Wavelet Frames on Groups and Hypergroups via Discretization of Calderón Formulas

Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we use Calderón-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in ⁿ, for an affine extension of the Heisenberg group, and on many commutative hypergroups.     

On multiresolution analysis of multiplicityd

In this paper we study some basic properties of multiresolution analysis of multiplicityd in several variables and discuss some examples related to the spaces of cardinal splines with respect to the unidiagonal or the crisscross partition of the plane. Furthermore, in analogy with [8], we show that if the scaling functions are compactly supported, then it is possible to find compactly supported mother wavelets _l ,l=1,...,2 ⁿ d–d, in such a way that the family {2 ^jn/2 _l (2 ^j x–v)} is a semiorthogonal basis ofL ² ( ⁿ ).     

Fractal penrose tiles II: Tiles with fractal boundary as duals of penrose triangles

Summary Suppose given a quasi-periodic tiling of some Euclidean space E _u which is self-similar under the linear expansiong: E_μ→E_μ. It is known that there is an embedding of E_μ into some higher-dimensional space ℝ ^N and a linear automorphism with integer coefficients such that E _u ⊂ ℝ ^N is invariant under andg is the restriction of to E _u . Let E _s be the -invariant complement of E _u , and . If certain conditions are fulfilled (e.g. must be a lattice automorphism,g ^* is an expansion), we construct a self-similar tiling of E _s whose expansion isg ^*, using the information contained in the original tiling of E_μ. The term “Galois duality” of tilings is motivated by the fact that the eigenvalues ofg ^* are Galois conjugates of those ofg. Our method can be applied to find the Galois duals which are given by Thurston, obtained in a somewhat other way for the case that dim E_μ=1, andg is the multiplication by a cubic Pisot unit. Bandt and Gummelt have found fractally shaped tilings which can be considered as strictly self-similar modifications of the kites-and-darts, and the rhombi tilings of Penrose. As one of the examples, we show that these fractal versions can be constructed by dualizing tilings by Penrose triangles.     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.     

On the sampling theorem for wavelet subspaces

In [13], Walter extended the classical Shannon sampling theorem to some wavelet subspaces. For any closed subspace V₀/L² (R), we present a necessary and sufficient condition under which there is a sampling expansion for everyf V₀-Several examples are given.     

Adjoints of slant Toeplitz operators II

Let be the unit circle {z|z|=1} and _n c _n e ⁱⁿ be a bounded measurable function on . Theslant Toeplitz operator A onL ² ( ) is defined by A e _n ,e _m =c _2m–n for allm, n wheree _n (z)=z ⁿ , . In this paper, we continue the study initiated in [6] onA ^* , the adjoint ofA . Specifically, we will show that for a certain dense set of continuous functions on ,A ^* is similar to some constant multiple of either a shift, or a shift plus a rank one operator.     

Kernels of Gramian operators for frames in shift-invariant subspaces

For an invertible n×n matrix B and Φ a finite or countable subset of L²(Rⁿ), consider the collection X={?(·-Bk):?∈Φ,k∈Zⁿ} generating the closed subspace M of L²(Rⁿ). Our main objects of interest in this paper are the kernel of the associated Gramian G(.) and dual Gramian operator-valued functions. We show in particular that the orthogonal complement of M in L²(Rⁿ) can be generated by a Parseval frame obtained from a shift-invariant system having m generators where . Furthermore, this Parseval frame can be taken to be an orthonormal basis exactly when almost everywhere. Analogous results in terms of dim(Ker(G(.))) are also obtained concerning the existence of a collection of m sequences in the orthogonal complement of the range of analysis operator associated with the frame X whose shifts either form a Parseval frame or an orthonormal basis for that orthogonal complement.     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.     

Finite-dimensional approximation of the inverse frame operator

A frame in a Hilbert space allows every element in to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of those coefficients and many other situations where frames occur, requires knowledge of the inverse frame operator. But usually it is hard to invert the frame operator if the underlying Hilbert space is infinite dimensional. In the present paper we introduce a method for approximation of the inverse frame operator using finite subsets of the frame. In particular this allows to approximate the frame coefficients (even inl ²) using finite-dimensional linear algebra. We show that the general method simplifies in the important cases of Weil-Heisenberg frames and wavelet frames.     

