Error analysis for Laplace transform ?Finite element solution of hyperbolic equations

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.

In this article, for a class of linear systems arising from the compact finite difference schemes, we apply Krylov subspace methods in combination the ADI, BLAGE,... preconditioners. We consider our scheme in solution of hyperbolic equations subject to appropriate initial and Dirichlet boundary conditions, where is constant. We show, the BLAGE preconditioner is extremely effective in achieving optimal convergence rates. Numerical results performed on model problems to confirm the efficiency of our approach.

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.

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.

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.

Following Sarason’s classification of the densely defined multiplication operators over the Hardy space, we classify the densely defined multipliers over the Sobolev space, \(W^{1,2}[a,b]\). In this paper we find that the collection of such multipliers for the Sobolev space is exactly the Sobolev space itself, and each is a densely defined multiplier is bounded. This sharpens a result of Shields concerning bounded multipliers. The densely defined multiplication operators over the spaces \(W_0(a,b) = \{ f \in W^{1,2}[a,b] : f(a)=f(b)=0 \}\) and \(W^{1,2}(\mathbb {R})\) are also classified. In the case of \(W_0(a,b)\), the densely defined multiplication operators can be written as a ratio of functions in \(W_0(a,b)\) where the denominator is non-vanishing. This is proved using a constructive argument.

Subgrid resolution of fluid discontinuities, II

Vector cascade algorithms and refinable function vectors in Sobolev spaces

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.

In this chapter, we describe several function spaces on homogeneous groups. The origins of the extensive use of homogeneous groups in analysis go back to the book [FS82] of Folland and Stein where Hardy spaces on homogeneous groups have been thoroughly analysed. It turns out that several other function spaces can be defined on homogeneous groups since their main structural properties essentially depend only on the group and dilation structures. Thus, in this chapter we carry out such a construction for Morrey and Campanato spaces and analyse their main properties. Moreover, we describe a version of Sobolev spaces associated to the Euler operator. We call such spaces the Euler–Hilbert–Sobolev spaces.

Advances in Adaptive Computational Methods in Mechanics