Simon J. Smith

Some integer-valued trigonometric sums

It is shown that for m = 1,2,3,…, the trigonometric sums and can be represented as integer-valued polynomials in n of degrees 2 m – 1 and 2 m , respectively. Properties of these polynomials are discussed, and recurrence relations for the coefficients are obtained. The proofs of the results depend on the representations of particular polynomials of degree n – 1 or less as their own Lagrange interpolation polynomials based on the zeros of the n th Chebyshev polynomial T n (x) = cos( n arccos x ), -1≤ x ≤1.

On the Lebesgue function for lagrange interpolation with equidistant nodes

Properties of the Lebesgue function associated with interpolation at the equidistant nodes , are investigated. In particular, it is proved that the relative maxima of the Lebesgue function are strictly decreasing from the outside towards the middle of the interval [0, n], and upper and lower bounds, and an asymptotic expansion, are obtained for the smallest maximum when n is odd.

The Lebesgue function for generalized Hermite-Fejér interpolation on the Chebyshev nodes

This paper presents a short survey of convergence results and properties of the Lebesgue function λ m,n (x) for(0, 1, …, m )Hermite-Fejer interpolation based on the zeros of the n th Chebyshev polynomial of the first kind. The limiting behaviour as n → ∞ of the Lebesgue constant Λ m,n = max{λ m,n (x) : −1 ≤ x ≤ 1} for even m is then studied, and new results are obtained for the asymptotic expansion of Λ m,n . Finally, graphical evidence is provided of an interesting and unexpected pattern in the distribution of the local maximum values of λ m,n (x) if m ≥ 2 is even.

MARKOFF TYPE INEQUALITIES FOR CURVED MAJORANTS

(x) = fo(x).We make the following remarks concerning Theorems 1 and 2.REMARK 1. For the parabolic majorant, the corresponding problems in the uniformnorm have been solved by Pierre and Rahman [11] and Rahman and Schmeisser [13].REMARK 2. Problems of this type also occur in approximation theory, most notablyin the work of Dzyadyk [3].For the second aim of this paper, we recall a well known inequality of S. Bern-stein [1]. According to this result, in(x)f p is a real algebraic polynomial of degree nor less that satisfies(1.12) \p

On Hermite-Fejér type interpolation on the Chebyshev nodes

Given f ∈ C [−1, 1], let H n , 3 ( f , x ) denote the (0,1,2) Hermite-Fejer interpolation polynomial of f based on the Chebyshev nodes. In this paper we develop a precise estimate for the magnitude of the approximation error | H n , 3 ( f , x ) − f ( x )|. Further, we demonstrate a method of combining the divergent Lagrange and (0,1,2) interpolation methods on the Chebyshev nodes to obtain a convergent rational interpolatory process.

ON THE DIVERGENCE OF HERMITE-FEJER TYPE INTERPOLATION WITH EQUIDISTANT NODES

If f ( x ) is defined on [−1, 1], let H¯ 1 n ( f, x ) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n +1 equally spaced points x k, n = −1 + (2 k )/ n (0 ≤ k ≤ n ). A famous example due to S . Bernstein shows that even for the simple function h ( x ) = │ x │, the sequence H¯ 1 n ( h, x ) diverges as n → ∞ for each x in 0 H¯ mn ( f, x ), which is the unique polynomial of degree no greater than m ( n + 1) – 1 which satisfies ( f, X k , n ) = δ o, p f ( x k , n ) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n ). In general terms, if m is an even number, the polynomials H¯ mn ( f, x ) seem to possess better convergence properties than the H¯ 1 n ( f, x ). Nevertheless, D.L. Berman was able to show that for g ( x ) ≡ x , the sequence H¯ 2n ( g, x ) diverges as n → ∞ for each x in 0 x │. In this paper we extend Bermans result by showing that for any even m, H¯ mn ( g, x ) diverges as n → ∞ for each x in 0 x │ H¯ mn ( g, x ) – g ( x )│.

A note on the Newton—Cotes integration formula

Abstract If Φ(x) is defined on [−1, 1], let Ln(Φ, x) denote the Lagrange interpolation polynomial of degree n (or less) which agrees with Φ(x) at the equidistant nodes x k,n = −1 + (2k) n (k = 0, 1, …, n) . The classical Newton-Cotes integration formula approximates ∝−11 Φ(x) dx by ∝−11 Ln(Φ, x) dx. In this paper we present a very simple example of an analytic function Φ(x) for which limn → ∞ ∝−11 Ln(Φ, x) dx ≠ ∝−11 Φ(x) dx.