In a Hilbert space H with an unconditional basis of reproducing kernels, we study the existence of a weighted integral norm with respect to an absolutely continuous measure, which is equivalent to the original H-norm. If the space H is defined via weighted integrals, the problem can be interpreted as restoring the original structure.
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N. Aronszajn, “Theory of reproducing kernels”, Trans. Am. Math. Soc. 68 No. 3, 337–404 (1950).
S. V. Khrushchev, N. K. Nikol’skij, and B. S. Pavlov, “Unconditional bases of exponentials and of reproductional kernels”, Lect. Notes Math. 864, 214–335 (1981).
K. Seip, “Density theorems for sampling and interpolation in the Bargmann–Fock space. I”, J. Reine Angew. Math. 429, 91–106 (1992).
K. Seip and R. Wallsten, “Density theorems for sampling and interpolation in the Bargmann–Fock space. II”, J. Reine Angew. Math. 429, 107–113 (1992).
A. Borichev and Yu. Lyubarskii, “Riesz bases of reproducing kernels in Fock type spaces”, J. Inst. Math. Jussieu 9, No. 3, 449–461 (2010).
A. Baranov, A. Dumont, A. Hartmann. amd K. Kellay, “Sampling, interpolation and Riesz bases in small Fock spaces”, J. Math. Pures Appl. 103, No. 6, 1358–1389 (2015).
A. Baranov, Yu. Belov, and A. Borichev, “Fock type spaces with Riesz bases of reproducing kernels and de Branges spaces”, Stud. Math. 236, No. 2, 127–142 (2017).
B. Ya. Levin and Yu. I. Lyubarskij, “Interpolation by entire functions of special classes and related expansions into series of exponentials” [in Russian], Izv. AN SSSR, Ser. Mat. 39, No. 3, 657–702 (1975).
K. P. Isaev, “Riesz bases of exponents in Bergman spaces on convex polygons” [in Russian], Ufim. Math. Zhurn. 2 (2010), No. 1, 71–86.
B. S. Pavlov, “Basicity of an exponential system and Muchkehoupt’s condition,” Sov. Math. Dokl. 20, 655–659 (1979).
K. P. Isaev and R. S. Yulmukhametov, “On unconditional bases of exponentials in Hilbert spaces” [in Russian], Ufim. Math. Zhurn. 3, No. 1, 3–15 (2011).
K. P. Isaev and R. S. Yulmukhametov, “Unconditional bases of reproducing kernels in Hilbert spaces of entire functions” [in Russian], Ufim. Math. Zhurn. 5, No. 3, 67–77 (2013).
K. P. Isaev, R. S. Yulmukhametov, “On Hilbert spaces of entire functions with unconditional bases of reproducing kernels”, Lobachevskii J. Math. 40, No. 9, 1283–1294 (2019).
V. I. Lutsenko and R. S. Yulmukhametov, “ A generalization of the Paley–Wiener theorem to functionals on Smirnov spaces,” Proc. Steklov Inst. Math. 200, 271–280 (1993).
K. P. Isaev and R. S. Yulmukhametov, “Laplace transform of functionals on Bergman spaces” Izv. Math. 68, No. 1, 3–41 (2004).
M. V. Fedoryuk, The Saddle-Point Method [in Russian], Nauka, Moscow (1977).
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Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 111-120.
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Isaev, K.P., Trunov, K.V. & Yulmukhametov, R.S. Equivalent Norms in Hilbert Spaces with Unconditional Bases of Reproducing Kernels. J Math Sci 250, 310–321 (2020). https://doi.org/10.1007/s10958-020-05017-3
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DOI: https://doi.org/10.1007/s10958-020-05017-3