Application of the real Hardy--Sobolev space on the line for finding the best rational approximations in $L_p$

T. S. Mardvilko

Trudy Instituta matematiki, Tome 32 (2024) no. 2, pp. 31-42.

Résumé

This work is dedicated to developing methods of the real Hardy–Sobolev space on the line for finding the best rational approximations in the $L_p$ space. The methods considered are based on representing a function of this space as a sum of simple functions and the application of a Cauchy-type integral. Sufficient conditions for a function's membership in the considered space have been obtained and inequalities for assessing the corresponding $\sigma$-norm have been proven. Using the obtained results, exact order estimates of the best rational approximations of certain functions have been found. In particular, from the obtained results, the well-known estimate of the best rational approximations of a function of bounded variation follows.

Détail

Keywords: Hardy space, Sobolev space, Hardy–Sobolev space, rational approximation, $L_p$-approximations, functions of bounded variation.

Bibliographie

[1] Garnett J. B., Bounded analytic function, Revised first edition, Springer, New York, 2007 | MR

[2] Coifman R. R., Weiss G., “Extensions of Hardy spaces and their use in analysis”, Bulletin of the American Mathematical Society, 83:4 (1977), 569–645 | DOI | MR

[3] Grafakos L., Modern Fourier analysis, Graduate Texts in Mathematics, 2nd ed., Springer, New York, 2009 | DOI | MR

[4] Krotov V. G., “Differential properties of boundary functions of Hardy spaces”, Mathematische Nachrichten, 126:1 (1986), 241–253 (in Russian) | DOI | MR

[5] Stelmakh A. I., “Direct and inverse theorems of rational approximation for Hardy spaces in the half-plane”, Doklady of the National Academy of Sciences of Belarus, 52:6 (2008), 36–41 (in Russian) | MR

[6] Mardvilko T. S., Pekarskii A. A., “Application of the real Hardy–Sobolev space on the line to study the order of uniform rational approximations of functions”, Journal of the Belarusian State University. Mathematics and Informatics, 2022, no. 3, 16–36 (in Russian) | MR

[7] Pekarskii A. A., “Estimates of the derivatives of rational functions in $L_p[-1,1]$”, Math. Notes, 39:3 (1986), 388–394 (in Russian) | MR

[8] Lorentz G. G., Golitschek M. V., Makovoz Y., Constructive Approximation. Advanced Problems, Springer-Verlag, New York–Berlin–Heidelberg, 1996 | MR

[9] Phillips G. M., “Error estimates for best polynomial approximation”, Approximation theory, Academic Press, London, 1979, 1–6 | MR

[10] DeVore R. A., Lorentz G. G., Constructive Approximation: polinomials and splines approximation, Springer-Verlag, New York–Berlin–Heidelberg, 1993 | MR

[11] Stechkin S. B., “On the order of the best approximations of continuous functions”, Izv. Akad. Nauk SSSR Ser. Mat., 15:3 (1951), 219–242 (in Russian)

[12] King F. W., Hilbert transform, v. 1, University Press, Cambridge, 2009 | MR

[13] Petrushev P. P., Popov V. A., Rational approximation of real functions, University Press, Cambridge, 1987 | MR

[14] Pekarskii A. A., “Rational approximation of singular functions”, Izv. NAS BSSR. Ser. fiz.-mat. navuk, 1980, no. 3, 32–40 (in Russian)