Hereditary completeness for systems of exponentials in weighted $L^2$-spaces
Andrei V. Semenov
TL;DR
The paper proves that for any weight $w$ with $1/w\in\mathcal{K}$ there exists a complete and minimal system of exponentials in $L^2(w)$ that is not hereditarily complete, extending the unweighted result of Baranov–Belov–Borichev to weighted spaces. The authors recast the problem in reproducing-kernel Hilbert spaces $\mathcal{F}_w$, construct a generating function $G$, and build normalized kernels $\mathbb{K}_{\lambda}$ along with a pair of functions $F$ and $H$ that enforce orthogonality on a partition $\Lambda=\Lambda_1\cup\Lambda_2$ while keeping $(H,F)\neq 0$. Through a careful balance of lacunary kernel sums and sine terms, they show that the mixed system is not complete, thereby demonstrating non-hereditary completeness in the weighted setting. The results rely on precise asymptotics for the kernel norm $K(y)$ and the Legendre transform $\tilde{h}$, and they bridge spectral synthesis considerations with weighted reproducing-kernel methods, indicating fundamental limits for reconstruction in $L^2(w)$ spaces with polynomially decaying weights.
Abstract
We prove that for a weight $w$, which has at least polynomial decay, there exists a complete and minimal system $\{e^{iλ_n t}\}_{n\in \mathbb{N}}$ of exponentials in weighted space $L^2(w)$ on $(-π,π)$, which is not hereditarily complete.