Recently in this Journal J. Esterlé gave a new proof of the Wiener Tauberian theorem for $L^1\left(\mathbf{R}\right)$ using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane. We here use essentially the same method to prove the analogous result for Beurling algebras $L_{\phi }^1\left(\mathbf{R}\right)$. Our estimates need a theorem of Hayman and Korenblum.

We construct two Banach algebras, one which contains analytic semigroups ${\left(a^z\right)}_{Rez>0}$ such that $|a^{1+iy}|\to \infty$ arbitrarily slowly as $\left|y\right|\to \infty$, the other which contains ones such that $|a^{1+iy}|\to \infty$ arbitrarily fast

In analogy to the analyticity condition $\parallel A{e}^{tA}\parallel \le C{t}^{-1}$, t > 0, for a continuous time semigroup ${\left(e^{tA}\right)}_{t\ge 0}$, a bounded operator T is called analytic if the discrete time semigroup ${\left(T^n\right)}_{n\in \mathbb{N}}$ satisfies $\parallel (T-I)T^n\parallel \le C{n}^{-1}$, n ∈ ℕ. We generalize O. Nevanlinna’s characterization of powerbounded and analytic operators T to the following perturbation result: if S is a perturbation of T such that $\parallel R({\lambda }_0,T)-R({\lambda }_0,S)\parallel$ is small enough for some ${\lambda }_0\in \varrho \left(T\right)\cap \varrho \left(S\right)$, then the type $\omega$ of the semigroup $\left(e^{t(S-I)}\right)$ also controls the analyticity of S in the sense that $\parallel (S-I)S^n\parallel \le C(\omega +n^{-1})e^{\omega n}$, n ∈ ℕ. As an application...

Let $𝒜$ be a homogeneous algebra on the circle and ${𝒜}^{+}$ the closed subalgebra of $𝒜$ of functions having analytic extensions into the unit disk $D$. This paper considers the structure of closed ideals of ${𝒜}^{+}$ under suitable restrictions on the synthesis properties of $𝒜$. In particular, completely characterized are the closed ideals in ${𝒜}^{+}$ whose zero sets meet the circle in a countable set of points. These results contain some previous results of Kahane and Taylor-Williams obtained independently. ...

Assume $f_1,...,f_N$ a finite set of functions in $H^{\infty }\left(D\right)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty }\left(D\right)$ to belong to the norm-closure of the ideal $I(f_1,...,f_N)$ generated by $f_1,...,f_N$, namely the property $$\left|f\right(z\left)\right|\le \alpha \left(\right|f_1\left(z\right)|+...+|f_N\left(z\right)\left|\right)\text{for}z\in D$$ for some function $\alpha$ : ${\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ satisfying ${lim}_{t\to 0}\alpha \left(t\right)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property $$\left|f\left(z\right)\right|\le C\underset{1\le j\le N}{max}\left|f_j\left(z\right)\right|\text{for}z\in D$$ ...

We consider a singularity perturbed nonlinear differential equation $\epsilon u^{\text{'}}=f\left(x\right)u++\epsilon P(x,u,\epsilon )$ which we suppose real analytic for $x$ near some interval $[a,b]$ and small $\left|u\right|$, $\left|\epsilon \right|$. We furthermore suppose that 0 is a turning point, namely that $xf\left(x\right)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $ϵ\to 0$) local (at $x=0$) or global, real analytic or $C^{\infty }$ solutions is equivalent to the existence of a formal series solution $\sum u_n\left(x\right){\epsilon }^n$ with $u_n$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for...

A scattered element of a Banach algebra is an element with at most countable spectrum. The set of all scattered elements is denoted by (). The scattered radical ${}_{sc}\left(\right)$ is the largest ideal consisting of scattered elements. We characterize in several ways central elements of modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) () + () ⊂ (); (ii) ()() ⊂ (); (iii) $[\left(\right),]{\subset }_{sc}\left(\right)$.