We establish dilation theorems for non-tight frames with additional structure, i.e., frames generated by unitary groups of operators and projective unitary representations. This generalizes previous dilation results for Parseval frames due to Han and Larson and Gabardo and Han. We also extend the dilation theorem for Parseval wavelets, due to Dutkay, Han, Picioroaga, and Sun , by identifying the optimal class of frame wavelets for which dilation into an orthonormal wavelet is possible.
We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=< u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics. We show that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics. We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we show that there are examples of Par…
We consider the construction of orthonormal directional wavelet bases in the multi-resolution analysis (MRA) framework with quincunx dilation downsampling. We show that the Parseval frame property in MRA is equivalent to the identity summation and shift cancellation conditions on M functions, which essentially characterize the scaling (father) function and all directional (mother) wavelets. Based on these two conditions, we further derive sufficient conditions for orthonormal bases and build a family of quasishearlet orthonormal bases, that has the same frequency support as that of the least r…
In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $π(\G)ψ$, where $π$ is a unitary representation of a wavelet group and $\G$ is the abstract pseudo-lattice $\G$. We prove a condition in order that a Parseval frame $π(\G)ψ$ can be dilated to an orthonormal basis of the form $τ(\G)Ψ$ where $τ$ is a super-representation of $π$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite fam…
Spectral representations of the dilation and translation operators on $L^2({\mathbb R})$ are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions defined on the functional spectral spaces. The approach is useful for computational purposes.
A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there …
The measure supported on the Cantor-4 set constructed by Jorgensen-Pedersen is known to have a Fourier basis, i.e. that it possess a sequence of exponentials which form an orthonormal basis. We construct Fourier frames for this measure via a dilation theory type construction. We expand the Cantor-4 set to a 2 dimensional fractal which admits a representation of a Cuntz algebra. Using the action of this algebra, an orthonormal set is generated on the larger fractal, which is then projected onto the Cantor-4 set to produce a Fourier frame.
The orthonormal basis generated by a wavelet of $L^2(\mathbb R)$ has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis of $L^2({\mathbb R}^d)$ is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further, we show how to construct various orthonormal bases of $L^2({\mathbb R}^d)$ from the multiwavelet packets.
We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. A result of N.J. Kalton is included which shows that this is best possible in that: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. We further show that every frame can be written as a (multiple of a) sum of two normalized tight frames or as a sum of an orthonormal basis and a Riesz basis for H. Finally, every frame can be represented as a (multiple of a) average of two orthonormal bases for a larger Hilbert spa…
We investigate the topological and metric structure of the set of idempotent operators and projections which have prescribed diagonal entries with respect to a fixed orthonormal basis of a Hilbert space. As an application, we settle some cases of conjectures of Larson, Dykema, and Strawn on the connectedness of the set of unit-norm tight frames.
We will show that tight frames satisfying the restricted isometry property give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equi-isoclinic. We will also show how to replace parts of the RIP frame with orthonormal sets while maintaining the RIP property.
Configurations of subspaces like equichordal and equiisoclinic tight fusion frames, which are in some sense optimally spread apart and which also have reconstruction properties emulating those of orthonormal bases, are useful in various applications, such as wireless communications and quantum information theory. In this paper, a new construction of infinite classes of equichordal tight fusion frames built on semiregular divisible difference sets is presented. Sometimes this construction yields an equiisoclinic packing. Each of the constructed fusion frames is shown to have both a flat represe…
A Weyl-Heisenberg frame for L^2(R) is a frame consisting of translates and modulates of a fixed function. In this paper we give necessary and sufficient conditions for this family to form a tight WH-frame. This allows us to write down explicitly all functions g for which all translates and modulates of g form an orthonormal basis for L^2(R). There are a number of consequences of this classification, including a simple direct classification of the alternate dual frames to a WH-frame (A result originally due to Janssen).
The local trace function introduced in \cite{Dut} is used to derive equations that relate multiwavelets and multiscaling functions in the context of a generalized multiresolution analysis, without appealing to filters. A construction of normalized tight frame wavelets is given. Particular instances of the construction include normalized tight frame and orthonormal wavelet sets.
