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In a recent paper R. Dudley gave a characterization of those sequences of independent and identically distributed random variables which are l p {l_p} -compatible for p ≧ 1 p \geqq 1 . In the present note we extend his result into p ∈ ( 0 , 1 ] p \in (0,1] and provide some conditions (necessary or sufficient) for l φ {l_\varphi } -compatibility of a sequence of independent random variables not necessarily identically distributed.

This paper is concerned with linear partial difference operators L having constant coefficients. The functions considered are defined only on the lattice points of the complex plane. It is shown that with any two solutions of Lu(z) = 0 there is associated a new solution which is represented as a convolution product. This product may be considered as a type of line integral and is based upon a discrete analogue of Green's formula. This development may be regarded as an anology to the pioneering work of H. Lewy concerning the composition of solutions of partial differential equations. It may also be considered a continuation of the investigation pursued by Duffin and Duris in the introduction of a convolution product for discrete analytic functions. HUNT LIBRARY CARNEGIE-MELLON UNIVERSITY A Convolution Product for the Solutions of Partial Difference Equations* by R. J. Duffin and Joan Rohrer CARNEGIE INSTITUTE OF TECHNOLOGY

The present paper is motivated by the observation that Weyl's equidistribution theorem for real sequences on a bounded interval can be formulated in a way which is also meaningful for sequences of selfadjoint operators on a Hilbert space. We shall provide general results on weak convergence of operator measures which yield this version of Weyl's theorem as a corollary. Further, by combining the above results with the von Neumann ergodic theorem, we will obtain a Cesaro convergence property, equivalently, an "ergodic theorem", which is valid for all (projectionvalued) spectral measures whose support is in a bounded interval, as well as for the more general class of positive operator-valued measures. Within the same circle of ideas we deduce a convergence property which completely characterizes those spectral measures associated with "strongly mixing" unitary transformations. The final sections are devoted to applications of the preceding results in the study of complexvalued Borel measures as well as to an extension of our results to summability methods other than CesAro convergence. In particular, we obtain a complete characterization, in purely measure theoretic terms, of those complex measures on a bounded interval whose Fourier-Stieltjes coefficients converge to zero.

or one of its rotations. The conjecture was proved for n = 2, 3, and 4 by Bieberbach [1], Loewner [2] and Garabedian and Schiffer [3], respectively. In recent papers, it has been proved to be true if f is sufficiently close to K in one or more of the topologies defined by n (l) ~~~~~E I ak- k | k=2 (2) Re(2 - a2), (3) Re(3 - a3).