Some bases in spaces of regular functions and their application to interpolation

V. A. Oskolkov

Sbornik. Mathematics, Tome 34 (1978) no. 2, pp. 215-234

Résumé

Systems of functions $\{\underset tL{}_n[\Phi(tz)]\}_0^\infty$ are considered, where $\Phi(z)=\sum_0^\infty a_nz^n$ ($a_n\ne0$, $n=0,1,\dots$) is an entire function, $$ L_n[F]=\frac{n!}{2\pi i}\int_{|z|=r_n>\max\limits_{0\leqslant k\leqslant n}|\lambda_{k,n}|}\frac{F(z)\,dz}{(z-\lambda_{0,n})\cdots (z-\lambda_{n,n})}\qquad(n=0,1,\dots), $$ and the matrix $(\lambda_{k,n})$, $k=0,1,\dots,n$, $n=0,1,\dots$, is given. Under various assumptions on the matrix, theorems are proved which deal with the question of whether the systems $\{\underset tL{}_n[\Phi(tz)]\}_0^\infty$ form a basis in the spaces $A(|z|<R)$. They are conclusive in the sense that they cannot be improved without changing the hypotheses. The basis theorems are applied to Gel'fond and Abel–Goncharov interpolation problems, which makes it possible to study the distribution of zeros of sequences of derivatives of certain classes of entire functions. Bibliography: 16 titles.

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