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This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $$\Psi $$ on domains or manifolds $$\Gamma $$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ , provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces $$b^\alpha _{p,q}(\nabla )$$ of Besov type is developed.

The present paper is a survey aiming at characterizing all solutions of the discrete phase retrieval problem. Restricting ourselves to discrete signals with finite support, this problem can be stated as follows. We want to recover a complex-valued discrete signal $$\mathbf{x} :\mathbb {Z}\rightarrow \mathbb {C}$$ with support $$\{ 0, \ldots , N-1 \}$$ from the modulus of its discrete-time Fourier transform $$\widehat{x}(\omega )$$ . We will give a full classification of all trivial and nontrivial ambiguities of the discrete phase retrieval problem. In our classification, trivial ambiguities are caused either by signal shifts in space, by multiplication with a rotation factor $$\mathrm {e}^{\mathrm {i}\alpha }$$ , $$\alpha \in [-\pi , \pi )$$ , or by conjugation and reflection of the signal. Furthermore, we show that all nontrivial ambiguities of the finite discrete phase retrieval problem can be characterized by signal convolutions. In the second part of the paper, we examine the usually employed a priori conditions regarding their ability to reduce the number of ambiguities of the phase retrieval problem or even to ensure uniqueness. For the corresponding proofs we can employ our findings on the ambiguity classification. The considerations on the structure of ambiguities also show clearly the ill-posedness of the phase retrieval problem even in cases where uniqueness is theoretically shown.

Let \(\Gamma \) be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group \(G\). In this paper we study translation-invariant (TI) subspaces of \(L^2(G)\) by elements of \(\Gamma \). We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that \(\Gamma \) be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of \(G=\mathbb {R}^n\). We also extend the notion of the spectral function in \(\mathbb {R}^n\) to the LCA setting. It is shown that spectral functions, initially defined in terms of \(\Gamma \), do not depend on \(\Gamma \). Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of \(G\) with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit.

In our previous paper, the Haar multiresolution analysis (MRA) $\{V_{j}\}_{j\in \mathbb {Z}}$ in $L^{2}(\mathbb {A})$ was constructed, where $\mathbb {A}$ is the adele ring. Since $L^{2}(\mathbb {A})$ is the infinite tensor product of the spaces $L^{2}({\mathbb {Q}}_{p})$ , p=∞,2,3,…, the adelic MRA has some specific properties different from the corresponding finite-dimensional ones. Nevertheless, this infinite-dimensional MRA inherits almost all basic properties of the finite-dimensional case. In this paper we derive explicit formulas for bases in V j , $j\in \mathbb {Z}$ , and for the wavelet bases generated by the above-mentioned adelic MRA. In view of the specific properties of the adelic MRA, there arise some technical problems in the construction of wavelet bases. These problems were solved with the aid of the operator formalization of the process of generation of wavelet bases. We study the spectral properties of the fractional operator introduced by S. Torba and W.A. Zuniga-Galindo. We prove that the constructed wavelet functions are eigenfunctions of this fractional operator. This paper, as well as our previous paper, introduces new ideas to construct different infinite-dimensional MRAs. Our results can be used in the theory of adelic pseudo-differential operators and equations over the ring of adeles and in adelic models in physics.

In this short note we respond to some concerns raised by Y. Censor, G. Herman, and M. Jiang about the randomized Kaczmarz method that we proposed in [5]. The Kaczmarz method is a well-known iterative algorithm for solving a linear system of equations Ax = b. For more than seven decades, this method was useful in practical applications, and it was studied in many research papers. Despite of this, little is known about the rate of convergence of this method. The classical scheme of Kaczmarz’s method sweeps through the rows of A in a cyclic manner, projecting in each substep the last iterate orthogonally onto a hyperplane associated with a row of A. One variation of Kaczmarz’s method consists of randomly choosing in each iteration the row for the projection. Our algorithm in [5] (labeled Algorithm 1 there) is based on this approach. The idea of choosing the rows randomly is certainly not new. It has been mentioned for instance by Natterer [4], and later also by Feichtinger et al. [1] and by G. Herman and L. Meyer [3]. In these papers, improvement of performance is observed in numerical experiments, but none of these papers contain any proof of the rate of convergence. We consider the main contribution of [5] that (i) it contains the first proof for a rate of convergence for the Kaczmarz’s method that is applicable to general matrices (and not just to very restricted special cases); (ii) the algorithm achieves an exponential rate of convergence; and (iii) the rate

