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The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.

We give a concrete realization of the Plancherel measure for a semi-direct product $$N \rtimes H$$ where N and H are vector groups for which the linear action of H on N is almost everywhere regular. A procedure using matrix reductions produces explicit (orbital) parameters by which a continuous field of unitary irreducible representations is realized and the almost all of the dual space of $$N \rtimes H$$ naturally has the structure of a smooth manifold. Using the simplest possible field of positive semi-invariant operators, the Plancherel measure is obtained via an explicit volume form on a smooth cross-section $$\Sigma $$ for almost all H-orbits. The associated trace characters are also shown to be tempered distributions.

The paper presents a new scheme for the representation of tensors which is well-suited for high-order tensors. The construction is based on a hierarchy of tensor product subspaces spanned by orthonormal bases. The underlying binary tree structure makes it possible to apply standard Linear Algebra tools for performing arithmetical operations and for the computation of data-sparse approximations. In particular, a truncation algorithm can be implemented which is based on the standard matrix singular value decomposition (SVD) method.

We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say $$x_0\in \mathbb {C}^{d}$$ ) are illuminated one at a time, often with a physical mask between the sample and light source. The corresponding measurements are the noisy magnitudes of the Fourier transform coefficients resulting from the pointwise product of the mask and the sample. The goal is to recover the original signal from such measurements. The algorithmic framework we study herein relies on first inverting a linear system of equations to recover a fraction of the entries in $$x_0 x_0^*$$ and then using non-linear techniques to recover the magnitudes and phases of the entries of $$x_0$$ . Thus, this paper’s contributions are three-fold. First, focusing on the linear part, it expands the theory studying which measurement schemes (i.e., masks, shifts of the sample) yield invertible linear systems, including an analysis of the conditioning of the resulting systems. Second, it analyzes a class of improved magnitude recovery algorithms and, third, it proposes and analyzes algorithms for phase recovery in the ptychographic setting where large shifts—up to $$50\%$$ the size of the mask—are permitted.

The Turán problem for an open ball of radius r centered at the origin in $${\mathbb {R}}^d$$ consists in computing the supremum of the integrals of positive definite functions compactly supported on that ball and taking the value 1 at the origin. Siegel proved, in the 1930s that this supremum is equal to $$2^{-d}$$ mutiplied by the Lebesgue measure of the ball and is reached by a multiple of the self-convolution of the indicator function of the ball of radius r/2. Several proofs of this result are known and, in this paper, we will provide a new proof of it based on the notion of “dual Turán problem”, a related maximization problem involving positive definite distributions. We provide, in particular, an explicit construction of the Fourier transform of a maximizer for the dual Turán problem. This approach to the problem provides a direct link between certain aspects of the theory of frames in Fourier analysis and the Turán problem. In particular, as an intermediary step needed for our main result, we construct new families of Parseval frames, involving Bessel functions, on the interval [0, 1].

We prove $$L^p$$ boundedness in $$A_\infty $$ weighted spaces for operators defined by almost-orthogonal expansions indexed over the dyadic cubes. The constituent functions in the almost-orthogonal families satisfy weak decay, smoothness, and cancellation conditions. We prove that these expansions are stable (with respect to the $$L^p$$ operator norm) when the constituent functions suffer small dilation and translation errors.

This article gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function inL p (ℝ d , ρ) where ρ is a Gaussian measure. The proof relies on a family of inversion formulas for the Segal-Bargmann transform, which can be “tuned” to give the best estimates for a given value of p. This article also gives a single necessary-and-sufficient condition for a holomorphic function to be the transform of a function f such that any derivative of f multiplied by any polynomial is in Lp ( d , ρ). Finally, I give some weaker but dimension-independent conditions.

We consider Hörmander type symbols on a family of spaces associated with non-negative self-adjoint operators, and we prove boundedness of the corresponding pseudodifferential operators on both classical and non-classical Besov and Triebel–Lizorkin spaces. Consequently, this also covers the case of Sobolev spaces. As an application, we obtain boundedness of spectral multipliers on the mentioned spaces.

Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on $$\mathbb {R}\hspace{0.5pt}^d$$ . In particular, we classify all periodic eigenmeasures on $$\mathbb {R}\hspace{0.5pt}$$ , which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on $$\mathbb {R}\hspace{0.5pt}$$ with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around 0.

Analogues of Slepian vectors are defined for finite-dimensional Boolean hypercubes. These vectors are the most concentrated in neighborhoods of the origin among bandlimited vectors. Spaces of bandlimited vectors are defined as spans of eigenvectors of the Laplacian of the hypercube graph with lowest eigenvalues. A difference operator that almost commutes with space and band limiting is used to initialize computation of the Slepian vectors.