The "positivity phenomenon" for Bessel sequences, frames and Riesz bases $\left(u_k\right)$ are studied in $L^2$ spaces over the compacts of homogeneous (Coifman-Weiss) type $\Omega=(\Omega, \rho, \mu)$. Under some relations between three basic metric-measure dimensions of $\Omega$, we obtain asymptotics for the mass moving norms $\left\|u_k\right\|_{K R}$ (Kantorovich-Rubinstein), as well as for singular numbers of the Lipschitz and Hajlasz-Sobolev embeddings. Our main observation shows that, quantitatively, the rate of the convergence $\left\|u_k\right\|_{K R} \longrightarrow 0$ depends on an interplay between geometric doubling and measure doubling/halving exponents. The "more homogeneous" is the space, the sharper are the results.[-]

I shall discuss the theory of multi-norms. This has connections with norms on tensor products and with absolutely summing operators. There are many examples, some of which will be mentioned. In particular we shall describe multi-norms based on Banach lattices, define multi-bounded operators, and explain their connections with regular operators on lattices. We have new results on the equivalences of multi-norms. The theory of decompositions of Banach lattices with respect to the canonical 'Banach-lattice multi-norm' has a pleasing form because of a substantial theorem of Nigel Kalton that I shall state and discuss. I shall also discuss brie y a generalization that gives 'pmulti-norms' (for $1\leq p\leq1$) and an extension of a representation theorem of Pisier that shows that many pmulti-norms are 'sous-espaces de treillis'. The theory is based on joint work with Maxim Polyakov (deceasead), Hung Le Pham (Wellington), Matt Daws (Leeds), Paul Ramsden (Leeds), Oscar Blasco (Valencia), Niels Laustsen (Lancaster), Timur Oikhberg (Illinois), and Vladimir Troitsky (Edmonton).

multi-norms - equivalences - absolutely summing operators - tensor products[-]

Let $X$ be a Banach space of holomorphic functions on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_{n} :X\rightarrow X$ with the property that, for each $f\in X$, the functions $T_{n}\left ( f \right )$ are polynomials converging to $f$ in the norm of the space. We completely characterize those spaces $X$ that admit a linear polynomial approximation scheme. In particular, we show that it is not sufficient merely that polynomials be dense in $X$. (Joint work with Javad Mashreghi).[-]