H.L Loeb

On the existence of optimal integration formulas for analytic functions

SummaryIn this paper we prove the existence of quadrature formulas that are optimal with respect to a Hilbert space of analytic functions solving a problem unsuccessfully attacked so far. Although we allow the formulas to be of type (2) the optimal formulas will be of the form (1).

Zur Eindeutigkeit der rationalen Tschebyscheff Approximationen an stetig differenzierbare Funktionen

ZusammenfassungIn der vorliegenden Arbeit untersuchen wir die Aufgabe, eine stetig differenzierbare Funktion durch verallgemeinerte rationale Funktionen im Sinne vonTschebyscheff zu approximieren. Es wird ein Kriterium abgeleitet, das notwendig und hinreichend dafür ist, daß jede stetig differenzierbare Funktion höchstens eine Minimallösung besitzt.

Oscillating Tchebycheff systems

Abstract As is well known the Tchebycheff polynomial of degree n minimizes the sup norm over all monic polynomials with n simple zeros in [−1,+1). B. D. Bojanov [J. Approx. Theory, 26 (1979) , 293–300] recently investigated the situation for polynomials with a full set of zeros of higher multiplicities. In this paper we generalize these results to extended complete Tchebycheff systems.

Generalized polynomials of minimal norm

We consider the problem of finding optimal generalized polynomials of minimal Lp norm (1 ⩽ p < ∞). As an application of our results we obtain Gaussian quadrature formulas for extended Tchebycheff systems.

A necessary condition for controlled approximation

A necessary condition is given on the Fourier transforms of a finite family of functions in Rsso that the finite family and their translates will approximate an arbitrary function within certain precision.

The optimal L 1 problem for generalized polynomial monosplines and a related problem

We investigate generalized polynomial monosplines with fixed multiple knots and free multiple zeros which have minimal L 1 -norm. We call this subject the Optimal L 1 Problem for Generalized Polynomial Monosplines. We prove that there exists a unique monospline of this type. Related to this problem is a problem for monosplines with free knots which have a set of prescribed zeros. Also in this case the existence of a unique solution is shown

Multiple node splines with boundary conditions: The fundamental theorem of algebra for monosplines and Gaussian quadrature formulae for splines

In this paper our main goal is to establish a “fundamental theorem of algebra” for monosplines having multiple knots and satisfying boundary conditions and to show the existence and uniqueness of multiple node Gaussian quadrature formulas for classes of splines where linear boundary functionals are included in the formulas. The known results on the “fundamental theorem” are due to the follow- ing individuals. For simple knots and multiple zeros, the results were proven by Karlin and Schumaker [4]. In the multiple knot and simple zero setting Micchelli [9] established the results. For multiple knots and multiple zeros the theorems were developed by Barrar and Loeb [ 11. For the simple knot, multiple zero, and the boundary condition case the results were derived by Karlin and Micchelli [IS]. With respect to the quadrature formulas the principal investigators were Karlin [3], Karlin and Pinkus [7], Melkman [8], Michelli and Pinkus [lo], and Karlin and Micchelli [S]. They developed the simple node case. Our point of departure is the paper of Micchelli and Pinkus. They exhibited the duality properties of the monospline and quadrature problems. Using this linkage and the topological methods developed in [l] we will establish similar results for multinle nodes and multiple zeros. 90