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The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space $\mathbf H_2$ of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and…
In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator $\mathcal S_a$ with the decomposition process $$f=e_a\langle f, e_a\rangle+B_{a}*\mathcal S_a f,$$ where $e_a$ denotes the…
We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…
Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real…
Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such…
In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the…
Let $p\in(0,1]$ and $W$ be an $A_p$-matrix weight, which in scalar case is exactly a Muckenhoupt $A_1$ weight. In this article, we introduce matrix-weighted Hardy spaces $H^p_W$ via the matrix-weighted grand non-tangential maximal function…
We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly…
We investigate Hardy spaces $H^1_L(X)$ corresponding to self-adjoint operators $L$. Our main aim is to obtain a description of $H^1_L(X)$ in terms of atomic decompositions similar to such characterisation of the classical Hardy spaces…
Let $p(\cdot)$ be a measurable function defined on a probability space satisfying $0<p_-:={\rm ess}\inf_{x\in \Omega}p(x)\leq {\rm ess}\sup_{x\in\Omega}p(x)=:p_+<\infty$. We investigate five types of martingale Hardy spaces $H_{p(\cdot)}$…
We propose a simple and effective method for designing approximation formulas for weighted analytic functions. We consider spaces of such functions according to weight functions expressing the decay properties of the functions. Then, we…
A new Hardy space Hardy space approach of Dirichlet type problem based on Tikhonov regularization and Reproducing Hilbert kernel space is discussed in this paper, which turns out to be a typical extremal problem located on the upper…
A quaternionic matrix-valued regular function is a map $F: \Omega \rightarrow M_n(\mathbb{H})$ whose entries are (left) regular functions of a quaternion variable, where $\Omega$ is a domain in $\mathbb{H}$. Our aim is to bring out some…
A novel compressed matrix format is proposed that combines an adaptive hierarchical partitioning of the matrix with low-rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the…
We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel…
In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this…
We consider orthogonal decompositions of invariant subspaces of Hardy spaces, these relate to the Blaschke based phase unwinding decompositions. We prove convergence in Lp. In particular we build an explicit multiscale wavelet basis. We…
An inspiration at the origin of wavelet analysis (when Grossmann, Morlet, Meyer and collaborators were interacting and exploring versions of multiscale representations) was provided by the analysis of holomorphic signals, for which the…
A class is studied of complex valued functions defined on the unit disk (with a possible exception of a discrete set) with the property that all their Pick matrices have not more than a prescribed number of negative eigenvalues. Functions…
We present a dimension-incremental algorithm for the nonlinear approximation of high-dimensional functions in an arbitrary bounded orthonormal product basis. Our goal is to detect a suitable truncation of the basis expansion of the…