A composite dilation wavelet is a collection of functions generating an orthonormal basis for L2(ℝn) under the actions of translations from a full rank lattice and dilations by products of elements of non-commuting groups A and B. A minimally supported frequency composite dilation wavelet has generating functions whose Fourier transforms are characteristic functions of a lattice tiling set. In this paper, we study the case where A is the group of integer powers of some expanding matrix while B is a finite subgroup of the invertible n×n matrices. This paper establishes that with any finite group B together with almost any full rank lattice, one can generate a minimally supported frequency composite dilation wavelet system. The paper proceeds by demonstrating the ability to find such minimally supported frequency composite dilation wavelets with a single generator.

Recently, a large deviation multifractal formalism based on histograms of wavelet leader coefficients, compared to some other wavelet-based formalisms, was proved to be efficient for uniform Holder functions. In this paper, we extend this efficiency for non-uniform Holder functions. We first obtain optimal bounds for both wavelet and wavelet leader histograms for all functions in the critical Besov space $$B^{m/t,q}_t({\mathbb {T}})$$ , where $$t,q>0$$ and $${\mathbb {T}}$$ is the unit torus of $$\mathbb {R}^m$$ . We then compute these histograms for quasi-all functions in $$B^{m/t,q}_t({\mathbb {T}})$$ , in the sense of Baire Category. Although, increasing parts of these histograms have increasing visibility, they coincide only if $$00$$ , however wavelet histograms method covers it only if $$0

In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying $$\begin{aligned} \sum \limits _{k=n}^{2n} |a_k - a_{k+1}| \le C \sum \limits _{k=[{n}/{\gamma }]}^{[\gamma n]} \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$ for some $$C \ge 1$$ and $$\gamma >1$$ . We first prove the Lebesgue-type inequalities for such series $$\begin{aligned} n|a_n|\le C \omega (f,1/n). \end{aligned}$$ Moreover, we obtain necessary and sufficient conditions for the sum of such series to belong to the generalized Lipschitz, Nikolskii, and Zygmund spaces. We also prove similar results for trigonometric series with weak monotone coefficients, i.e., satisfying $$\begin{aligned} |a_n | \le C \sum \limits _{k=[{n}/{\gamma }]}^{\infty } \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$ for some $$C \ge 1$$ and $$\gamma >1$$ . Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem–Hardy type.

In this paper, bilinear pseudo-differential operators with symbols in the bilinear Hormander symbol class $$BS^{m}_{1,1}$$ on Triebel–Lizorkin spaces are discussed. As a result, we can obtain the Kato–Ponce inequality in local Hardy spaces.

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.

The purpose of this paper is to show that the Rudin–Carleson interpolation theorem is a direct corollary of Fatou’s much older interpolation theorem (of 1906).

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of \(L_1\) signals. This \(L_1\) assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that \(d/2+\varepsilon \) derivatives of the signal are bounded by some polynomial in the \(L_2\) sense.

We construct a polyanalytic extension of analytic (Poisson) wavelet frames, leading to a new system of wavelet superframes. Superframes have been introduced in signal analysis as a tool for the multiplexing of signals—encoding several signals as a single one with the purpose of sharing a communication channel. In this paper, a new system of vector-valued wavelets is constructed by selecting as elements of the analyzing vector the first n elements from an explicit orthogonal basis of the space of admissible functions depending on a parameter \(\alpha \). We show that if the resulting discrete affine system indexed by the set \(\Gamma (a,b)=\{a^{m}bk,a^{m}\}_{k,m\in \mathbb {Z} }\) is a wavelet superframe, then the estimate $$\begin{aligned} b\log a

The concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in 1990 and later by Kaniuth and Ulger. This abbreviation refers to a famous theorem proved by Bochner and Schoenberg for $$L^1({\mathbb {R}})$$ , where $${\mathbb {R}}$$ is the additive group of real numbers, and by Eberlein for $$L^1(G)$$ of a locally compact abelian group G. In this paper we investigate this property for the Banach algebra $$L^p(S,\mu )\;(1\le p

