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 Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 6, Pages 1322–1384 (Mi izv1968)

Basicity of some biorthogonal systems and the solution of a multiple interpolation problbm in the $H^p$ classes in the half-plane

M. M. Dzhrbashyan

Abstract: Let $\{\lambda_k\}_1^\infty$ be a sequence in $G^{(+)}=ż:\operatorname{Im}z>0\}$, and $s_k$ the multiplicity of the occurrences of $\lambda_k$ in the segment $\{\lambda_1,…,\lambda_k\}$. Also let $H_+^p$ $(1<p<+\infty)$ be the space of functions $f(z)$ holomorphic in $G^{(+)}$ that obey
$$\|f\|_p=\sup_{0<y<+\infty}\{\int^{+\infty}_{-\infty}|f(x+iy)|^p dx\}^{1/p}<\infty.$$
The article gives a completely internal characterization of systems of the form $\{r_k(z)=\frac{(s_k-1)!}{(z-\overline\lambda_k)^{s_k})}\}^\infty_{k+1}$ that are not total in $H^p_+$ and of the biorthogonal systems $\{\Omega_k(z)\}_1^\infty$ constructed for such nontotal systems. The closed linear hulls of the systems $\{r_k(z)\}_1^\infty$ and $\{\Omega_k(z)\}_1^\infty$ are also characterized. Criteria for these systems to be bases in their closed linear hulls in the metric of $H^p_+$ are obtained. A complete and effective solution of the multiple interpolation problem in the classes $H_+^p$ is given. In addition it is proved that functions with given interpolation data can be represented both as an interpolation series in the system $\{\Omega_k(z)\}_1^\infty$ and as a series in the system $\{r_k(z)\}_1^\infty$.
Bibliography: 20 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1979, 13:3, 589–646

Bibliographic databases:

UDC: 517.5
MSC: Primary 30B60; Secondary 30D55, 30E05

Citation: M. M. Dzhrbashyan, “Basicity of some biorthogonal systems and the solution of a multiple interpolation problbm in the $H^p$ classes in the half-plane”, Izv. Akad. Nauk SSSR Ser. Mat., 42:6 (1978), 1322–1384; Math. USSR-Izv., 13:3 (1979), 589–646

Citation in format AMSBIB
\Bibitem{Dzh78} \by M.~M.~Dzhrbashyan \paper Basicity of some biorthogonal systems and the solution of a~multiple interpolation problbm in the $H^p$ classes in the half-plane \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1978 \vol 42 \issue 6 \pages 1322--1384 \mathnet{http://mi.mathnet.ru/izv1968} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=522941} \zmath{https://zbmath.org/?q=an:0443.30042|0425.30027} \transl \jour Math. USSR-Izv. \yr 1979 \vol 13 \issue 3 \pages 589--646 \crossref{https://doi.org/10.1070/IM1979v013n03ABEH002078} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1979JG49100004} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. N. U. Arakelian, A. G. Vitushkin, V. S. Vladimirov, A. A. Gonchar, “Mkhitar Mkrtichevich Dzhrbashyan (on his sixtieth birthday)”, Russian Math. Surveys, 34:2 (1979), 269–275
2. M. M. Dzhrbashyan, “A characterization of the closed linear spans of two families of incomplete systems of analitic functions”, Math. USSR-Sb., 42:1 (1982), 1–70
3. M. M. Dzhrbashyan, V. M. Martirosyan, “Integral representations and best approximation by generalized polynomials in systems of Mittag-Leffler type”, Math. USSR-Izv., 23:3 (1984), 449–471
4. G. M. Gubreev, “Spectral analysis of biorthogonal expansions of functions, and exponential series”, Math. USSR-Izv., 35:3 (1990), 573–605
5. V. B. Dybin, “The Wiener–Hopf equation and Blaschke products”, Math. USSR-Sb., 70:1 (1991), 205–230
6. M. S. Martirosyan, “Summirovanie biortogonalnogo razlozheniya po nepolnoi sisteme ratsionalnykh funktsii v poluploskosti”, Uch. zapiski EGU, ser. Fizika i Matematika, 1998, no. 2, 3–11
7. M. S. Martirosyan, S. V. Samarchyan, “$q$-Bounded systems: Common approach to Fisher–Micchelli's and Bernstein–Walsh's type problems”, Lobachevskii J. Math., 25 (2007), 197–216
8. Eugenia G. Rodikova, “On interpolation in the class of analytic functions in the unit disk with the Nevanlinna characteristic from Lp-spaces”, Zhurn. SFU. Ser. Matem. i fiz., 9:1 (2016), 69–78
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