Bases in invariant subspaces (Q2455216)

If a convolution equation is a linear differential equation with constant coefficients, then each solution to the homogeneous equation can be represented as a linear combination of elementary solutions of the form \(z^ne^{\lambda_kz}\). The possibility of such representation is known as the fundamental principle of Euler. In a more general setting, we can consider \(H^*(D)\) the strong dual of \(H(D)\), \(W^0=\{ \overline{\mu} \in H^*(D): \overline{\mu} (\varphi )=0\), \(\varphi \in W\}\), \( \Lambda = \{ \lambda_k\), \(m_k \}\) the spectrum of \(W\) and \(E_\Lambda =\{ z^n e^{\lambda_k z}\}_{k=1,n=0}^{\infty, m_k-1} \). The fundamental principle problem asks for conditions under which each function from \(W\) can be expanded in elements of \(E_\Lambda\). This problem has been studied by several authors and, under a natural constraint on \(\Lambda \), it has been completely solved for all closed invariant subspaces admiting spectral synthesis. However it is easy to provide examples for which \(E_\Lambda\) is not a basis in \(W\), but any function from \(W\) can be expanded in a ``parenthesized'' series. Then the following problem arises: Divide \(\{\lambda_k\}_1^\infty\) into finite pairwise disjoint groups \(U_p=\{\lambda_{k_p}, \dots , \lambda_{k_{p+1}-1} \}\) and let \(M_p\) be the sum of multiplicities \(m_i\) of all points \(\lambda_i \in U_p\). Under what conditions every function \(\varphi \in W\) can be expanded in a series \[ \varphi (z)=\sum{p=1,\;n=1}^{\infty , \;M_p} d_{p,n} e_{p,n}(z) \tag{1} \] where \(e_{p,n}(z) = \sum_{i=1}^{k_{p+1}-k_p} \sum_{j=1}^{m_{i-1+k_p}} c_{pnij} z^{j-1} e^{\lambda_{i-1+k_p} z}\) does not depend on \(\varphi\). Solving this problem requires answering the questions: how to construct the groups \(U_p\)?, how to determinate the coefficients \(c_{pnij}\)? and the space of sequences \((d_{pn})\)?, in what topology the series (1) must converge?. Summarizing results of other papers the ``size'' of \(U_p\) was determined and all the authors used the same method to construct \(U_p\): by using some Laplace transform of an element of \(W^0\). Also partial solutions for the other questions were provided. In this paper the author, under a natural constraint on \(\{M_p\}\), solves completely the questions. At the same time an intrinsic description of \(U_p\) in terms of \(\lambda_k\) is given, without use of any Laplace transform of an element of \(W^0\).