In this paper we study band truncation approximations for operators in uniform Roe algebras of countable discrete groups. Under conditions on certain growth rates for discrete groups, we find large classes of dense subspaces of uniform Roe algebras whose elements can be approximated by their band truncations in the operator norm. We apply these results to construct a nested family of spectral invariant Banach algebras on discrete groups. For a group with polynomial growth, the intersection of these Banach algebras is a spectral invariant dense subalgebra of the uniform Roe algebra. For a group with subexponential growth, we show that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra.

In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $$\mathcal {M}$$ introduced in Lerner et al. (Adv Math 220:1222–1264, 2009) and for multilinear Calderon–Zygmund operators. In particular we obtain a sharp mixed “ $$A_p-A_{\infty }$$ ” bound for $$\mathcal {M}$$ , some partial results related to a Buckley-type estimate for $$\mathcal {M}$$ , and a sufficient condition for the boundedness of $$\mathcal {M}$$ between weighted $$L^p$$ spaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calderon–Zygmund operators in terms of dyadic positive multilinear operators in the spirit of the recent work (Lerner, J Anal Math 121:141–161, 2013). Then we obtain a multilinear version of the “ $$A_2$$ conjecture”. Several open problems are posed.

In this paper we prove and discuss some new \(\left( H_{p},weak-L_{p}\right) \) type inequalities of maximal operators of Vilenkin–Norlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such Vilenkin–Norlund means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

Various properties of the system \({\mathcal {B}}_{\psi }\) of integer translates of a square integrable function \(\psi \in L^2({\mathbb {R}})\) can be completely described in terms of the periodization function \(p_{\psi }(\xi )=\sum _{k\in {\mathbb {Z}}}|\widehat{\psi }(\xi +k)|^2\). In this paper, we consider the problem of \(\ell ^p\)-linear independence, where \(p>2\). The results we present include the method of construction for one type of counterexamples to several naturally taken conjectures, a new sufficient condition for \(\ell ^p\)-linear independence and a characterization theorem having an additional assumption on \({\mathcal {B}}_{\psi }\). In the latter, we obtain the characterization in terms of the sets of multiplicity of Lebesgue measure zero.

The classical Bochner-Schoenberg-Eberlein theorem characterizes the continuous functions on the dual group of a locally compact abelian group G which arise as Fourier-Stieltjes transforms of elements of the measure algebra M(G) of G. This has led to the study of the concept of a BSE-algebra as introduced by Takahasi and Hatori in 1990. In the present paper we establish affirmatively a question raised by Takahasi and Hatori that whether \(L^{1}({\mathbb{R}}^{+})\) is a BSE-algebra.

Ramanujan’s Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the “Master Theorem”. In this paper we prove an analogue of Ramanujan’s Master theorem for the hypergeometric Fourier transform associated with root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.

In this paper we investigate encoding the bit-stream resulting from coarse Sigma-Delta quantization of finite frame expansions (i.e., overdetermined representations) of vectors. We show that for a wide range of finite-frames, including random frames and piecewise smooth frames, there exists a simple encoding algorithm—acting only on the Sigma-Delta bit stream—and an associated decoding algorithm that together yield an approximation error which decays exponentially in the number of bits used. The encoding strategy consists of applying a discrete random operator to the Sigma-Delta bit stream and assigning a binary codeword to the result. The reconstruction procedure is essentially linear and equivalent to solving a least squares minimization problem.

