Tensor products of approximation spaces associated with regular elliptic operators on compact manifolds

M. I. Dmytryshyn

doi:10.30970/ms.64.2.170-178

M. I. Dmytryshyn Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0002-3248-7736

DOI: https://doi.org/10.30970/ms.64.2.170-178

Keywords: approximation spaces, regular elliptic operators, tensor products

Abstract

The article describes the tensor products of approximation spaces associated with regular elliptic operators on tensor products of Lebesgue spaces $L_2(\partial\Omega)$, where $\partial \Omega$ considers as smooth manifold that describes in the usual way by local system of local coordinates. We use the quasi-normed approximation spaces and subspaces of exponential type functions associated with such operators.} A connection between the tensor products of approximation spaces and interpolation spaces obtained by the real method of interpolation is showed. We prove the direct and inverse approximation theorems for Bernstein–Jackson type inequalities as well as we give the explicit dependence of constants on parameters of approximation spaces. Such constants are expressed via some normalization factor. Application to spectral approximations on tensor products of interpolation spaces associated with regular elliptic operators on compact manifolds is shown. In the article also consider the spectral approximations (Theorem 2), since the subspaces of entire functions of exponential type of regular elliptic operators on compact manifolds coincide with their spectral subspaces (Lemma 3).

References

J. Bergh, J. Lofstrom, Interpolation spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

S. Chandler-Wilde, D. Hewett, A. Moiola, Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples, Mathematika, 61 (2015), №2, 414–443. https://doi.org/10.1112/S0025579314000278

M. Dmytryshyn, Tensor products of exponential type vectors of unbounded operators, Int. J. Math. Anal., 8 (2014), 529–538. https://doi.org/10.12988/ijma.2014.4233

M. Dmytryshyn, Besov-Lorentz-type spaces and best approximations by exponential type vectors, Int. J. Math. Anal., 9 (2015), 779–786. https://doi.org/10.12988/ijma.2015.5233

M. Dmytryshyn, Approximation by interpolation spectral subspaces of operators with discrete spectrum, Mat. Stud., 55 (2021), №2, 162–170. https://doi.org/10.30970/ms.55.2.162-170

M. Dmytryshyn, L. Dmytryshyn, Bernstein-Jackson-type inequalities with exact constants in Orlicz spaces, Carpathian Math. Publ., 14 (2022), №2, 364–370. https://doi.org/10.15330/cmp.14.2.364-370

M. Dmytryshyn, L. Dmytryshyn, Projective tensor products of approximation spaces associated with positive operators, Res. Math., 32 (2024), №1, 33–44. https://doi.org/10.15421/242403

M. Dmytryshyn, O. Lopushansky, Bernstein-Jackson-type inequalities and Besov spaces associated with unbounded operators, J. Inequal. Appl., 2014 (2014), №105, 1–12. https://doi.org/10.1186/1029-242X-2014-105

M. Dmytryshyn, O. Lopushansky, On spectral approximations of unbounded operators, Complex Anal. Oper. Theory, 13 (2019), №8, 3659–3673. https://doi.org/ 10.1007/s11785-019-00923-0

S.I. Fedynyak, P.V. Filevych, Bernstein-type inequalities for analytic functions represented by power series, Mat. Stud., 62 (2024), №2, 121–131. https://doi.org/10.30970/ms.62.2.121-131

P.V. Filevych, Asymptotic estimates for entire functions of minimal growth with given zeros, Mat. Stud., 62 (2024), №1, 54–59. https://doi.org/10.30970/ms.62.1.54-59

S. Giulini, Bernstein and Jackson theorems for the Heisenberg group, J. Austral. Math. Soc., 38 (1985), 241–254. https://doi.org/10.1017/S1446788700023107

M.L. Gorbachuk, Ya.I. Hrushka, S.M. Torba, Direct and inverse theorems in the theory of approximation by the Ritz method, Ukrainian Math. J., 57 (2005), №5, 751–764. https://doi.org/10.1007/s11253-005-0225-4

E. Hille, R.S. Phillips, Functional analysis and semigroups, Providence, Rhode Island: American Mathematical Society, Colloquium Publications, V.31, 1957.

L. Hormander, Linear partial differential operators, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

J.L. Lions, E. Magenes, Problemes aux limites non homogenes et applications. V.1, Dunod, Paris, 1968.

O. Lopushansky, Bernstein-Jackson inequalities on Gaussian Hilbert spaces, J. Fourier Anal. Appl., 29:58 (2023). https://doi.org/10.1007/s00041-023-10035-1

O.V. Lopushansky, M.I. Dmytryshyn, Vectors of exponential type of operators with discrete spectrum, Mat. Stud., 9 (1998), №1, 70–77.

S. Nikolskii, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, Berlin-Heidelberg-New York, 1975.

J. Peetre, G. Sparr, Interpolation of normed abelian groups, Ann. Mat. Pura ed Appl., 92 (1972), №1, 217–262. https://doi.org/10.1007/BF02417949

J. Prestin, V.V. Savchuk, A.L. Shidlich, Direct and inverse theorems on the approximation of 2π-periodic functions by Taylor–Abel–Poisson operators, Ukrainian Math. J., 69 (2017), №5, 766—781. https://doi.org/10.1007/s11253-017-1394-7

G.V. Radzievskii, On the best approximations and rate of convergence of decompositions in the root vectors of an operator, Ukrainian Math. J., 49 (1997), №6, 844–864. https://doi.org/10.1007/BF02513425

H. Triebel, Interpolation theory. Function spaces. Differential operators, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1978.