On optimal polynomial interpolation of analytic functions

This paper discusses the problem of choosing the Lagrange interpolation points T = (t₀, t₁,…, t_n) in the interval −1 t 1 to minimize the norm of the error, considered as an operator from the Hardy space H²(R) of analytic functions to the space C[−1, 1]. It is shown that such optimal choices converge for fixed n, as R → ∞, to the zeros of a Chebyshev polynomial. Asymptotic estimates are given for the norm of the error for these optimal interpolations, as n → ∞ for fixed R. These results are then related to the problem of choosing optimal interpolation points with respect to the Eberlein integral. This integral is based on a probability measure over certain classes of analytic functions, and is used to provide an average interpolation error over these classes. The Chebyshev points are seen to be limits of optimal choices in this case also.     

Optimal interpolation of analytic functions

We construct an optimal interpolation formula for a particular class of analytic functions, optimization being over a set of interpolation methods which are not necessarily linear. Optimal nodes and the norm of the error are found for the optimal interpolation formula.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 465–476, October, 1972.     

On the Lebesgue constants for interpolation of analytic functions

Дль сИстЕМы РАжлИЧНы х тОЧЕкΤ=(t ₁,...,t _n ) Иж ОтРЕ жкА [?1,1] Иk?[0,1) ВВОДИтсь ВЕлИЧ ИНА $$L_n (\tau ,p,k) = \mathop {\max }\limits_{t \in [ - 1,1]} (\mathop \Sigma \limits_{j = 1}^n |D_j (t)|^p )^{1/p} ,$$ где $$D_j (t) = \frac{{\omega _j (t)}}{{\omega _j (t_j )}}[1 - kW_j^2 (t)],{\mathbf{ }}\omega _j (t) = \mathop \prod \limits_{\begin{array}{*{20}c} {m = 1} \\ {m \ne 1} \\ \end{array} }^n W_m (t),{\mathbf{ }}W_m (t) = \frac{{t - t_m }}{{1 - kt_m t}}.$$ пРИk=0 ОНА сОВпАДАЕт с кОНс тАНтОИ лЕБЕгА, сВьжАН НОИ с ИНтЕРпОльцИЕИ МНОгО ЧлЕНОМ лАгРАНжА. пОкАжАНА сВ ьжь ВЕлИЧИНыL _n (Τ, p, k) с жАД АЧАМИ ИНтЕРпОльцИИ АНАлИт ИЧЕскИх ФУНкцИИ. Дль сИстЕМы $$Z = \left\{ {sn\left[ {\left( {\frac{{2j - 1}}{n} - 1} \right)K,k} \right]} \right\}_{j = 1}^n ,$$ ьВльУЩЕИсь АНАлОгОМ ЧЕБышЕВскОИ сИстЕМы, пОлУЧЕНы ОцЕНкИL _n (Z, p, k) пРИp≧2 Иp≧1.     

On the existence of optimal integration formulas for analytic functions

Summary In this paper we prove the existence of quadrature formulas that are optimal with respect to a Hilbert space of analytic functions solving a problem unsuccessfully attacked so far. Although we allow the formulas to be of type (2) the optimal formulas will be of the form (1).Supported in part by N.S.F. Grant G.P.-28111.Supported in part by Sonderforschungsbereich 72 at Institute for Applied Mathematics, University of Bonn and N.S.F. Grant G.P.-18609.     

On the interpolation of analytic mappings

Let (E ₀,E ₁) and (H ₀,H ₁) be two pairs of complex Banach spaces densely and continuously embedded into each other, E ₁ ? E ₀ and H ₁ ? H ₀ and also let $\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } $ . By E _θ = [E ₀, E ₁]_θ and H _θ = [H ₀, H ₁]_θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by B _θ(0,R) we denote an open ball of radius R in the space E _θ. Let Φ: B ₀(0,R) → H ₀ be an analytic mapping taking B ₁(0,R) into H ₁, and let the estimates $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_\theta \left\| x \right\|_{H_\theta } for allx \in B_\theta (0,R)$ hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball B _θ(0,r) of radius r ∈ (0,R) in the space E _θ into H _θ and $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_0^{1 - \theta } C_1^\theta \frac{R} {{R - r}}\left\| x \right\|_{E_\theta } ,x \in B_\theta (0,r). $ .     

Free interpolation in some banach spaces of analytic functions

We state a series of results regarding the interpolation in the spaces of analytic functions being the Taylor coefficient of the expansion at O, and. One asserts that a sequence with one condensation point, having a structure similar to a geometric progression, is an interpolation sequence for these spaces, i.e., the restriction operator on these sets maps these spaces onto the corresponding collection of sequences. In this case the restriction operator has a continuous right inverse which is explicitly constructed. This note is a continuation of the author's paper. Ref. Zh. Mat. 1973, 4B164.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 186–187, 1976.     

Limit problems for interpolation by analytic radial basis functions

Interpolation problems for analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become increasingly flat, or the data points coalesce in the limit while the radial basis functions stay fixed. Both cases call for a careful regularization, which, if carried out explicitly, yields a preconditioning technique for the degenerating linear systems behind these interpolation problems. This paper deals with both cases. For the increasingly flat limit, we recover results by Larsson and Fornberg together with Lee, Yoon, and Yoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique can also handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite–Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.     

Widths and optimal sampling in spaces of analytic functions

Let Ω be a finitely-connected planar domain and μ be a positive measure with compact supportE in Ω. LetA _p be the unit ball of the Hardy spaceH ^p. The main result of this paper is that Kolmogorov, Gelfand, and linearn-widths ofA _p inL ^q are comparable in size to each other and to the sampling error ifq≤p. Moreover, ifp=q=2 andE is small enough, then all these quantities are equal.     

Free interpolation of germs of analytic functions in Hardy spaces

One proves theorems on the interpolation of germs of analytic functions, defined in the neighborhoods of the interpolation nodes, in the Hardy spaces H^P(0 < p +), generalizing the corresponding results of N. K. Nikol'skii and V. I. Vasyunin for the classes H and H². One obtains estimates of the norms of the interpolating functions in terms of the parameter of the set on which the interpolation is performed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 36–45, 1982.     

Recursive interpolation,extrapolation and projection

The recursive projection algorithm derived in a previous paper is related to several well-known methods of numerical analysis such as the conjugate gradient method, Rosen's method and Henrici's. It is connected with the general interpolation problem, with extrapolation methods, with orthogonal projection on a subspace and with Fourier expansions. Several other connections and applications are presented.