Let A be a d×d expansive matrix with |detA|=2. An A-wavelet is a function \(\psi\in L^{2}(\mathbb{R}^{d})\) such that \(\{2^{\frac{j}{2}}\psi(A\cdot-k):\,j\in \mathbb{Z},\,k\in \mathbb{Z}^{d}\}\) is an orthonormal basis for \(L^{2}(\mathbb{R}^{d})\). A measurable function f is called an A-wavelet multiplier if the inverse Fourier transform of \(f\hat{\psi}\) is an A-wavelet whenever ψ is an A-wavelet, where \(\hat{\psi}\) denotes the Fourier transform of ψ. A-scaling function multiplier, A-PFW multiplier, semi-orthogonal A-PFW multiplier, MRAA-wavelet multiplier, MRAA-PFW multiplier and semi-orthogonal MRAA-PFW multiplier are defined similarly. In this paper, we prove that the above seven classes of multipliers are equivalent, and obtain a characterization of them. We then prove that if the set of all A-wavelet multipliers acts on some A-scaling function (A-wavelet, A-PFW, semi-orthogonal A-PFW, MRAA-wavelet, MRAA-PFW, semi-orthogonal MRAA-PFW), the orbit is arcwise connected in \(L^{2}(\mathbb{R}^{d})\), and that if the generator of an orbit is an MRAA-PFW, the orbit is equal to the set of all MRAA-PFWs whose Fourier transforms have same module, and is also equal to the set of all MRAA-PFWs with corresponding pseudo-scaling functions having the same module of their Fourier transforms.

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ . We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.

The problem of estimating the AP-norm of univariate almost-periodic functions using a Gabor system or a wavelet system was studied by several authors, and culminated in the characterization given in a recent paper of the present author. The present article unifies and generalizes the various efforts via the study of this problem in the context of (multivariate) generalized shift-invariant (GSI) systems. The main result shows that the sought-for norm estimation of the AP functions is valid if and only if the given GSI system is an \(L_{2}({\mathbb{R}}^{d})\)-frame. Moreover, the frame bounds of the system are also the sharpest bounds in our estimation.

International audience; Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's $3/2$-problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the adéle ring $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to reformulate our results as estimation problems for character sums.

Interpolating refinable function vectors with compact support are of interest in applications such as sampling theory, numerical algorithm, and signal processing. Han et al. (J. Comput. Appl. Math. 227:254–270, 2009), constructed a class of compactly supported refinable function vectors with (d,r)-interpolating property. A continuous d-refinable function vector ϕ=(ϕ 1,…,ϕ r ) T is (d,r)-interpolating if $$\phi_{\ell}\biggl(\frac{m}{r}+k\biggr)=\delta_{k}\delta_{\ell-1-m},\quad \forall k\in\mathbb{Z},\ m=0,1,\ldots,r-1,\ \ell=1,\ldots,r.$$ In this paper, based on the (d,r)-interpolating refinable function vector ϕ∈(C 1(ℝ)) r , we shall construct r functions ϕ r+1,…,ϕ 2r such that the new d-refinable function vector ϕ ♮=(ϕ T ,ϕ r+1,…,ϕ 2r ) T belongs to (C 1(ℝ))2r and has the Hermite-like interpolating property: Then any function f∈C 1(ℝ) can be interpolated and approximated by That is, $\widetilde{f}^{(\kappa)}(k+\frac{m}{r})=f^{(\kappa)}(k+\frac{m}{r})$ , ∀κ∈{0,1}, ∀k∈ℤ, and m=0,1,…,r−1. When ϕ has symmetry, it is proved that so does ϕ ♮ by appropriately selecting some parameters. Moreover, we address the approximation order of ϕ ♮. A class of Hermite-like interpolating refinable function vectors with symmetry are constructed from ϕ such that they have higher approximation order than it. Several examples of Hermite-like interpolating refinable function vectors are given to illustrate our results. The truncated error estimate of the interpolating series above is given in Sect. 3. A numerical example of recovering signal is given in Sect. 5 to check the efficiency of the interpolating formula above.

Let (S,d,ρ) be the affine group ℝ n ⋉ℝ+ endowed with the left-invariant Riemannian metric d and the right Haar measure ρ, which is of exponential growth at infinity. In this paper, for any linear operator T on (S,d,ρ) associated with a kernel K satisfying certain integral size condition and Hormander’s condition, the authors prove that the following four statements regarding the corresponding maximal singular integral T ∗ are equivalent: T ∗ is bounded from $L_{c}^{\infty}$ to BMO, T ∗ is bounded on L p for all p∈(1,∞), T ∗ is bounded on L p for some p∈(1,∞) and T ∗ is bounded from L 1 to L 1,∞. As applications of these results, for spectral multipliers of a distinguished Laplacian on (S,d,ρ) satisfying certain Mihlin-Hormander type condition, the authors obtain that their maximal singular integrals are bounded from $L_{c}^{\infty}$ to BMO, from L 1 to L 1,∞, and on L p for all p∈(1,∞).

