×

Asymptotics of Kolmogorov diameters for some classes of harmonic functions of spheroids. (English) Zbl 1030.46024

Author’s abstract: “Let \(\Gamma^K_D\) be the unit ball of the space of all bounded harmonic functions in a domain \(D\) in \(\mathbb{R}^3\), considered as a compact subset of the Banach space \(C(K)\), where \(K\) is a compact subset of \(D\). The old problem about the exact asymptotics for Kolmogorov diameters (widths) of this set, \[ \ln d_k(\Gamma^K_D)\sim -\tau k^{1/2},\qquad k\to\infty, \] is solved positively in the case when \(K\) and \(D\) are closed and open confocal spheroids, respectively (i.e., prolate or oblate ellipsoids of revolution). Using some special asymptotic formulas for the associated Legendre functions \(P^m_n(\cosh\sigma)\) as \(n\to\infty\) and \(m/n\to\gamma\in [0, 1]\) (considered earlier by the second author), we show that the constant \(\tau\) is some averaged characteristic of the pair of spheroids, expressed by means of a certain function of the variable \(\gamma\), which appears within those asymptotics. Unlike the corresponding problem for analytic functions, quite well investigated, the harmonic functions case has been studied, up to now, only in the case of concentric balls.”
Some remarks: The holomorphic case is well investigated only in \(\mathbb{C}\). In \(\mathbb{C}^n\) and for Reinhardt domains, the problem was solved recently by Aytouna, Rashkovskii and Zahariuta. For harmonic functions in \(\mathbb{R}^2\), one can solve the problem for rather general condensers \((\Omega,K)\) (for instance, \(K\) is a compact of Jordan type) by use of a“good” common Schauder base.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babenko, K. I., On the entropy of a class of analytic functions, Nauchn. Dokl. Vyssh. Shkoly Ser. Fiz.-Mat. Nauk, 2, 9-13 (1958)
[2] Brelot, M., Eléments de la Théorie Classique du Potentiel (1959), Les cours de Sorbonne: Les cours de Sorbonne Paris · Zbl 0084.30903
[3] de Bruijn, N. G., Asymptotic Methods in Analysis (1970), North-Holland: North-Holland Amsterdam · Zbl 0082.04202
[4] Erdelyi, A., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York
[5] Erokhin, V. D., Estimates of \(ε\)-entropy and linear widths of some classes of analytic functions, Researches in Modern Problems of Function Theory of a Complex Variable (1961), Fizmatgiz: Fizmatgiz Moscow, p. 159-167
[6] Erokhin, V. D., On the best linear approximation of functions, extended analytically from a given continuum to a given domain, Russian Math. Surveys, 23, 91-119 (1968)
[7] Fisher, S. D.; Micchelli, Ch. A., The n-widths of sets of analytic functions, Duke Math. J., 47, 789-801 (1980) · Zbl 0451.30032
[8] Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics (1931), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0004.21001
[9] Kolmogorov, A. N., Über die Annäherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math., 37, 107-110 (1936) · Zbl 0013.34903
[10] Kolmogorov, A. N., Information Theory and Theory of Algorithms. Information Theory and Theory of Algorithms, Collected Works (1987), Nauka: Nauka Moscow · Zbl 0642.94001
[11] Landkof, N. S., Foundations of Modern Potential Theory (1966), Nauka: Nauka Moscow · Zbl 0253.31001
[12] Levin, A. L.; Tikhomirov, V. M., On theorem of V. D. Erokhin, Russian Math. Surveys, 23, 119-132 (1968)
[13] Mityagin, B. S., Approximative dimension and bases in nuclear spaces, Russian Math. Surveys, 16, 59-127 (1961) · Zbl 0104.08601
[14] Mityagin, B. S.; Tikhomirov, V. M., Asymptotical characteristics of compacta in linear spaces, Proceedings of Fourth Mathematical Congress (1965), Nauka: Nauka Moscow, p. 299-308
[15] Nguyen, T. V., Bases de Schauder dans certains espaces de fonctions holomorphes, Ann. Inst. Fourier, 22, 169-253 (1972) · Zbl 0226.46025
[16] Nguyen, T. V., Bases communes pour certains espaces de fonctions harmoniques, Bull. Sci. Math. Ser. 2, 97, 33-49 (1973) · Zbl 0265.46025
[17] Nguyen, T. V.; Djebbar, B., Propriétés asymptotiques d’une suite orthonormale de polynomes harmoniques, Bull. Sci. Math. Ser. 2, 113, 239-261 (1989) · Zbl 0685.42010
[18] Pinkus, A., \(n\)-Widths in Approximation Theory (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0551.41001
[19] Skiba, N. I., On Hilbert scales of spaces of harmonic functions of two real variables, Actual Problems of Mathematical Analysis (1978), Rostov State University: Rostov State University Rostov-on-Don, p. 161-166
[20] Tikhomirov, V. M., Approximation theory, Analysis 2. Analysis 2, Encyclopaedia of Math. Sciences, 14 (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0728.41016
[21] Tikhomirov, V. M., Some Problems of Approximation Theory (1976), Moscow State University: Moscow State University Moscow · Zbl 0156.13602
[22] Vitushkin, A. G., Estimation of Tabulation Complexity (1959), Fizmatgiz: Fizmatgiz Moscow · Zbl 0087.07103
[23] Widom, H., Rational approximation and \(n\)-dimensional diameter, J. Approx. Theory, 5, 343-361 (1972) · Zbl 0234.30023
[24] Zahariuta, V. P., On extandable bases in spaces of analytic functions, Siberian Math. J., 8, 204-216 (1967)
[25] Zahariuta, V. P., Isomorphism of spaces of harmonic functions, Math. Anal. and Appl. (1971), Rostov State University: Rostov State University Rostov-on-Don, p. 152-158
[26] Zahariuta, V. P., Spaces of harmonic functions, Functional Analysis. Functional Analysis, Lecture Notes in Pure and Applied Mathematics, 150 (1993), Dekker: Dekker New York, p. 497-522 · Zbl 0792.31001
[27] Zahariuta, V. P., Spaces of analytic functions and complex potential theory, Linear Topological Spaces and Complex Analysis, I (1994), METU-TÜBITAK: METU-TÜBITAK Ankara, p. 74-146 · Zbl 0859.30041
[28] V. P. Zahariuta, Hadamard-type inequalities for harmonic functions and their applications, to appear.; V. P. Zahariuta, Hadamard-type inequalities for harmonic functions and their applications, to appear. · Zbl 1032.31004
[29] Zahariuta, V. P.; Skiba, N. I., Estimates of \(n\)-diameters of some classes of functions, analytic on Riemann surfaces, Math. Notes, 19, 525-532 (1976) · Zbl 0375.46028
[30] Zeriahi, A., Bases communes dans certains espaces de fonctions harmoniques et fonctions separement harmoniques sur certains ensembles de \(C^n\), Ann. Fac. Sci. Toulouse, 6, 75-102 (1982) · Zbl 0504.31002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.