Gradient estimates for positive harmonic functions by stochastic analysis

We prove Cheng–Yau type inequalities for positive harmonic functions on Riemannian manifolds by using methods of Stochastic Analysis. Rather than evaluating an exact Bismut formula for the differential of a harmonic function, our method relies on a Bismut type inequality which is derived by an elementary integration by parts argument from an underlying submartingale. It is the monotonicity inherited in this submartingale which allows us to establish the pointwise estimates.     

Convergence of simultaneous Hermite-Padé approximants to then-tuple ofq-hypergeometric series\{ {}_1\Phi _1 (_{(c,\gamma _j )}^{(1,1)} ;z)\} _{j = 1}^n

We investigate the convergence of simultaneous Hermite-Padé approximants to then-tuple of power series $$f_i (z) = \sum\limits_{k = 0}^\infty {C_k^{(i)} z^k ,} i = 1,2,...,n,$$ where $$C_0^{(i)} = 1;C_k^{(i)} = \prod\limits_{p = 0}^{k - 1} {\frac{1}{{(C - q^{\gamma i + p} )}},} k \ge 1.$$ HereC, q∈?, γ _i ∈?,i=1, 2,...,n. For |C|≠1, ifq=e^iθ, θ∈(0, 2π) and θ/2π is irrational, eachf _i (z),i=1,...,n, has a natural boundary on its circle of convergence. We show that “close-to-diagonal” and other sequences of Hermite-Padé approximants converge in capacity to (f ₁(z),..., f_n (z)) inside the common circle of convergence of eachf _i (z),i=1,...,n.     

A note on the interlacing of zeros and orthogonality

Let be a sequence of monic polynomials with deg(t_n)=n such that, for each n∈N, the zeros of t_n are real and simple and t_n and t_n+1 have no common zeros. We discuss the connection between the orthogonality of the sequence, the positivity of a certain ratio, and the interlacing of the zeros of t_n and t_n+1 for n≥1, n∈N.     

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L ²(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup.     

Zeros of Jacobi polynomials Pn( α, β)$ P_{n}^{(\alpha ,\beta)} $, − 2 < α, β < − 1

The sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) is orthogonal on (??1,1) with respect to the weight function (1 ? x)^α(1 + x)^β provided α > ??1,β > ??1. When the parameters α and β lie in the narrow range ??2 < α, β < ??1, the sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) is quasi-orthogonal of order 2 with respect to the weight function (1 ? x)^α+?1(1 + x)^β+?1 and each polynomial of degree n,n ≥?2, in such a Jacobi sequence has n real zeros. We show that any sequence of Jacobi polynomials \(\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }\) with ??2 < α, β < ??1, cannot be orthogonal with respect to any positive measure by proving that the zeros of Pn??1(α,β) do not interlace with the zeros of Pn(α,β) for any \(n \in \mathbb {N},\)n ≥?2, and any α,β lying in the range ??2 < α, β < ??1. We also investigate interlacing properties satisfied by the zeros of equal degree Jacobi polynomials Pn(α,β),Pn(α,β+?1) and Pn(α+?1,β+?1) where ??2 < α, β < ??1. Upper and lower bounds for the extreme zeros of quasi-orthogonal order 2 Jacobi polynomials Pn(α,β) with ??2 < α, β < ??1 are derived.