We generalize to several variables Kurbanov and Maksimov's result that all linear polynomial operators can be expressed as a formal sum P 1 k=0 ak(X)D k in terms of the derivative D (or any degree reducing operator) and multiplication by x. In contrast, we characterize those linear operators that can be expressed as P 1 k=0 fk(D)X k and give several examples. Generalizations to several variable...

We consider the multiplication operator, M, in Sobolev spaces with respect to general measures and give a characterization for M to be bounded, in terms of sequentially dominated measures. This has important consequences for the asymptotic behaviour of Sobolev orthogonal polynomials. Also, we study properties of Sobolev spaces with respect to measures. 2001 Academic Press

INVARIANT SUBSPACES OF ABSTRACT MULTIPLICATION OPERATORS by Hermann Flaschka We describe a class of operators on a Banach space ft whose members behave, in a sense, like multiplication operators, and consequently leave invariant a proper closed subspace of IB. One of the sufficient conditions for an operator to be such an "abstract multiplication" bears a striking resemblence to an assumption m...

We use a simulated evolution search (genetic programming) for the automatic synthesis of small iterative machine-language programs. For an integer register machine with an addition instruction as its sole arithmetic operator, we show that genetic programming can produce exact and general multiplication routines by synthesizing the necessary iterative control structures from primitive machine-la...