Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = Qp, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H = Zp, the ring of p-adic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient ̂G/H ⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to so-called wavelet sets in the dual group ̂G. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.

Sets K in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transforms of the characteristic functions 1 K 1 ; : : : ; 1 K L are a family of orthonormal wavelets is treated in [Leo99].

. In the context of a general lattice \Gamma in R n and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the multiwavelets associated with a Multiresolution Analysis (MRA) of multiplicity d 1; and all the scaling functions. Moreover, we give several examples: in particular, we construct a single, MRA and C 1 (R n ) wavelet, which is nonseparable and with compactly supported Fourier transform. 1. Introduction An orthonormal wavelet is a function / 2 L 2 (R) such that the set \Phi / j;k j 2 j=2 /(2 j x \Gamma k) : j; k 2 Zg (1) is an orthonormal basis for L 2 (R): A complete characterization of these wavelets is given by the equations a) X j2Z j b /(2 j )j 2 = 1 a.e. 2 R; b) 1 X j=0 b /(2 j ) b /(2 j ( + 2k)) = 0 a.e. 2 R; k 2 2Z + 1; (I) together with the assumption k/k 2 1: These two equations have been known since the beginning of the theory of wavelets (see [L1] ...

Abstract. It is well known that the Haar and Shannon wavelets in L 2 (R) are at opposite extremes, in the sense that the Haar wavelet is localized in time but not in frequency, whereas the Shannon wavelet is localized in freqency but not in time. We present a rich setting where the Haar and Shannon wavelets coincide and are localized both in time and in frequency. More generally, if R is replaced by a group G with certain properties, J. Benedetto and the author have proposed a theory of wavelets on G, including the construction of wavelet sets [2]. Examples of such groups G include the p-adic rational group G = Qp, which is simply the completion of Q with respect to a certain natural metric topology, and the Cantor dyadic group F2((t)) of formal Laurent series with coefficients 0 or 1. In this expository paper, we consider some specific examples of the wavelet theory on such groups G. In particular, we show that Shannon wavelets on G are the same as Haar wavelets on G. We also give several examples of specific groups (such as Qp and Fp((t)), for any prime number p) and of various

In this paper we give a necessary and sufficient condition for a pair of wavelet families \Psi = f/ 1 ; : : : ; / L g; e \Psi = f ~ / 1 ; : : : ; ~ / L g; in L 2 (R n ), to arise from a pair of biorthogonal MRA's. The condition is given in terms of simple equations involving the functions / ` and ~ / ` . To work in greater generality, we allow multiresolution analyses of arbitrary multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L 2 (R), and includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemari'e in the biorthogonal situation. 1

We give a characterization of all (quasi) affine frames in L²(R^n) which have a (quasi) affine dual in terms of the two simple equations in the Fourier transform domain. In particular, if the dual frame is the same as the original system, i.e. it is a tight frame, we obtain the well known characterization of wavelets. Although these equations have already been proven under some special conditions we show that these characterizations are valid without any decay assumptions on the generators of the affine system.

Abstract. Given a function ψ in L 2 (R d), the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions { | deta | j/2 ψ(a j x −γ) : j ∈ Z,γ ∈ Γ}. In this article we prove that the set of functions generating affine systems that are a Riesz basis of L 2 (R d) is dense in L 2 (R d). We also prove that a stronger result is true for affine systems that are a frame of L 2 (R d). In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of L 2 (R d) with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems. 1.

The first wavelet system was discovered by Alfréd Haar one hundred years ago. Since then the field has grown enormously. In 1952, Richard Duffin and Albert Schaeffer synthesized the earlier ideas of a number of illustrious mathematicians into a unified theory, the theory of frames. Interest in frames as intriguing objects in their own right arose when wavelet theory began to surge in popularity. Wavelet and frame analysis is found in such diverse fields as data compression, pseudo-differential operator theory and applied statistics. We shall explore five areas of frame and wavelet theory: frame bound gaps, smooth Parseval wavelet frames, generalized shearlets, Grassmannian fusion frames, and p-adic wavlets. The phenomenon of a frame bound gap occurs when certain sequences of functions, converging in L2 to a Parseval frame wavelet, generate systems with frame bounds that are uniformly bounded away from 1. In the 90’s, Bin Han proved the existence of Parseval wavelet frames which are smooth and compactlysupported on the frequency domain and also approximate wavelet set wavelets. We discuss problems that arise when one attempts to generalize his results to higher dimensions. A shearlet

Abstract. Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers of a matrix. In this paper we consider the following simultaneous tiling problem: Given a lattice in ̷L ∈ R d and a matrix A ∈ GL(d, R), does there exist a measurable set T such that both {T + α: α ∈ ̷L} and {A n T: n ∈ Z} are tilings of R d? This problem comes directly from the study of wavelets and wavelet sets. Such a T is known to exist if A is expanding. When A is not expanding the problem becomes much more subtle. Speegle [24] exhibited examples in which such a T exists for some ̷L and nonexpanding A in R 2. In this paper we give a complete solution to this problem in R 2. 1.

Abstract. It is proven that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves affirmatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets. Dedicated to Larry Baggett for his great friendship, his love of mathematics, and his continued support of young mathematicians. 1.