In signal processing and system identification for $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are constructed from rational Blaschke functions, which form a group with respect to function composition that is isomorphic to the Blaschke group, respectively to the hyperbolic matrix group. Consequently, the background theory uses tools from non-commutative harmonic analysis over groups and the generalization of Fourier transform uses concepts from the theory of the voice transform. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may cause a drawback of the method. In this paper we give a set of poles and using them we will generate a multiresolution in $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ . The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the space $H^{2}(\Bbb{T})$ . The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties of the hyperbolic wavelet representation will be studied.

Let S⊂ℝd be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PWS, is defined to be the set of all square-integrable functions on ℝd whose Fourier transforms vanish outside S. A sequence (xj:j∈ℕ) in ℝd is said to be a Riesz-basis sequence for L2(S) (equivalently, a complete interpolating sequence for PWS) if the sequence \((e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N})\) of exponential functions forms a Riesz basis for L2(S). Let (xj:j∈ℕ) be a Riesz-basis sequence for L2(S). Given λ>0 and f∈PWS, there is a unique sequence (aj) in l2 such that the function $$I_\lambda(f)(x):=\sum_{j\in \mathbb {N}}a_je^{-\lambda \|x-x_j\|_2^2},\quad x\in \mathbb {R}^d,$$ is continuous and square integrable on ℝd, and satisfies the condition Iλ(f)(xn)=f(xn) for every n∈ℕ. This paper studies the convergence of the interpolant Iλ(f) as λ tends to zero, i.e., as the variance of the underlying Gaussian tends to infinity. The following result is obtained: Let \(\delta\in(\sqrt{2/3},1]\) and \(0

Let A be any 2×2 real expansive matrix. For any A-dilation wavelet ψ, let \(\widehat{\psi}\) be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of \((f\widehat{\psi})\) is an A-dilation wavelet for any A-dilation wavelet ψ. In this paper, we give a complete characterization of all A-dilation wavelet multipliers under the condition that A is a 2×2 matrix with integer entries and |{det }(A)|=2. Using this result, we are able to characterize the phases of A-dilation wavelets and prove that the set of all A-dilation MRA wavelets is path-connected under the L2(ℝ2) norm topology for any such matrix A.

In this paper we study properties of generalized multiresolution analyses (GMRAs) and wavelets associated with rational dilations. We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and Merrill. As a consequence, we introduce and derive the properties of the dimension function of rationally dilated wavelets. In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity) extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also characterize all 3 interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension function for which the conventional algorithm for constructing integer dilated wavelet sets fails.

Given a set of vectors (the data) in a Hilbert space ℋ, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of ℝN and to infinite dimensional shift-invariant spaces in L2(ℝd). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.

As a particular wavelet subspace, the Paley-Wiener space \(B_{\pi}\) has both regular and irregular sampling theorems. A regular sampling theorem in general wavelet subspaces has been established for several years. In this paper, we discuss the irregular sampling problem in wavelet subspaces.

This paper is devoted to a study of the Hausdorff-Young theorems from a historical perspective, beginning with the F. Riesz-Fischer theorem. Introduced by W. H. Young (1912), these theorems were considered and extended by F. Hausdorff (1923), F. Riesz (1923), E.C. Titchmarsh (1924), G. H. Hardy and J.E. Littlewood (1926), M. Riesz (1927), and O. Thorin (1939/48). Special emphasis is placed upon the development of the proofs of the two Hausdorff-Young inequalities and their impact upon Fourier analysis as a whole, in particular on the M. Riesz-Thorin convexity theoremand on the interpolation of operators. The golden thread connecting the various extensions and generalizations is the concept of logarithmic convexity, one that goes back to the work of J. Hadamard (1896), A. Liapounoff (1901), J.L.W.V. Jensen (1906), and O. Blumenthal (1907).