Showing papers in "Tohoku Mathematical Journal in 1982"

Helicoidal surfaces with constant mean curvature

Abstract: We describe the space ƒ°H of all surfaces in R3 that have constant mean curvature H•‚0 and are invariant by helicoidal motions, with a fixed axis, of R3. Similar to the case ƒ°0 of minimal surfaces ƒ°H behaves roughly like a circular cylinder where a certain generator corresponds to the rotation surfaces and each parallel corresponds to a periodic family of isometric helicoidal surfaces.

A criterion for uniform integrability of exponential martingales

Abstract: Let (Ω, F, P) be a complete probability space equipped with a nondecreasing right continuous family (Ft) of sub σ-fields of F such that Fo contains all null sets. We shall use the notations given in Meyer [5]. Let M be a local martingale with Mo = 0, M c its continuous part and (M) the increasing process associated with M. We put ΔM9 = Af. — M,_ and assume the condition AM. > — 1 throughout this note. Denote the exponential martingale of M by ί?(Λf), that is, &(M)t = exp{ikft — (l/2)t + (log(l + x) — x)-μt}9 where μ is the integer valued random measure associated with jumps of M. As is well-known, l?(Λf) is a positive supermartingale with S?(Λf)o = 1 but it is not always a uniformly integrable martingale. Girsanov [1] raised the problem of finding a sufficient condition for the process i?(ΛΓ) to be a uniformly integrable martingale. The purpose of this paper is to establish the following.

A local property of absolutely convergent Jacobi polynomial series

Abstract: ^We then form the normalized polynomials R{na>β)(x) = P^β){x)IP^β\l), so that sup.^^i \R{f>β)(x)\ = 1, Vw ^ 0. We let AJ(α, /9, 0) denote the space of series f(x) = Σ»=o αJ^'^OiO subject to the condition Σ*=o|αJ < °° The main result of Chapter 2 of this paper states that if /e AJ(α, βf 0) and if 0 < e < τr/2 then on [e, π — e] we can write