One of a key problems in signal reconstruction process with the use of frames is to find a dual frame. Typically, a canonical dual frame is used. However, there are many applications where this choice appears to be unfortunate. Due to that fact, it is necessary to develop a tool, which helps to find a suitable dual frame. In this paper we give a method to find every dual frames. The proposed method is based on Naimark's dilation theorem and the obtained description of dual frames involves parameters that characterize extension of a Parseval frame to an orthonormal basis. These formulas are sim…
We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere approximately form a finite unit norm tight frame (FUNTF). In the present paper, we develop probabilistic versions of tight frames and FUNTFs to significantly weaken the requirements on the random choice of points to obtain an approximate finite tight frame. Namely, points can be chosen from any probabilistic tight frame, they do not have to be identically distributed,…
Motivated by the idea of $J$-frame for a Krein space $\textbf{\textit{K}}$, introduced by Giribet \textit{et al.} (J. I. Giribet, A. Maestripieri, F. Martínez Pería, P. G. Massey, \textit{On frames for Krein spaces}, J. Math. Anal. Appl. (1), {\bf 393} (2012), 122--137.), we introduce the notion of $ζ-J$-tight frame for a Krein space $\textbf{\textit{K}}$. In this paper we characterize $J$-orthonormal basis for $\textbf{\textit{K}}$ in terms of $ζ-J$-Parseval frame. We show that a Krein space is richly supplied with $ζ-J$-Parseval frames. We also provide a necessary and sufficient condition wh…
Hilbert space frames generalize orthonormal bases to allow redundancy in representations of vectors while keeping good reconstruction properties. A frame comes with an associated frame operator encoding essential properties of the frame. We study a polytope that arises in an algorithm for constructing all finite frames with given lengths of frame vectors and spectrum of the frame operator, which is a Gelfand-Tsetlin polytope. For equal norm tight frames, we give a non-redundant description of the polytope in terms of equations and inequalities. From this we obtain the dimension and number of f…
This work characterizes (dyadic) wavelet frames for $L^2({\mathbb R})$ by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated to the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis and extension principles, which are described here on the basis of the periodized Fourier transform. In a second paper of this series, we shall show how the spectral formulas obtained here permit us to calculate all the tight wavelet frames for $L^2({\mathbb…
In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start with defining the quasi-affine systems on a given manifold $\cM$ that is formed by generalized dilations and shifts of a finite collection of wavelet functions $Ψ:=\{ψ_j: 1\le j\le r\}\subset L_2(\R)$. We further require that $ψ_j$ is generated by some refinable function $…
Given a window $ϕ\in L^2(\mathbb R),$ and lattice parameters $α, β>0,$ we introduce a bimodal Wilson system $\mathcal{W}(ϕ, α, β)$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(ϕ, α, β)$. For a class of window functions $ϕ,$ we show that the Gabor system $\mathcal{G}(ϕ, α, β)$ is a tight frame of redundancy $β^{-1}$ if and only if the Wilson system $\mathcal{W}(ϕ, α, β)$ is Parseval system for $L^2(\mathbb R).$ Examples of smooth rapidly decaying generators $ϕ$ are constructed. In addition, when $3\leq β^{-1}\in \mathbb N$, we prove that it is …
A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to be scalable if there exist nonnegative scalars $\{c_i\}_{i\in I}$ such that $\{c_if_i\}_{i\in I}$ is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame $\{f_i\}_{i\in I}$ to be a collection of subsets of $I$…
Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $Δ$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f \in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 Δ)$. Suppose $f$ satisfies Daubechies' criterion, and $b > 0$. For each $j$, write ${\bf M}$ as a disjoint union of measurable sets $E_{j,k}$ with diameter at most $ba^j$, and comparable to $ba^j$ if $ba^j$ is sufficiently small. Take $x_{j,k} \in E_{j,k}$. We then show that the functions $ϕ_{j,k}(x)=[μ(E_{j,k})]^{1/2} \bar{K_{a^j}}(x_{j,k},x)$ form a frame for…
We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,dν_λ) and L^2_h(C^4_+,d\tildeν_λ) of square integrable h…
