Lawrence A. Harris

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We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange poly-nomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the(More)
Let X and Y be real normed linear spaces and let φ : X → R be a non-negative function satisfying φ(x + y) ≤ φ(x) + y for all x, y ∈ X. We show that there exist optimal constants c m,k such that if P : X → Y is any polynomial satisfying P (x) ≤ φ(x) m for all x ∈ X, thenˆD k P (x) ≤ c m,k φ(x) m−k whenever x ∈ X and 0 ≤ k ≤ m. We obtain estimates for these(More)
Let φ : (−∞, ∞) → (0, ∞) be a given continuous even function and let m be a positive integer. We show that, with some additional restrictions on φ, there exist decreasing sequences x where equality holds with alternating signs at the corresponding sequence of points (and also at ±∞ for Q). Moreover, for any polynomial p of degree at most m, a) if |p(x j)| ≤(More)