By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, is called inner or outer if multiplication by this defines an isometry dense range, respectively, as linear operator on Hardy Space, H2, functions with square-summable Taylor series. This factorization can be further refined; θ decomposes uniquely p...

We prove that if G is an Abelian p-group of length not exceeding ω and H is its pω+n-projective subgroup for n ∈ N ∪ {0} such that G/H is countable, then G is also pω+n-projective. This enlarges results of ours in (Arch. Math. (Brno), 2005, 2006 and 2007) as well as a classical result due to Wallace (J. Algebra, 1971). Unless we do not specify some else, by the term “group” we mean “an Abelian ...

It is obvious that lp (/> 2:1) or any Hubert space satisfies (A). In §3 it is shown that Lp (p>l) does likewise. The method of proof is entirely different from that of Clarkson. In §4 it is shown that if X is any Banach space having the property that every function on a linear interval to X, which satisfies a Lipschitz condition, is differentiable almost everywhere, then also every function of ...

Fourier orthogonal series with respect to the weight function (1 − |x|2)μ−1/2 on the unit ball in Rd are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourie...

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions x α1−1/2 1 · · · xαd−1/2 d (1 − |x|1)αd+1−1/2 on the standard simplex Σd in Rd. It is proved that such an expansion is uniformly (C, δ) summable on the simplex for any continuous function if and only if δ > |α|1 + (d − 1)/2. Moreover, it is shown that (C, |α|1 + (d + 1)/2) means def...