Related papers: Equality of uniform and Carleman spectra for bound…

We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions $F \in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$ and compare them with other spectra including the weak…

Let $G$ be a locally compact abelian topological group. For locally bounded measurable functions $\varphi: G\to\Bbb {C}$ we discuss notions of spectra for $\varphi$ relative to subalgebras of $L^{1}(G)$. In particular we study polynomials…

We revisit the notion of reduced spectra $sp_{\Cal {F}} (\phi)$ for bounded measurable functions $\phi \in L^{\infty} (J,\Bbb{X})$, ${\Cal {F}}\subset L^1_{loc}(J,\Bbb{X})$. We show that it can not be obtained via Carleman spectra unless…

In this paper, we describe families of those bounded linear operators on a separable Hilbert space that are simultaneously unitarily equivalent to integral operators on $L_2(R)$ with bounded and arbitrarily smooth Carleman kernels. The main…

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large…

For a metric space $X$, we study the space $D^{\infty}(X)$ of bounded functions on $X$ whose infinitesimal Lipschitz constant is uniformly bounded. $D^{\infty}(X)$ is compared with the space $\LIP^{\infty}(X)$ of bounded Lipschitz functions…

On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large…

Given a finite measure space $(\Omega,\Sigma,\mu)$, we show that any Banach space $X(\mu)$ consisting of (equivalence classes of) real measurable functions defined on $\Omega$ such that $f \chi_A \in X(\mu) $ and $ \|f \chi_A \| \leq \|f\|,…

We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators.…

A charge space $(X,\mathcal{A},\mu)$ is a generalisation of a measure space, consisting of a sample space $X$, a field of subsets $\mathcal{A}$ and a finitely additive measure $\mu$, also known as a charge. Key properties a real-valued…

Let $\varphi: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb R^n)$ weight uniformly in $t$. In this article, the authors introduce…

It is shown that there exist Banach spaces $X,Y$, a $1$-net $\mathscr{N}$ of $X$ and a Lipschitz function $f:\mathscr{N}\to Y$ such that every $F:X\to Y$ that extends $f$ is not uniformly continuous.

We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions $F \in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$, where $\jj$ is $\r_+$ or $\r$ and $X$ is a Banach…

A bounded function $\phi: G\to \C$ on an LCA group $G$ is called Hartman measurable if it can be extended to a Riemann integrable function $\phi^*: X\to \C$ on some group compactification $(\iota_X,X)$, i.e. on a compact group $X$ such that…

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the…

Let $X$ be a Banach space, let $(\Omega,\mu)$ be a $\sigma$-finite measure space and let $A,B\colon\Omega\to B(X)$ be strongly measurable $\gamma$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert…

We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known…

The concept of measurability of functions on a charge space is generalised for functions taking values in a uniform space. Several existing forms of measurability generalise naturally in this context, and new forms of measurability are…