On optimal extrapolation and interpolation of fuzzy analytic functions

Для класса ? аналитич еских в единичном кру ге функций, ограниченны х по модулю единицей, погрешност ью наилучшего прибли жения в точкеz ₀ по значениям в точкахz ₁,..., z_n, заданным с погрешнос тьюδ, называется вели чинаr(z ₀, z₁ z..., z_n, α)=inf sup sup ¦f(z₀)-S(f₁, ...f_n)¦, где нижняя грань бере тся по всевозможным ф ункциям S: Сⁿ→С. ДляE～((?1,1) иz ₀∈ ∈(-1,1)Е рассматривается задача о нахождении п орядка информативности мно жестваЕ, т.е. минимальногоп, на котором достигается нижняя грань в равенстве $$R(z_0 ,\delta ,E) = \mathop {\inf }\limits_n {\text{ }}\mathop {\inf }\limits_{z_1 , \ldots ,z_n \in E} {\text{ }}r(z_0 ,z_1 , \ldots ,z_n ,\delta ).$$ Кроме того, приδ, близ ких к 1, решена задача о нахождении величины $$r_n (\delta ,E) = \mathop {\inf }\limits_{z_1 , \ldots ,z_n \in Ez_0 \in E} \sup r(z_0 ,z_1 , \ldots ,z_n ,\delta )$$ и найдены узлы, на кото рых достигается нижн яя грань.     

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.     

Free interpolation of harmonic functions by analytic functions

In the paper one describes the subsets E of the unit circle, for which L¦_E=H¦_E (the functions from the Lebesgue space on the circumference are assumed to be harmonically continued into the circle). The condition consists of two parts: the Carleson-Newman conditions and some geometric condition. One obtains partial results for the sets E such that L _E ¹ =H¹¦_E. It is shown that if the set E does not satisfy the Blaschke condition, then for 1 < p < we have L^P¦_E=H^P¦_E if and only if E lies on the Lobachevskii line.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 209–212, 1982.The author expresses his deep gratitude to his scientific adviser V. P. Khavin for useful discussions.     

Free interpolation by nonvanishing analytic functions

We are concerned with interpolation problems in where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence in the unit disk, we ask whether there exists a nontrivial minorant (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem has a nonvanishing solution whenever for all . The sequences with this property are completely characterized. Namely, we identify them as `` thin" sequences, a class that arose earlier in Wolff's work on free interpolation in VMO.

     

On multivariate polynomial interpolation

We provide a map which associates each finite set in complexs-space with a polynomial space from which interpolation to arbitrary data given at the points in is possible and uniquely so. Among all polynomial spacesQ from which interpolation at is uniquely possible, our is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq, there is associated a polynomial space P, and, for given smoothf, a polynomialqQ is sought for which

On partial polynomial interpolation

The Alexander-Hirschowitz theorem says that a general collection of k double points in Pⁿ imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. We generalize this theorem to arbitrary zero-dimensional schemes contained in a general union of double points. We work in the polynomial interpolation setting. In this framework our main result says that the affine space of polynomials of degree ?d in n variables, with assigned values of any number of general linear combinations of first partial derivatives, has the expected dimension if d≠2 with only five exceptional cases. If d=2 the exceptional cases are fully described.     

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.     

Approximating exponential and logarithmic functions using polynomial interpolation

This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is analysed. The results of interpolating polynomials are compared with those of Taylor polynomials.     

On polynomial interpolation of two variables

Polynomial interpolation of two variables based on points that are located on multiple circles is studied. First, the poisedness of a Birkhoff interpolation on points that are located on several concentric circles is established. Second, using a factorization method, the poisedness of a Hermite interpolation based on points located on various circles, not necessarily concentric, is established. Even in the case of Lagrange interpolation, this gives many new sets of poised interpolation points.     

On the divergence of polynomial interpolation

Consider a triangular interpolation scheme on a continuous piecewise C¹ curve of the complex plane, and let Γ be the closure of this triangular scheme. Given a meromorphic function f with no singularities on Γ, we are interested in the region of convergence of the sequence of interpolating polynomials to the function f. In particular, we focus on the case in which Γ is not fully contained in the interior of the region of convergence defined by the standard logarithmic potential. Let us call Γ_out the subset of Γ outside of the convergence region.In the paper we show that the sequence of interpolating polynomials, {P_n}_n, is divergent on all the points of Γ_out, except on a set of zero Lebesgue measure. Moreover, the structure of the set of divergence is also discussed: the subset of values z for which there exists a partial sequence of {P_n(z)}_n that converges to f(z) has zero Hausdorff dimension (so it also has zero Lebesgue measure), while the subset of values for which all the partials are divergent has full Lebesgue measure.The classical Runge example is also considered. In this case we show that, for all z in the part of the interval (−5,5) outside the region of convergence, the sequence {P_n(z)}_n is divergent.     

Faber polynomial coefficient estimates for analytic bi-close-to-convex functions

Using the Faber polynomials, we obtain coefficient expansions for analytic bi-close-to-convex functions and determine coefficient estimates for such functions. We also demonstrate the unpredictable behavior of the early coefficients of subclasses of bi-univalent functions. A function is said to be bi-univalent in a domain if both the function and its inverse map are univalent there.     

On optimal polynomial meshes

Let $P_n^d$ be the space of real algebraic polynomials of $d$ variables and degree at most $n$, $K?{\mathbb{R}}^d$ a compact set, ${6p6}_K:={sup}_{x\in K}\left|p\left(x\right)\right|$ the usual supremum norm on $K$, and $card\left(Y\right)$ the cardinality of a finite set $Y$. A family of sets $Y=\{Y_n?K,n\in \mathbb{N}\}$ is called an admissible mesh in $K$ if there exists a constant $c_1>0$ depending only on $K$ such that ${6p6}_K\le c_1{6p6}_{Y_n}\text{,}p\in P_n^d,n\in \mathbb{N}\text{,}$ where the cardinality of $Y_n$ grows at most polynomially. If $card\left(Y_n\right)\le c_2{n}^d,n\in \mathbb{N}$ with some $c_2>0$ depending only on $K$ then we say that the admissible mesh is optimal. This notion of admissible meshes is related to norming sets which are widely used in the literature. In this paper we present some general families of sets possessing admissible meshes which are optimal or near optimal in the sense that the cardinality of sets $Y_n$ does not grow too fast. In particular, it will be shown that graph domains bounded by polynomial graphs, convex polytopes and star like sets with $C^2$ boundary possess optimal admissible meshes. In addition, graph domains with piecewise analytic boundary and any convex sets in ${\mathbb{R}}^2$ possess almost optimal admissible meshes in the sense that the cardinality of admissible meshes is larger than optimal only by a $logn$ factor.     

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.