Stephen D. Fisher

Spline solutions to L1 extremal problems in one and several variables

Let a = x0 < s1 < ‘. -< x,,, = h be a fixed partition of the closed interval [a, 61 and let 0’ be the closed flat in the Sobolcv space P,*(a, b), 1 :g n ‘:: 171 -I- 1, defined by the interpolation to specified values ri at the points xi , i = 0, I ,..., ITT. If then the minimization problem (I) does not, in general, have an interpolating solution in the class Wn~l(a, b). More generally, if U is a closed flat in a Banach space A’, and R is a continuous linear mapping of X onto Y with finite-dimensional null space, then it is possible that inf{i/ Ru I( Y: ZI E Uj is not attained in U; for example, if Y is not reflexive. In his recent paper [6], Holmes has discussed, with a both a literature survey and new results, the technique of embedding such optimization problems in dual spaces. By considering the extended problem of minimizing (1 R**p iiy.* over the flat JU in X* *. where J denotes the natural injection of X into X**, Holmes has shown that a solution ‘p exists in JU, achieving the same norm extremal value as in the original problem under a natural poisedness hypothesis. Thus, the problem has a solution in the sense of roffe and Tihomirov [7]. In this paper we shall discuss concrete ways in which such problems can be extended to possess natural solutions. In particular, for the problem (l), we expand the class JP,l(a, b) to include functions whose nth derivatives

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL2(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleHb(g)=(I−P)bg.This paper continues earlier investigations of the authors and others by determining conditions under whichHb is bounded, compact, or lies in the Schatten-von Neumann idealSp, 1

Stable and unstable elastica equilibrium and the problem of minimum curvature

The Euler-Bernoulli equations governing a stable equilibrium configuration of an elastica, subject to pin supports, can be derived by a formal calculus of variations via a local minimum principle for the strain energy of the elastica. Such a derivation was carried out by Lee and Forsythe [4] in their analysis of open and closed nonlinear spline curves. The basic equation satisfied for each unloaded span between supports by the curvature (or normalized bending moment) k is the familiar equation in the arc length variable

Envelopes, widths, and landau problems for analytic functions

AbstractFix a positive integerr and letA denote those analytic functions on the open unit disk whoserth derivative is bounded by 1 and which are real-valued on the interval (−1, 1). For a positive integern letx0,⋯,xn+r be distinct points in (−1, 1) and set

A quadratic extremal problem on the dirichlet space

\Lambda = \{ (f(x_0 ), \ldots ,f(x_{n + r} )):f \in A\} .

The decomposition of Cr(K) into the direct sum of subalgebras

. This paper contains results which (a) describe the boundary points of Λ, (b) determine the “envelope”: max{|f(z0)|:f(xj)=0,1≤j≤r+n}, and (c) determine the Kolmogorov widths ofA in the space of continuous real-valued functions on a compact setE in (−1, 1).

Algebras of bounded functions invariant under the restricted backward shift

STEPHEN D. FISHER: I received my bachelors degree from MIT in 1963. My graduate work was done at the University of Wisconsin, Madison, where I was a student of Frank Forelli. I was an instructor at MIT from 1967 to 1969, at which time I joined the faculty at Northwestern University. My research focuses on the interaction of complex analysis with functional | analysis and approximation theory. In addition to a number of research papers, I have written a graduate-level book, Function Theory on Planar Domains, John Wiley and Sons, 1983, and an undergraduate text, Complex Variables, Wadsworth, 1986.

Widths and optimal sampling in spaces of analytic functions

It is shown that there is a unique solution Fto the problem The function F is entire with a number of special properties. The number λ is the reciprocal of the smallest zero of the 0th Bessel function of the first kind.