On Configurations of Points on the Sphere and Applications to Approximation of Holomorphic Functions by Lagrange Interpo
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On Configurations of Points on the Sphere and Applications to Approximation of Holomorphic Functions by Lagrange Interpolants Phung Van Manh
Received: 8 January 2014 / Revised: 15 December 2014 / Accepted: 20 December 2014 © Springer-Verlag Berlin Heidelberg 2015
Abstract We study certain configurations of points on the unit sphere in R N . As an application, we prove that the sequence of Lagrange interpolation polynomials of holomorphic functions at certain Chung–Yao lattices converge uniformly to the interpolated functions. Keywords
Lagrange interpolation · Chung–Yao lattices · Configurations on spheres
Mathematics Subject Classification
Primary 41A05 · 41A63 · 52C35
1 Introduction Let Pd (C N ) denote the space of polynomials of degree at most d in N complex variables. A subset A of C N that consists of N d+d distinct points is said to be unisolvent of degree d if, for every function f defined on A, there exists a unique polynomial P ∈ Pd (C N ) such that P(z) = f (z) for all z ∈ A. This polynomial is called the Lagrange polynomial interpolation of f at A and is denoted by L[A; f ]. We are concerned with the problem of approximation of holomorphic functions. Problem 1 Let F be a subclass of entire functions in C N and Ad a unisolvent set of degree d for d = 1, 2, . . .. Under what conditions does L[Ad ; f ] converge to f uniformly on every compact subset of C N for every f ∈ F?
Communicated by Norman Levenberg. P. Van Manh (B) Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy street, Cau Giay, Hanoi, Vietnam e-mail: [email protected]
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It is well known that if N = 1 then a sufficient condition is the boundedness of ∪∞ d=1 Ad . This is an immediate consequence of the Hermite Remainder Formula (see [11, p. 59]). Moreover, when the interpolation sets are unbounded, there exists a function f ∈ H (C) for which convergence does not hold. In this case, the problem is valid for a subclass F of H (C) in which the modulus of the interpolation points is controlled by the order of f (see [1] for more details). It is also proved in [1] that the same results also hold true for Kergin interpolation in C N , a natural generalization of the univariate Lagrange interpolation. In contrast to the univariate case, Bloom and Levenberg showed in [5] that the boundedness of the interpolation array (Ad ) does not guarantee the uniform convergence of every entire function as soon as N ≥ 2. The trouble here is that the interpolation operator has bad behavior when interpolation points tend to an algebraic hypersurface of degree d. Problem 2 Let E be a compact subset of C N and F the class of functions which are holomorphic in a neighborhood of E (which can depend on the functions). Let Ad ⊂ C N be a unisolvent set of degree d for d = 1, 2, . . .. Under what conditions does L[Ad ; f ] converge to f uniformly on E for every f ∈ F? We mention that tools from (pluri)potential theory can be used to solve Problem 2. The sufficient conditions are related to the Lebesgue constants, the tra
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