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.

We prove a sharp quantitative version of recent Faber–Krahn inequalities for the continuous Wavelet transforms associated to a certain family of Cauchy wavelet windows [Ramos and Tilli, Soc. 55 (2023), no. 4, 2018–2034]. Our results are uniform on the parameters of the family of Cauchy wavelets, and asymptotically sharp in both directions . As a corollary of our results, we are able to recover not only the original result for the short‐time Fourier transform as a limiting procedure, but also a new concentration result for functions in Hardy spaces. This is a completely novel result about optimal concentration of Poisson extensions, and our proof automatically comes with a sharp stability version of that inequality. Our techniques highlight the intertwining of geometric and complex‐analytic arguments involved in the context of concentration inequalities. In particular, in the process of deriving uniform results, we obtain a refinement over the proof of the result in [Gómez et al., Invent. Math. 236 (2024), no. 2, 779–836], further improving the current understanding of the geometry of near extremals in all contexts under consideration.

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.

Every semigroup $$\{\varphi _t\}_{t\ge 0}$$ of self-maps of the disc defines a semigroup $$\{C_{\varphi _t}\}_{t\ge 0}$$ of compositions operators on the space of holomorphic functions on the disc. We characterize the (uniform) mean ergodicity (in the sense of continuous means) and the asymptotic behaviour of these operators when they define a $$C_0$$ -semigroup on the disc algebra, in terms of the Denjoy–Wolff point and the associated planar domain in the sense of Berkson and Porta. Finally we deal with the case of Hardy and Bergman spaces.

We show that the space of all bounded derivations from the disc algebra into its dual can be identified with the Hardy space $H^1$; using this, we infer that all such derivations are compact. Also, given a fixed derivation $D$, we construct a finite, positive Borel measure $\mu_D$ on the closed disc, such that $D$ factors through $L^2(\mu_D)$. Such a measure is known to exist, for any bounded linear map from the disc algebra to its dual, by results of Bourgain and Pietsch, but these results are highly non-constructive.

Since the transfer function (or matrix) of a stable linear system is analytic in |z| ≥ 1, the transformation z → z −1 guarantees its analyticity in the unit disk |z| ≤ 1. In this chapter we will study approximation by polynomials and “stable” rational functions in |z| < 1, using the Hardy spaces H 2 and H ∞ norms. Here, due to the reciprocal transformation, “stability” requires location of poles in |z| > 1. Approximation in the Hankel norm will be considered in the next chapter. In order to achieve a firm understanding of the rudiments of these important mathematical tools for optimal system reduction, filter design, model matching, feedback control, and disturbance rejection, etc., we will first consider SISO systems. Standard methods are sometimes applicable to generalize our discussions to MIMO systems which will be studied in Chap. 6. Here and throughout, P n will always denote the collection of all polynomials of degree no greater than n and R n the collection of all rational functions of degree n.KeywordsHardy SpaceFilter DesignBlaschke ProductExtremal FunctionInterpolation ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

A general summability method, the so called θ-summability is considered for multi-dimensional Fourier series, where θ is in the Wiener algebraW(C,ℓ1)($\mathbb{R}$d). It is based on the use of Wiener amalgam spaces, weighted Feichtinger's algebra, Herz and Hardy spaces. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from theHpHardy space toLp(orHp). This implies some norm and almost everywhere convergence results for the θ-means. Large number of special cases of the θ-summation are considered.

Sampling theorem for bandlimited Hardy space functions generated by Regge problem

This paper proposes a new adaptive algorithm for identification Hammerstein systems by using the rational orthogonal basis functions. By adopting an adaptive decomposition algorithm for the Hardy space functions, identification of the linear part can be achieved with the sampling input and output data by selecting poles for the rational orthogonal basis functions in order to get optimal approximations. After that the nonlinear part can be estimated with general least-squares method. Example shows the proposed algorithm is efficient.