A sharper stability bound of Fourier frames

Given a real sequence {_n}_n. Suppose that is a frame for L²[–, ] with bounds A, B. The problem is to find a positive constant L such that for any real sequence {_n}_n with ¦_n –_n¦ <L, is also a frame for L²[–, ]. Balan [1] obtained arcsin . This value is a good stability bound of Fourier frames because it covers Kadec's 1/4-theorem and is better than (see Duffin and Schaefer [3]). In this paper, a sharper estimate is given.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Totally complex submanifolds inH P m (1)

Let (resp.K) be the second fundamental form (resp. the sectional curvature) of a compact submanifoldM of the quaternion projective spaceH P ^m (1). We determine all compact totally complex submanifolds of complex dimensionn inH P ^m (1) satisfying either ||² n orK 1/8.Supported by the JSPS postdoctoral fellowship.     

From continuous to discrete Weyl-Heisenberg frames through sampling

In this article we consider the question when one can generate a Weyl- Heisenberg frame for l ² (ℤ) with shift parameters N, M ⁻¹ (integer N, M) by sampling a Weyl-Heisenberg frame for L ² (ℝ) with the same shift parameters at the integers. It is shown that this is possible when the window g ε L ² (ℝ) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition, the Tolimieri-Orr condition A is satisfied, the minimum energy dual window ^o γ ε L ² (ℝ) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. The results of this article also provide a rigorous basis for the engineering practice of computing dual functions by writing the Wexler-Raz biorthogonality condition in the time-domain as a collection of decoupled linear systems involving samples of g and ^o γ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the space of K-periodic sequences, where K=LCM (N, M), by periodization of a Weyl-Heisenberg frame for ℓ ² ℤ with shift parameters N, M ⁻¹ .     

Idempotents of Fourier multiplier algebra

We prove that if the indicator-function1 _E of a measurable setE is a Fourier multiplier in the spaceE ^p () for somep2 thenE is an open set (up to a set of measure zero).     

Orthogonal polynomials on the disc

We consider the space P_n of orthogonal polynomials of degree n on the unit disc for a general radially symmetric weight function. We show that there exists a single orthogonal polynomial whose rotations through the angles , j=0,1,…,n forms an orthonormal basis for P_n, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Legendre polynomials on the disc.Furthermore, such a polynomial reflects the rotational symmetry of the weight in a deeper way: its rotations under other subgroups of the group of rotations forms a tight frame for P_n, with a continuous version also holding. Along the way, we show that other frame decompositions with natural symmetries exist, and consider a number of structural properties of P_n including the form of the monomial orthogonal polynomials, and whether or not P_n contains ridge functions.     

Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on ℝⁿ. Assume that the associated Bochner-Riesz kernel s _R ^δ satisfies the estimate, |s _R ^δ (x, y)| ≤ C R^n/d(1+R^1/d|x - y|^-αδ+β)for some fixed constants a>0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S _p ^−α . We prove that m(P) is bounded on L^P if . We also study multipliers associated to the Hermite operator H on ℝⁿ and the special Hermite operator L on ℂⁿ given by the symbols . As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.     

Orthogonal exponentials on the generalized plane Sierpinski gasket

The self-affine measure μ_Mp,D corresponding tois supported on the the generalized plane Sierpinski gasket T(M_p,D). In the present paper we show that there exist at most 3 mutually orthogonal exponential functions in L²(μ_Mp,D), and the number 3 is the best. This generalizes several known results on the non-spectral self-affine measure problem.     

Isochronicity conditions for some planar polynomial systems

We study the isochronicity of centers at O∈R² for systems , , where A,B∈R[x,y], which can be reduced to the Liénard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.     

Cube-tilings of ℝ n and nonlinear codesand nonlinear codes

Families of nonlattice tilings of ℝ ⁿ by unit cubes are constructed. These tilings are specializations of certain families of nonlinear codes overGF(2). These cube-tilings provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common. We construct cube-tilings of ℝ ⁿ such that no two cubes have a common face of dimension exceeding .     

Metric normal and distance function in the Heisenberg group

We introduce a notion which is equivalent in the Heisenberg group to that of segment normal to a surface. Then, we study some regularity properties of the function measuring the Carnot-Carathéodory distance from an Euclidean surface S in in terms of the regularity of S.