Abstract:
The problem of multiple interpolation is considered in the class [ρ(r),∞)+ of functions of at most normal type for the proximate order ρ(r) in the upper half-plane
C+:f(k−1)(an)=bn,k, k=1,…,qn, n=1,2,…, where the divisor D={an,qn} has limit points only on the real axis, and the numbers {bn,k} satisfy the condition
¯limn→∞r−ρ(rn)nlnsup1⩽k⩽qn(Λn)k−1|bn,k|(k−1)!<∞.
The following result is valid.
Theorem. {\it D is an interpolation divisor in the class [ρ(r),∞)+ if and only if
¯limn→∞r−ρ(rn)nlnqn!|E(qn)(an)|(Λn)k<∞,
where E(z) is the canonical product of the set D}.
Necessary and sufficient conditions are also found in terms of the measure determined by the divisor D: μ(G)=∑an∈Gqnsin(argan).
Citation:
K. G. Malyutin, “The problem of multiple interpolation in the half-plane in the class of analytic functions of finite order and normal type”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 253–266
\Bibitem{Mal93}
\by K.~G.~Malyutin
\paper The problem of multiple interpolation in the~half-plane in the~class of analytic functions of finite order and normal type
\jour Russian Acad. Sci. Sb. Math.
\yr 1994
\vol 78
\issue 1
\pages 253--266
\mathnet{http://mi.mathnet.ru/eng/sm968}
\crossref{https://doi.org/10.1070/SM1994v078n01ABEH003468}
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Linking options:
https://www.mathnet.ru/eng/sm968
https://doi.org/10.1070/SM1994v078n01ABEH003468
https://www.mathnet.ru/eng/sm/v184/i2/p129
This publication is cited in the following 11 articles:
M. V. Kabanko, K. G. Malyutin, “Interpolation sets in spaces of functions of finite order in half–plane”, Ufa Math. J., 16:3 (2024), 40–53
M. V. Kabanko, K. G. Malyutin, “Interpolirovanie metodom Erla v prostranstve funktsii poluformalnogo poryadka”, Trudy Voronezhskoi zimnei matematicheskoi shkoly S. G. Kreina — 2024, SMFN, 70, no. 4, Rossiiskii universitet druzhby narodov, M., 2024, 597–609
K. G. Malyutin, A. A. Naumova, “Predstavlenie subgarmonicheskikh funktsii v polukoltse i v polukruge”, Chebyshevskii sb., 24:5 (2023), 136–152
Malyutin K. Kabanko M., “Multiple Interpolation By the Functions of Finite Order in the Half-Plane”, Lobachevskii J. Math., 41:11, SI (2020), 2211–2222
B. V. Vynnyt'skyi, V. L. Sharan, I. B. Sheparovych, “On an interpolation problem in the class of functions of exponential type in a half-plane”, Ufa Math. J., 11:1 (2019), 19–26
K. G. Malyutin, A. L. Gusev, “Geometric meaning of the interpolation conditions in the class of functions of finite order in the half-plane”, Probl. anal. Issues Anal., 8(26):3 (2019), 96–104
K. G. Malyutin, A. L. Gusev, “The interpolation problem in the spaces of analytical functions of finite order in the half-plane”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 113–123
K. G. Malyutin, “Interpolation Problems of A. F. Leontiev Type”, J. Math. Sci. (N. Y.), 252:3 (2021), 399–419
O. A. Bozhenko, K. G. Malyutin, “Problem of multiple interpolation in class of analytical functions of zero order in half-plane”, Ufa Math. J., 6:1 (2014), 18–28
K. G. Malyutin, “Sets of regular growth of functions in the half-plane. I”, Izv. Math., 59:4 (1995), 785–814
K. G. Malyutin, “Sets of regular growth of functions in a half-plane. II”, Izv. Math., 59:5 (1995), 983–1006