This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication by i. It is natural to consider other kinds of complex multiplications and here we shall make a start on the case when the ring of complex multiplications is isomorphic to the ring Z[σ], where σ2 = − 2.

Let G be a locally compact Abelian group; Γ the dual group of G; CB(Γ) the algebra of continuous, bounded functions on Γ C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; M(G) the algebra of bounded Borel measures on G; M(G)^ the algebra of Fourier–Stieltjes transforms; and M(G)^− the completion of M(G)^ in the sup-norm topology on Γ. The object of this paper is to study the natural pairing between M(G)^ and M(Γ).

Suppose that

belongs to L2( − π, π). If most of the coefficients vanish then f (x) cannot be too small in a certain interval without being small generally. More precisely Ingham ((2), Theorem 1) has proved the following

THEOREM A. Suppose that f (x) is given by (1·1) and that an = 0, except for a sequence n = nν, where nν+1 − nν ≥ C. Then given ∈ > 0 there exists a constant A (∈), such that we have for any real x1

We denote by M(G) the space of bounded regular borel measures on a non-discrete locally compact Abelian topological group G. Under the convolution product and norm of total mass M(G) becomes a complex commutative banach algebra. We denote by Φ the space of m.l.f.s (multiplicative linear functionals) on M(G) and by σ the subset of Φ consisting of functionals symmetric in the sense that

where * denotes the usual involution on M(G). These matters are discussed more fully in Williamson (1).

In 1966, Hewitt and Zuckerman(3,4) proved that if G is a non-discrete locally compact Abelian group with Haar measure λ, then there exists a non-negative, continuous regular measure μon G that is singular to λ(μ ┴ λ) such that μ(G)= 1, μ * μ is absolutely continuous with respect to λ(μ * μ ≪ λ), and the Lebesgue-Radon-Nikodym derivative of μ * μ with respect to λ is in (G, λ) for all real p > 1. They showed also that such a μ can be chosen so that the support of μ * μ contains any preassigned σ-compact subset of G. It is the purpose of the present paper to extend this result to obtain large independent sets of such measures. Among other things the present results show that, for such groups, the radical of the measure algebra modulo the -algebra has large dimension. This answers a question (6.4) left open in (3).

A paranormed space X = (X, g) is a topological linear space in which the topology is given by paranorm g—a real subadditive function on X such that g(θ) = 0, g(x) = g(−x) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space X and continuity of multiplication means that λn → λ, xn → x(i.e. g(xn − x) → 0) imply λnxn → λx, for scalars λ and vectors x. We shall use the term semimetric function to describe a real subadditive function g on X such that g(θ) = 0, g(x) = g(−x). Two familiar paranormed sequence spaces, which have been extensively studied (3), are l(p) and m(p). For a given sequence p = (gk) of strictly positive numbers, l;(p) is the set of all complex sequences x = (xk) such that and m(p) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 < pk ≤ 1 so that natural paranorms would seem to be in m(p). In fact Simons showed that g1 was a paranorm for l(p), but that g2 did not satisfy the continuity of multiplication axiom.

We use J(a, b) to denote a Jordan curve of positive two-dimensional measure in the plane, with end-points a and b. If υ is a point of J(a, b), we define the right lower arc density at υ by

where J( υ, υ′) is the largest arc, whose left-end point is υ, which is contained in the disc c(υ, r).

The object of this paper is to prove a conjecture made by Robinson (1) concerning cyclotomic integers, that is, integers that are rationally expressible in terms of roots of unity.

Some years back, Slater (4) discussed the approximations, based on the expansion in series, for the cases 1F1(a; b; x) = 0, when either of b and x or a and x are fixed. These approximations were based essentially on the well-known Newton's method of approximation and were helpful in the numerical evaluation of the small real zeros of the confluent hypergeometric function 1F1(a; b; x;). In this note, we deal with the corresponding problem for the basic confluent hypergeometric function 1Φ1(a; b; x;).

Given a locally convex Hausdorff linear topological space, we construct and examine the following topologies in the space:

the finest locally convex topology with the same convergent sequences as the initial topology, and the finest locally convex topology with the same precompact sets as the initial topology.

In a recent paper (1) the authors considered some generalizations of Cauchy's inequality. The method of approach was by the construction of certain convex functions (where by a convex function we mean a function f(x) satisfying

for every pair of unequal values x1 and x2). For example, it was shown that if (a), (b) are the sets of non-negative real numbers a1, …,an; bl, …, bn, the function

which had been used earlier by Callebaut(2), is a convex function with minimum at x = 0, except if the sets (a) and (b) are proportional, in which case the function is a constant. This function takes the value (σab)2 when x = 0 and (σa2) (Σb2) when x = 1, and the property of convexity gives Cauchy's inequality.

1. Suppose that f is a function analytic on a region G in complex n-space Cn, and that f(m)(w(m)) = 0 for each m = (m1, m2,…, mn), mi = 0, 1, 2,…, where

We obtain a simple new proof of Bonsall's characterization theorem for type 1 semi-algebras of continuous functions. An extension of our methods produces several new results on stability properties of wedges.

Let f(θ) be integrable L in (−π, π) and periodic with period 2π and let

The writer in his paper (4) has shown that

where the Jacobi polynomial is defined as ((5), p. 255)

.

We will here prove the following special case of an (unpublished) conjecture for a polynomial of degree ndue to L. Ilieff:

Theorem. If all zeros of the cubic polynomial p3(z) lie in |z| ≤ 1, then at least one zero of p3′(z) lies in or on the boundary of a circle of radius unity around each zero of p′3(z).

Little is known about definite integrals for the Whittaker functions. The object of this paper is to establish such results and the following formulae are established

where the symbol means that in the expression following it i is to be replaced by –i and the two expressions are to be added.

Suppose that λ > − 1 and that

It is easy to show that

With Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the series

is convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.

This paper is concerned with estimates of the coefficients Cn for normalized polynomials, inverse to univalent functions in the unit disc, i.e. univalent polynomials of the form

considered in the largest domain D such that 0 ∈ D and |pn(w)| < 1 for w ∈ D; this domain will be denoted by . Polynomials of the type (1) have been introduced by Charzyński in (1), where it was proved among other results that the domain is simply connected.

The object of this paper is to obtain some new generating functions for Jacobi polynomials. The results obtained, are of general character.