Rational interpolants with preassigned poles, theoretical aspects

Amiran Ambroladze; Hans Wallin

Displaying similar documents to “Rational interpolants with preassigned poles, theoretical aspects”

Pick-Nevanlinna interpolation on finitely-connected domains

Stephen Fisher (1992)

Studia Mathematica

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Let Ω be a domain in the complex plane bounded by m+1 disjoint, analytic simple closed curves and let $z_{0}, . . ., z_{n}$ be n+1 distinct points in Ω. We show that for each (n+1)-tuple $(w_{0}, . . ., w_{n})$ of complex numbers, there is a unique analytic function B such that: (a) B is continuous on the closure of Ω and has constant modulus on each component of the boundary of Ω; (b) B has n or fewer zeros in Ω; and (c) $B (z_{j}) = w_{j}$ , 0 ≤ j ≤ n.

On the decomposition of a positive real function into positive real summands

Jiří Gregor (1969)

Aplikace matematiky

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Analytic functions of one variable with positive real part in the right half-plane, assuming real values on the real positive half-axis, are called positive real functions. In the paper necessary and sufficient conditions for a positive real function to be a sum of two positive real functions are given. Further the structure of any positive real function $f$ is shown when written in the form $f = f_{0} + g + h$ where $f_{0}, g, h$ are positive real functions and $f_{0}$ has all the pure imaginary poles of the function $f$ . ...

Approximation problems and representations of Hardy spaces in circular domains

I. Chalendar, J. Partington (1999)

Studia Mathematica

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We derive various approximation results in the theory of Hardy spaces on circular domains G. Two applications are given, one to operators which admit a nice representation of $H^{\infty} (G)$ , and the other to extremal problems with links to the theory of differential equations.

Convergence of rational interpolants.

Stahl, Herbert (1996)

Bulletin of the Belgian Mathematical Society - Simon Stevin

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On absolutely representing systems in spaces of infinitely differentiable functions

Yu. Korobeĭnik (2000)

Studia Mathematica

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The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^{\infty} (G)$ and $C^{\infty} (K)$ of infinitely differentiable functions where G is an arbitrary domain in $ℝ^{p}$ , p≥1, while K is a compact set in $ℝ^{p}$ with non-void interior K̇ such that $\bar{K} ̇ = K$ . Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G \subseteq ℂ^{p}$ are also investigated.

Approximation on the sphere by Besov analytic functions

Evgueni Doubtsov (1997)

Studia Mathematica

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Boundary values of zero-smooth Besov analytic functions in the unit ball of $ℂ^{n}$ are investigated. Bounded Besov functions with prescribed lower semicontinuous modulus are constructed. Correction theorems for continuous Besov functions are proved. An approximation problem on great circles is studied.

Sequence spaces and interpolation problems for analytic functions

A. Snyder (1971)

Studia Mathematica

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Two-sided estimates of the approximation numbers of certain Volterra integral operators

D. Edmunds, W. Evans, D. Harris (1997)

Studia Mathematica

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We consider the Volterra integral operator $T : L^{p} (ℝ^{+}) \to L^{p} (ℝ^{+})$ defined by $(T f) (x) = v (x) ʃ_{0}^{x} u (t) f (t) d t$ . Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_{n} (T)$ of T are established when 1 < p < ∞. When p = 2 these yield $l i m_{n \to \infty} n a_{n} (T) = π^{- 1} ʃ_{0}^{\infty} | u (t) v (t) | d t$ . We also provide upper and lower estimates for the $ℓ^{α}$ and weak $ℓ^{α}$ norms of (an(T)) when 1 < α < ∞.

Partial differential operators depending analytically on a parameter

Frank Mantlik (1991)

Annales de l'institut Fourier

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Let $P (λ, D) = \sum_{| α | \leq m} a_{α} (λ) D^{α}$ be a differential operator with constant coefficients $a_{α}$ depending analytically on a parameter $λ$ . Assume that the family ${$ P( $λ$ ,D) $}$ is of constant strength. We investigate the equation $P (λ, D) 𝔣_{λ} \equiv g_{λ}$ where $𝔤_{λ}$ is a given analytic function of $λ$ with values in some space of distributions and the solution $𝔣_{λ}$ is required to depend analytically on $λ$ , too. As a special case we obtain a regular fundamental solution of P( $λ$ ,D) which depends analytically on $λ$ . This result answers a question of L. Hörmander. ...

Notes on interpolation of Hardy spaces

Quanhua Xu (1992)

Annales de l'institut Fourier

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Let $H_{p}$ denote the usual Hardy space of analytic functions on the unit disc $(0 < p \leq \infty)$ . We prove that for every function $f \in H_{1}$ there exists a linear operator $T$ defined on $L_{1} (T)$ which is simultaneously bounded from $L_{1} (T)$ to $H_{1}$ and from $L_{\infty} (T)$ to $H_{\infty}$ such that $T (f) = f$ . Consequently, we get the following results $(1 \leq p_{0}, p_{1} \leq \infty)$ : 1) $(H_{p_{0}}, H_{p_{1}})$ is a Calderon-Mitjagin couple; 2) for any interpolation functor $F$ , we have $F (H_{p_{0}}, H_{p_{1}}) = H (F (L_{p_{0}} (T), L_{p_{1}} (T)))$ , where $H (F (L_{p_{0}} (T), L_{p_{1}} (T)))$ denotes the closed subspace of $F (L_{p_{0}} (T), L_{p_{1}} (T))$ of all functions whose Fourier coefficients...