Lawrence A. Harris

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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 cm,k such that if P : X → Y is any polynomial satisfying ‖P (x)‖ ≤ φ(x)m for all x ∈ X, then ‖D̂kP (x)‖ ≤ cm,kφ(x) whenever x ∈ X and 0 ≤ k ≤ m. We obtain estimates for these(More)
The Hille-Yosida and Lumer-Phillips theorems play an important role in the theory of linear operators and its applications to evolution equations, probability and ergodic theory. (See, for example, [17] and [9].) Different nonlinear generalizations and analogues of these theorems can be found, for instance, in [13] and [2]. We are interested in establishing(More)
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 polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the(More)
In this paper we show that any Fréchet holomorphic function mapping the open unit ball of one normed linear space into the closed unit ball of another must be a linear mapping if the Fréchet derivative of the function at zero is a surjective isometry. From this fact we deduce a Banach-Stone theorem for operator algebras which generalizes that of R. V.(More)