Spaces of series in system of functions - Sheremeta, M. M. (Matematychni Studii, 2023)
The Banach and Fr\’echet spaces of series A(z)=_n=1^∞a_nf(λ_nz) regularly converging in C, where f is an entire transcendental function and (λ_n) is a sequence of positive numbers increasing to +∞, are studied. Let M_f(r)={|f(z)|: |z|=r}, Γ_f(r)=dM_f(r)dr, h be positive continuous function on [0,+∞) increasing to +∞ and S_h(f,Λ) be a class of the function A such that |a_n|M_f(λ_nh(λ_n)) → 0 as n→+∞. Define \|A\|_h={|a_n|M_f(λ_nh(λ_n)):n1}. It is proved that if n=o(Γ_f(λ_n)) as n→∞ then (S_h(f,Λ),\|·\|_h) is a non-uniformly convex Banach space which is also separable. In terms of generalized orders, the relationship between the growth of M r,A)=_n=1^∞ |a_n|M_f(rλ_n), the maximal term μ(r,A)= {|a_n|M_f(rλ_n)n1} and the central index u(r,A)= {n1|a_n|M_f(rλ_n)=μ(r,A)} and the decrease of the coefficients a_n. The results obtained are used to construct Fr\’echet spaces of series in systems of functions.