This paper deals with a class of interpolation series of the form

called R-series. It is equiconvergent with the Dirichlet series

If the nth Legendre polynomial for the interval (0,1) is denoted by (−1)nLn(t), then the bilinear formula

serves as generating function for the Rn(z). It also leads in easy steps to R-series expansions for rational functions.

Lagrange [7] has shown that a function holomorphic and of finite rate of growth in a right half-plane can be expanded in an R-series whose abscissa of convergence is limited by the rate of growth of f(z). The converse problem is attacked in Theorem 2 below where it is shown that

Following an earlier paper (Curle 1978) we consider a compressible laminar boundary layer with uniform pressure when the distance x along the wall satisfies x < x0 and a prescribed large adverse pressure gradient when x > x0. The viscosity and absolute temperature are again taken to be proportional, but the Prandtl number is no longer assumed to be unity. After applying the Illingworth-Stewartson transformation, the transformed external velocity u1(x) is chosen so that

is large and constant, where Ts is the stagnation temperature, Tw is the (constant) Wall temperature and u0 is the upstream value of u1(x).

The flow reacts to this sharp pressure rise mainly in a thin inner sublayer, so inner and outer asymptotic expansions are derived and matched for functions F and S which determine the stream function and the temperature.

The skin friction, heat transfer, displacement thickness and momentum thickness are determined as functions of , andinvolve two parameters B1, B2, which depend upon the Mach number and the walltemperature. Detailed numerical calculations are presented here for σ = 0.72. In particular, it is seen that the heat transfer rate varies roughly like σ⅓ except near to separation, where it varies like σ¼.

We give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.

We consider the operator L[y] = y(4) + ((ax2 + bx + c)y′)′ + dy on the half-line [0, ∞). This paper shows that the deficiency indices are independent of the real numbers b, c and d when a ≠ 0. They depend only on the sign of a and are (2,2) if a < 0 and (3, 3) if a > 0. In the case a =0 the sign of b must be considered.

This paper is devoted to the study of some non linear Schrödinger equations in two dimensions, arising in non linear optics; in particular, it is concerned with solutions to the Cauchy problem. The problem of global existence and regularity of the solutions, the asymptotic behaviour of global solutions, and the blow-up of non global solutions are studied.

We consider the expansion of a function in Lr (the class of measurable functions whose rth powers are Lebesgue integrable over some interval) in terms of the eigenfunctions arising from a singular Sturm-Liouville problem defined over an infinite or semi-infinite interval. We show that if l ≦ r ≦ inline1 or if r ≧ 4 there exists f in Lr whose eigenfunction expansion is divergent in the rth mean sense, and that the terms of the series form an unbounded sequence in Lr The result extends some work of Askey and Wainger concerning the Hermite series expansions of functions in Lr(–∞, ∞).