Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

Stieltjes polynomials and Lagrange interpolation

Bounds are proved for the Stieltjes polynomial , and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials . This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials . Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of , and for the extended Lagrange interpolation process with respect to the zeros of in the uniform and weighted norms. The corresponding Lebesgue constants are of optimal order.

     

Lagrange interpolation polynomials and orthogonal Fourier-Jacobi series

Let >–1 and > –1. Then a function f(x), continuous on the segment [–1; 1], exists such that the sequence of Lagrange interpolation polynomials constructed from the roots of Jacobi polynomials diverges almost everywhere on [–1; 1], and, at the same time, the Fourier-Jacobi series of function f(x) converges uniformly to f(x) on any segment [a; b] (1; 1).Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 215–226, August, 1976.     

Pointwise convergence of derivatives of Lagrange interpolation polynomials for exponential weights

For a general class of exponential weights on the line and on (−1,1), we study pointwise convergence of the derivatives of Lagrange interpolation. Our weights include even weights of smooth polynomial decay near ±∞ (Freud weights), even weights of faster than smooth polynomial decay near ±∞ (Erdös weights) and even weights which vanish strongly near ±1, for example Pollaczek type weights.     

Lebesgue functions corresponding to a family of Lagrange interpolation polynomials

In this paper we obtain various explicit forms of the Lebesgue function corresponding to a family of Lagrange interpolation polynomials defined at an even number of nodes. We study these forms by using the derivatives up to the second order inclusive. We estimate exact values of Lebesgue constants for this family from below and above in terms of known parameters. In a particular case we obtain new convenient formulas for calculating these estimates.     

Asymptotic formulas for Lebesgue functions corresponding to the family of Lagrange interpolation polynomials

Mathematical Notes - The asymptotic behavior of Lebesgue functions of trigonometric Lagrange interpolation polynomials constructed on an even number of nodes is studied. For these functions,...     

Zeros of quaternion polynomials

It is well known that, over a division ring, every zero of a polynomial f(x) = (x − x₁)…(x − x_n) is congruent to x_r for some r. In this note, we show further that, over the quaternion field, there exists at least one quaternion q_r congruent to each x_r, and that, through this result, a constructive method for determining the zeros of quaternion polynomials can be established.     

On the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials

The aim of this paper is to investigate the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials. Starting from the Lagrange interpolation polynomials, we shall construct a wide class of discrete processes which are uniformly convergent in a suitable weighted space of continuous functions. Error estimates for the approximation will also be considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

On roots of quaternion polynomials

The topological structure of the zero-sets of quaternion polynomials is discussed. As was earlier proved by the author, such a zero-set consists of several points and two-dimensional spheres with centers on the real line. We also show that one can define multiplicities of components of each type in such way that their sum is equal to the algebraic degree of the polynomial considered. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Lagrange interpolation with constraints

Uniform convergence of Lagrange interpolation at the zeros of Jacobi polynomials in the presence of constraints is investigated. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial respecting the given constraints well approximates a given function and its derivatives.     

Intertwining unisolvent arrays for multivariate Lagrange interpolation

Generalizing a classical idea of Biermann, we study a way of constructing a unisolvent array for Lagrange interpolation in C^n+m out of two suitably ordered unisolvent arrays respectively in Cⁿ and C^m. For this new array, important objects of Lagrange interpolation theory (fundamental Lagrange polynomials, Newton polynomials, divided difference operator, vandermondian, etc.) are computed. AMS subject classification 41A05, 41A63     

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP ^* (x; q, t, s) which replace theA _n-series interpolation Macdonald polynomialsP ^* (x; q, t) for theBC _n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP ^* (x; q, t, s) becomeP ^* (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.