In the early 1930s, Wiener proved that if f(x) is a strictly positive periodic function whose Fourier series is absolutely convergent, then the Fourier series of g(x)=1/f(x) is also absolutely convergent [8, pp. 10–14]. This phenomenon can be easily understood nowadays using Banach algebra techniques (see, for example, [4, pp. 202–203]). In fact, these techniques allow us to study the absolute convergence of g(x)=F(f(x)), where F is holomorphic in an open subset of [Copf ] that contains the range of f(x) (for x∈ℝ). In this context, Wiener's original problem corresponds to the choice F(z)=1/z.

In this work we want to analyse the constraints on the simultaneous rate of vanishing of the Fourier coefficients fˆ(n) and ĝ(n) as n→∞. We shall focus on g=1/f, but we shall also study the general case g=F(f). In either case, there are obviously no constraints when f is a constant function.

Although this problem does not seem to be directly related to uncertainty inequalities for the Fourier Transform, we observe that there are some analogies, both in the nature of the results and in the proof techniques. The general fact with which we are dealing is that fˆ(n) and ĝ(n) cannot vanish too quickly at the same time as n→∞, unless f(x) is constant. The general fact that underlies uncertainty inequalities is that a non-periodic function ϕ(x) and its Fourier Transform ϕcirc;(u) cannot vanish too quickly at the same time as x→∞ and u→∞, unless ϕ(x) is zero (almost everywhere). For a simple introduction to some aspects of uncertainty inequalities, see [5]; for a thorough and recent introduction to this vast subject, see [3].

Let G be either a free product with amalgamation A[midast ]CB or an HNN group A[midast ]C, where all normal subgroups of C are finitely generated. Suppose that both A and B have no non-trivial finitely generated normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which intersects A or B non-trivially but is not contained in C, then the index of N in G is finite.

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central.

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

Let Ω be the set {1, 2, [ctdot ], n}, and let [emptyv ] be the empty set. Let [Gscr ] be the family of all non-empty sets of subsets of Ω. For [Ascr ]∈[Gscr ] and X⊆Ω, put

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An important discovery is the following.

It is proved that, if $P$ is a polynomial with integer coefficients, having degree 2, and $1>\varepsilon>0$, then $n(n-1)\cdots(n-k+1)=P(m)$ has only finitely many natural solutions $(m,n,k)$, $n\ge k>n\varepsilon$, provided that the $abc$ conjecture is assumed to hold under Szpiro's formulation.

A unital C*-algebra $A$ is said to have cancellation of projections if the semigroup $D(A)$ of Murray–von Neumann equivalence classes of projections in matrices over $A$ is cancellative. It has long been known that stable rank one implies cancellation for any $A$, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras.