order Coiflets and 5 original lower order coiflets with high precision is listed. An application of higher order Coiflet to intervallic one is presented. [ABSTRACT FROM AUTHOR]

Abstract: In this work, kinds of Marr-type wavelets are obtained and an explicit expression for the wavelets is derived. Time–frequency analysis shows that, regarded as window functions in a short time Fourier transform (STFT), the wavelets have monotonically increasing supporting areas in terms of the size of the so-called time–frequency windows and the vanishing moment of is also monotonically regularly increasing with . [Copyright &y& Elsevier]

Abstract: For a wavelet frame with polynomial symbols generated by an MRA with a given approximation order, we present the method for parametric description and construction of all dual analysis operators (filter banks) with the maximum number of vanishing moments whose symbols are polynomials. The similar problems related to the maximum frame approximation order and some other wavelet frame properties are considered. [Copyright &y& Elsevier]

We shall prove that under a mild assumption, we can construct tight multiwavelet frames with the highest possible order of vanishing moments from any given refinable function vector. This paper generalizes Chui, He and Stöckler's work for dilation 2 and Chui, He, Stöckler and Sun's work for any integer dilation larger than 2 on the existence of tight wavelet frames derived from any given refinable function. [ABSTRACT FROM AUTHOR]

Abstract: 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 . 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 . [Copyright &y& Elsevier]

Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d×d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight M-wavelet frames and orthonormal M-wavelet bases in L2(Rd) of exponential decay, which are derived from compactly supported M-refinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several results in Battle (Comm. Math. Phys. 110 (1987) 601), Bownik (J. Fourier Anal. Appl. 7(2001) 489), Gro¨chenig and Ron (Proc. Amer. Math. Soc. 126 (1998) 1101), Lemarie (J. Math. Pures Appl. 67 (1988) 227), and Strichartz (Constr. Approx. 9 (1993) 327). [Copyright &y& Elsevier]

We construct locally supported, continuous wavelets on manifolds Γ that are given as the closure of a disjoint union of general smooth parametric images of an n‐simplex. The wavelets are proven to generate Riesz bases for Sobolev spaces Hs (Γ) when s ∈ (−1, 3/2), if not limited by the global smoothness of Γ. These results generalize the findings from Dahmen & Stevenson (1999) SIAM J. Numer. Anal., 37, 319–352, where it was assumed that each parametrization has a constant Jacobian determinant. The wavelets can be arranged to satisfy the cancellation property of, in principle, any order, except for wavelets with supports that extend to different patches, which generally satisfy the cancellation property of only order 1. [ABSTRACT FROM PUBLISHER]

In this paper, we construct a class of locally supported wavelet bases for C0 Lagrange finite element spaces on possibly nonuniform meshes on n-dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces Hs for |s| < 3⁄2 (|s| ≤ 1 on Lipschitz manifolds), and the wavelets can, in principle, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used, e.g., for constructing an optimal solver of discretized Hs-elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet bases can be implemented efficiently. [ABSTRACT FROM AUTHOR]