Uniform stability of concentration inequalities and applications

Approximation by rational functions in hardy space

Functions in Hardy spaces on multiply-connected domains in the plane are given an explicit characterization in terms of a boundary condition inspired by the two-dimensional Ising model. The key underlying property is the positivity of a certain operator constructed inductively on the number of components of the boundary.

ABSTRACTThis paper aims to obtain decompositions of higher dimensional functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range . In the one-dimensional case, Deng and Qian recently obtained such a Hardy space decomposition result: for any function , there exist functions and such that , where and are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions and discuss the uniqueness issue of such decomposition.

A sequence of special functions in Hardy space [Formula: see text] are constructed from Cauchy kernel on unit disk 𝔻. Applying projection operator of the sequence of functions leads to an analytic sampling approximation to f, any given function in [Formula: see text]. That is, f can be approximated by its analytic samples in 𝔻s. Under a mild condition, f is approximated exponentially by its analytic samples. By the analytic sampling approximation, a signal in [Formula: see text] can be approximately decomposed into components of positive instantaneous frequency. Using circular Hilbert transform, we apply the approximation scheme in [Formula: see text] to Ls(𝕋2) such that a signal in Ls(𝕋2) can be approximated by its analytic samples on ℂs. A numerical experiment is carried out to illustrate our results.

This study investigates the phase retrieval problem for wideband signals. More precisely, we solve the following problem: given f ∈ L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (ℝ) with Fourier transform in L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (ℝ, e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2c|x|</sup> dx) we determine all functions g ∈ L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (ℝ) with Fourier transform in L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (ℝ, e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2c|x|</sup> dx), such that |f (x)| = |g(x)| for all x ∈ ℝ. To do so, we translate the problem into a phase retrieval problem for functions in Hardy spaces on the disc and use the inner-outer factorization.

In Part I, a novel Galerkin-type method for finite dimensional approximations of transfer functions in Hardy space was developed based on approximation by simple poles. In Part II, this approximation is applied to system level synthesis, a recent approach based on a clever reparameterization, to develop a new technique for optimal control design. To solve system level synthesis problems, prior work relies on finite impulse response approximations that lead to deadbeat control, and that can experience infeasibility and increased suboptimality, especially in systems with large separation of time scales. The new design method does not result in deadbeat control, is convex and tractable, always feasible, can incorporate prior knowledge, and works well for systems with large separation of time scales. Suboptimality bounds with convergence rate depending on the geometry of the pole selection are provided. An example demonstrates superior performance of the method.

The finite element method with Laplace transform of time variable is proposed for the solution of hyperbolic equations. Error estimates in Hardy spaces of functions with values in Sobolev spaces are derived. Due to the isometric isomorphism of Hardy spaces with weighted Hilbert spaces these estimates are valid also for original formulations of hyperbolic equations.

We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight function $w$ that determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single and double exponential weights $w$, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions $w$. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.

Let be the Hardy space of functions that are analytic in the disk and let be the derivative of of order in the sense of Weyl. It is shown, for example, that if is a rational function of degree with all its poles in the domain , then , where , , and depends only on and .Bibliography: 32 titles.

Sampling theorem for bandlimited Hardy space functions generated by Regge problem

A sharp quantitative version of Hales' isoperimetric honeycomb theorem

We introduce and study Hardy spaces consisting of Clifford algebra-valued functions annihilated by perturbed Dirac operators in exterior uniformly rectifiable (UR) domains, and which radiate at infinity. In this context, we establish a higher dimensional version of Cauchy’s vanishing theorem, whose proof makes use of the properties of Cauchy-like operators in exterior UR domains, a sharp version of the Divergence Theorem in exterior Ahlfors regular domains, and a good understanding of the nature of various radiation conditions and properties of the far field pattern for Clifford algebra-valued null-solutions of the Helmholtz operator.

Analytic functions in the Hardy class H2 over the upper half‐plane ℍ+ are uniquely determined by their values on any curve Γ lying in the interior or on the boundary of ℍ+. The goal of this paper is to provide a sharp quantitative version of this statement. We answer the following question: Given f of a unit H2‐norm that is small on Γ (say, its L2‐norm is of order ϵ), how large can f be at a point z away from the curve? When Γ ⊂ ∂ℍ+, we give a sharp upper bound on ∣f(z)∣ of the form ϵγ, with an explicit exponent γ = γ(z) ∈ (0, 1) and explicit maximizer function attaining the upper bound. When Γ ⊂ ℍ+ we give an implicit sharp upper bound in terms of a solution of an integral equation on Γ. We conjecture and give evidence that this bound also behaves like ϵγ for some γ = γ(z) ∈ (0, 1). These results can also be transplanted to other domains conformally equivalent to the upper half‐plane. © 2020 Wiley Periodicals, Inc.

On a biased edge isoperimetric inequality for the discrete cube

Wavelet bases in Hardy-type spaces of analytic functions in a domain with circular boundaries are constructed from a wavelet basis in the Hardy space of functions analytic in the disk. An estimate of the rate of convergence of the partial sums of the constructed wavelets is also obtained.