On Stability of Bases Made from Perturbed Exponential Systems in Morrey Type Spaces

Perturbed exponential system {eiλkχ}keZ (where {λn} is some sequence of real numbers) isconsidered in Morrey spaces Lp,α (0, π) These spaces arenon-separable (except for exceptional cases), and thereforethe above system is not complete in them. Based on theshift operator, we define the subspace Mp,a (0, π)C Lp,α (0, π) where continuous functions aredense. We find a condition on the sequence {λn} which issufficient for the above system to form a basis for thesubspace Mp,a (0, π). Our results are the analogues ofthose obtained earlier for the Lebesgue spaces Lp. Wealso establish an analogue of classical Levinson theorem onthe completeness of above system in the spaces Lp,1 <= p <=+∞

MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE

For a class of sets with multiple terms$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots,\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1,$$ in weighted \(L^2 (-\infty, \infty)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,\, z\in \mathbb{C}. $$ The proof depends on our previous work where we characterized the closed span of the exponential system \(\{t^k e^{\lambda_n t}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,\mu_n-1\}\) in weighted \(L^2 (-\infty, \infty)\) spaces, and also derived a sharp upper bound for the norm of elements of a biorthogonal sequence to the exponential system. The proof also utilizes notions from Non-Harmonic Fourier series such as Bessel and Riesz–Fischer sequences.