Let $X$ be a complex Banach space with an unconditional Schauder basis. We prove that the monomials associated with this basis, endowed in each homogeneous degree with the square order and globally with any compatible ordering, form a Schauder basis for the space $(\calH(X),\tau_0)$ of entire holomorphic functions on $X$ with the compact-open topology. The proof relies on two main ideas. First, coordinate-tail conditions yield a fundamental system of compact solid subsets of $X$. Second, Fourier projections in the coordinates give the uniform estimate $c_{K,n}\leq n+1$ for the basis constant of the degree-$n$ monomials with respect to the supremum seminorm on each such compact set $K$. We derive applications to Banach sequence lattices with dense $c_{00}$, including classical, Lorentz, Orlicz-heart and Schreier-type sequence spaces, and to general vector-valued $E$-sums. In particular, the result covers mixed $c_0$- and $\ell_r$-sums of finite-dimensional $\ell_p$ spaces and the Lorentz predual $d_*(w,1)$.
We study the domains of absolute convergence of monomial expansions of holomorphic functions on Banach sequence spaces. We first show that, in every complex Banach sequence space $X$, each point outside the closed linear span of the coordinate unit vectors admits a continuous quadratic polynomial whose monomial expansi...
Let $M$ be a simple matroid and let $B(M)$ be the graded M\"obius algebra of its lattice of flats. The ordered-basis weights of flats define an inner product for which the adjoints of atom multiplication become ordinary coordinate derivatives under the basis-polynomial realization. From this, we construct a canonical g...
Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex $(n+1)$-s...
Xi-Nan Dai, W.-D. Deng, Yi-Dong Shi et al.· 0 citations
Let $\Gamma$ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For $f$ belonging to the reduced group $C^*$ algebra $C_r^*(\Gamma)$, we construct a rearrangement of its individual Fourier terms with a subsequence converging to $f$ in the operator norm. For every $2\le p<\infty$ and $f$...
We establish the uniform doubling property for $G$-invariant metrics on a wide class of compact Riemannian homogeneous spaces $(G/K,g)$, describing explicitly the volume growth of the metric balls $B_g(p,r)$. This property was previously known only for specific cases, including abelian Lie groups, the Lie group $SU(2)$...
We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings. For rational maps of fixed degree, the optimal centered Sobolev trace constant $A_f$ is a continuous proper function on M\"obius moduli space. Uniform bounds on $A_f$ therefore characterize relative compactness in mod...
Alastair N. Fletcher, I. Krishtal· 0 citations
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