Alan Horwitz

A version of Simpson's rule for multiple integrals

Abstract Let M(f) denote the midpoint rule and T(f) the trapezoidal rule for estimating ∫ a b f(x) d x . Then Simpsons rule =λM(f)+(1−λ)T(f), where λ= 2 3 . We generalize Simpsons rule to multiple integrals as follows. Let Dn be some polygonal region in R n , let P0,…,Pm denote the vertices of Dn, and let Pm+1 equal the center of mass of Dn. Define the linear functionals M(f)=Vol(Dn)f(Pn+1), which generalizes the midpoint rule, and T(f)=Vol(Dn)([1/(m+1)]∑j=0mf(Pj)), which generalizes the trapezoidal rule. Finally, our generalization of Simpsons rule is given by the cubature rule (CR) Lλ=λM(f)+(1−λ)T(f), for fixed λ, 0⩽λ⩽1 . We choose λ, depending on Dn, so that Lλ is exact for polynomials of as large a degree as possible. In particular, we derive CRs for the n simplex and unit n cube. We also use points Qj∈∂(Dn), other than the vertices Pj, to generate T(f). This sometimes leads to better CRs for certain regions — in particular, for quadrilaterals in the plane. We use Grobner bases to solve the system of equations which yield the coordinates of the Qjs.

When is the composition of two power series even

If h is the composition of two formal power series f and g , and if h is even, what can be said about f and g ? Some partial answers are given here.

Even compositions of entire functions and related matters

We examine when the composition of two entire functions f and g is even, and extend some of our results to cyclic compositions in general. We prove some theorems for the cases when f or g is a polynomial. Two of the key theorems we use are a well-known result of Borel, and a theorem of Baker and Gross concerning when f(p(z))=f(q(z)).

The space of totally bounded analytic functions

This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f . In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f . As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.

Restrictions on the zeroes of lagrange interpolants to analytic functions

We consider restrictions on the zeroes of Lagrange interpolants to analytic functions in bounded convex regions G. First we prove that if all linear interpolants (with distinct nodes in [Gbar]) vanish only in G, and if all quadratic Taylor interpolants to f have both zeroes in [Gbar] then f must be linear. The key to our proof is the fact that compact convex domains in the plane have the fixed point property. Our second result says that if all interpolants with nodes in G have all their zeroes in G, f is either linear or non-continuable past the boundary of G. Our proof relies on Jentschs Theorem and the Gauss-Lucas Theorem (which depends on the convexity of G). Finally we consider the class TUG of analytic function in G such that every Lagrange interpolant to f is univalent in G. Such functions turn out to be analytic in a region (E (G)), that properly contains G, where E(G) is the envelope of G (defined in Section 3). We also show that there are non-polynomials in TUG by investigating an appropriate Ba...

Homogeneous, isobaric, and autonomous algebraic differential equations

where P is a non-trivial polynomial in its n + 2 variables. We say that O(P) = order of P is n if (1) involves y’“’ in a non-trivial way, and we sometimes use the notation P = 0 in place of (1). An ADE (1) is called autonomous if P does not involve the independent variable x, while (1) is called homogeneous if P is a homogeneous polynomial in all its variables. Finally, (1) is isobaric if P( y,, zy , , . . . . t”y,) = t’P( y,, y, , . . . . y,), for some positive integer r. We prove (Theorem 1) that if u is C” and DA, then u must satisfy an algebraic differential equation which is autonomous, homogeneous, and isobaric. An example of such an equation is yy” (y’)* = 0, which is satisfied by e”. The autonomous part of Theorem 1 was first proven in [BR] using resultants, which are not used in this paper. In Theorem 2 we prove that if u is C” and satisfies an ADE P= 0, then u must satisfy an autonomous and homogeneous (or autonomous and isobaric) ADE & = 0 with O(Q) < O(P) + 2. This bound on the order of Q is sharp. The assumption that u is C” is necessary, for we construct in Theorem 4, for each positive integer n, a C” solution of an ADE on an interval Z which satisfies no homogeneous or autonomous ADE on I. Our proof of Theorem 1 does show, however, that an n-times differentiable solution of an algebraic differential equation is a generalized solution of an

Totally positive functions and totally bounded functions on U−1, 1e

Abstract A function ƒ on [−1, 1] is said to be totally positive if all its Lagrange interpolants are positive on [−1, 1]. It is said to be totally bounded if there is a uniform bound on all its Lagrange interpolants on [−1, 1]. These classes of functions are studied here.