Let Lp = Lp (X, μ), 1 ≤ p ≤ ∞, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let (T 1, . . . ,T K ) be L 2-contractions. Let 0 < ε < δ ≤ 1. Call a function f a δ-spanning function if ‖f‖2 = 1 and if ‖Tkf - Q k-1 Tkf‖2 ≥ δ for each k = 1, . . . ,K, where Q 0 = 0 and Qk is the orthogonal projection on the subspace spanned by (T 1 f , . . . ,Tkf). Call a function h a (δ, ε) -sweeping function if ‖h‖∞ ≤ 1, ‖h‖1 < ε, and if max1≤k≤K |Tk h| > δ-ε on a set of measure greater than 1 - ε. The following is the main technical result, which is obtained by elementary estimates. There is an integer K = K(δ, ε) ≥ 1 such that if f is a δ -spanning function, and if the joint distribution of (f , T 1 f , . . . ,TKf) is normal, then h = ((fΛM)Ꮩ(-M)/M is a (δ, ε)-sweeping function, for some M > 0. Furthermore, if Tk s are the averages of operators induced by the iterates of ameasure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence (Ti ) of these averages.Assume that for each K ≤ 1 there is a subsequence (Ti 1 , . . . ,Tik ) of length K, and a δ-spanning function fK for this subsequence. Then for each ε > 0 there is a function h, 0 ≥ h ≥ 1, ‖h‖1 < ε, such that lim supi Tih ≤ δ a.e.. Another application of the main result gives a refinement of a part of Bourgain’s “Entropy Theorem”, resulting in a different, self contained proof of that theorem.

We present a subordination theory for spatial branching processes. This theory is developed in three different settings, first for branching Markov processes, then for superprocesses and finally for the path-valued process called the Brownian snake. As a common feature of these three situations, subordination can be used to generate new branching mechanisms. As an application, we investigate the compact support property for superprocesses with a general branching mechanism.

For ameromorphic (or harmonic) function ƒ, let us call the dilation of ƒ at z the ratio of the (spherical)metric at ƒ(z) and the (hyperbolic)metric at z. Inequalities are knownwhich estimate the sup norm of the dilation in terms of its Lp norm, for p > 2, while capitalizing on the symmetries of ƒ. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which ƒ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p = 2.

Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.

This paper studies closed subspaces L of the Hardy spaces Hp which are g-invariant (i.e., g. L ⊆ L) where g is inner, g ≠ 1. If p = 2, theWold decomposition theorem implies that there is a countable “g-basis” f1, f2, . . . of L in the sense that L is a direct sum of spaces fj . H2[g] where H2[g] = {f o g | f ∈ H2}. The basis elements fj satisfy the additional property that ∫T |fj|2gk = 0, k = 1, 2, . . . . We call such functions g-2-inner. It also follows that any f ∈ H2 can be factored f = hf ,2 . (F2 o g) where hf,2 is g-2-inner and F is outer, generalizing the classical Riesz factorization. Using Lp estimates for the canonical decomposition of H2,we find a factorization f = hf ,p.(Fpog) for f ∈ Hp. If p ≤ 1 and g is a finite Blaschke product we obtain, for any g-invariant L ⊆ Hp, a finite g-basis of g-p-inner functions.

In a recent paper [2] we defined four classes of infinite dimensional simple Lie algebras over a field of characteristic 0 which we called W*, S*, H*, and K*. As the names suggest, these classes generalize the Lie algebras of Cartan type. A second paper [3] investigates the derivations of the algebras W* and S*, and the possible isomorphisms between these algebras and the algebras defined by Block [1]. In the present paper we investigate the automorphisms of the algebras of type W*.

Exterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction.

We have previously shown how to construct a deformation quantization of any locally compact space on which a vector group acts. Within this framework we show here that, for a natural class of Hamiltonians, the quantum evolutions will have the classical evolution as their classical limit.

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions |x 1|α 1 . . . |xd |αd on the unit sphere S d-1 in ℝd . The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.