The paper studies the properties of tight integral frames of a special kind. We are considering (p,μ)-frames (the definition is given in the article) in reproducing kernel Hilbert space (RKHS). The relationship between the properties of (p,μ)-frames with the condition of similarity of RKHS has been established. The case of a finite linear combination of such frames is studied. We get the necessary conditions for the equivalence of special-type RKHS.
Keywords: frame, tight frame, integral frame, reproducing kernel Hilbert space, orthosimilar decomposition system
Received September 24, 2025
Revised February 3, 2026
Accepted February 9, 2026
Funding Agency: The research of V.V. Napalkov (Jr.) was carried out with the support of the Ministry of Science and Higher Education of the Russian Federation as part of the state assignment (scientific topic code FMRS-2025-0010).
Valerii Valentinovich Napalkov, Dr. Phys.-Math. Sci., Institute of Mathematics, Ufa Federal Research Centre, RAS, Ufa, 450077 Russia, e-mail: vnap@mail.ru
Andrey Alexandrovich Nuyatov, Ph.D. Math., Nizhny Novgorod State Technical University, Nizhny Novgorod, 603950 Russia, e-mail: nuyatov1aa@rambler.ru
REFERENCES
1. Aronszajn N. Theory of reproducing kernels. Trans. Amer. Math. Soc., 1950, vol. 68, no. 3, pp. 337–404. https://doi.org/10.1090/S0002-9947-1950-0051437-7
2. Berlinet A., Thomas–Agnan C. Reproducing kernel Hilbert spaces in probability and statistics. NY, Springer, 2004. 355 p. https://doi.org/10.1007/978-1-4419-9096-9
3. Saitoh S., Sawano Y. Theory of reproducing kernel and application. Ser. Developments in Mathematics, vol. 44. Singapore, Springer, 2016, 452 p. https://doi.org/10.1007/978-981-10-0530-5
4. Lukashenko T.P. Properties of expansion systems similar to orthogonal ones. Izv. Math., 1998, vol. 62, no. 5, pp. 1035–1054. https://doi.org/10.1070/IM1998v062n05ABEH000215
5. Lukashenko T.P. Coefficients with respect to orthogonal-like decomposition systems. Sb. Math., 1997, vol. 188, no. 12, pp. 1783–1798. https://doi.org/10.1070/sm1997v188n12ABEH000273
6. Duffin R.J., Schaeffer A.C. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 1952, vol. 72, pp. 341–366. https://doi.org/10.1090/S0002-9947-1952-0047179-6
7. Young R.M. An introduction to nonharmonic Fourier series. NY, Acad. Press, 1980, 246 p. ISBN: 0127728503
8. Daubechies I., Grossmann A. Frames in the Bargmann space of entire functions. Comm. Pure Appl. Math., 1988, vol. 41, pp. 151–164.
9. Seip K. Density theorems for sampling and interpolation in the Bargmann–Fock space I. J. Reine Angew. Math., 1992, vol. 429, pp. 91–106. https://doi.org/10.1515/crll.1992.429.91
10. Kashin B.S., Kulikova T.Yu. A note on the description of frames of general form. Math. Notes, 2002, vol. 72, no. 6, pp. 863–867. https://doi.org/10.1023/A:1021402315905
11. Novikov S.Ya., Sevost’yanova V.V. Maltsev equal-norm tight frames. Izv. Math., 2022, vol. 86, no. 4, pp. 770–781. https://doi.org/10.4213/im9137e
12. Mallat S. A wavelet tour of signal processing: the sparse way. San Diego etc., Acad. Press, 1999, 637 p. ISBN: 0-12-466606-X . Translated to Russian under the title Veivlety v obrabotke signalov, Moscow, Mir Publ., 2005, 671 p. ISBN: 5-03-003691-1 .
13. Shayne F., Waldron D. An introduction to finite tight frames. Boston, Birkhäuser, 2018, 587 p. https://doi.org/10.1007/978-0-8176-4815-2
14. Zakharova A.A. On the properties of generalized frames. Math. Notes, 2008, vol. 83, iss. 2, pp. 190–200. https://doi.org/10.1134/S0001434608010215
15. Dunford N., Schwartz J. Linear operators: General theory, New York, London, Intersci. Publ., 1958, 872 p. ISBN: 9780470226056 . Translated to Russian under the title Lineinye operatory: Obshchaya teoriya, Moscow, Inostr. Liter. Publ., 1962, 895 p.
16. Napalkov V.V. (Jr.) Orthosimilar expansion systems in space with reproducing kernel. Ufa Math. J., 2013, vol. 5, no. 4, pp. 88–100. https://doi.org/10.13108/2013-5-4-88
17. Napalkov V.V. (Jr.), Nuyatov A.A. On a condition for the coincidence of transform spaces for functionals in a Hilbert space. Tr. Inst. Mat. Mekh. UrO RAN, 2022, vol. 28, no. 3, pp. 142–154 (in Russian).
https://doi.org/10.21538/0134-4889-2022-28-3-142-154
18. Napalkov V.V., Nuyatov A.A. On the question of the coincidence of Hilbert spaces square-integrable with respect to the measure of functions. Tr. MFTI, 2023, vol. 15, no. 3, pp. 39–49 (in Russian).
19. Napalkov V.V., Napalkov V.V. (Jr.) On isomorphism of reproducing kernel Hilbert spaces. Dokl. Math., 2017, vol. 95, no. 3, pp. 270–272. https://doi.org/10.1134/S1064562417030243
Cite this article as: V.V. Napalkov (Jr.), A.A. Nuyatov. Tight integral frames and similarity conditions of reproducing kernel Hilbert spaces. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 146–163.
