Approximation by simple partial fractions and the Hilbert transform

V. Yu. Protasov

Izvestiya. Mathematics , Tome 73 (2009) no. 2, pp. 333-349.

Résumé

We study the problem of approximation of functions in $L_p$ by simple partial fractions on the real axis and semi-axis. A simple partial fraction is a rational function of the form $g(t)=\sum_{k=1}^n\frac1{t-z_k}$, where $z_1,\dots,z_n$ are complex numbers. We describe the set of functions that can be approximated by simple partial fractions within any accuracy and the set of functions that can be approximated by convex combinations of them (the cone of simple partial fractions). We obtain estimates for the norms of simple partial fractions and conditions for the convergence of function series $\sum_{k=1}^\infty\frac1{t-z_k}$ in the space $L_p$. Our approach is based on the use of the Hilbert transform and the methods of convex analysis.

Détail

Keywords: approximation, simple partial fraction, convergence of function series, entire function, logarithmic derivative.
Mots-clés : Hilbert transform

Bibliographie

[1] V. I. Danchenko, “Estimates of the distances from the poles of logarithmic derivatives of polynomials to lines and circles”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 425–440 | DOI | MR | Zbl

[2] V. I. Danchenko, D. Ya. Danchenko, “Approximation by simplest fractions”, Math. Notes, 70:3–4 (2001), 502–507 | DOI | MR | Zbl

[3] O. N. Kosukhin, “Approximation properties of the most simple fractions”, Moscow Univ. Math. Bull., 56:4 (2001), 36–40 | MR | Zbl

[4] P. A. Borodin, O. N. Kosukhin, “Approximation by the simplest fractions on the real axis”, Moscow Univ. Math. Bull., 60:1 (2005), 1–6 | MR | Zbl

[5] A. D. Polyanin, A. V. Manzhirov, Handbook of integral equations, 2nd edition, CRC, Boca Raton, FL, 2008 | MR | Zbl | Zbl

[6] S. K. Pichorides, “On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov”, Studia Math., 44 (1972), 165–179 | MR | Zbl

[7] F. D. Gakhov, Boundary value problems, Pergamon, Oxford–New York–Paris; Addison-Wesley, Reading, MA–London, 1966 | MR | MR | Zbl | Zbl

[8] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., 32, Princeton Univ. Press, Princeton, NJ, 1971 | MR | Zbl | Zbl

[9] J. M. Borwein, A. S. Lewis, Convex analysis and nonlinear optimization. Theory and examples, CMS Books Math./Ouvrages Math. SMC, 3, Springer-Verlag, New York, 2000 | MR | Zbl

[10] G. G. Magaril-Il'yaev, V. M. Tikhomirov, Convex analysis: theory and applications, Transl. Math. Monogr., 222, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[11] A. Barvinok, A course in convexity, Grad. Stud. Math., 54, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[12] J. B. Garnett, Bounded analytic functions, Pure Appl. Math., 96, Academic Press, New York–London, 1981 | MR | MR | Zbl | Zbl

[13] A. N. Kolmogorov, S. V. Fomin, Introductory real analysis, Prentice-Hall, Englewood Cliffs, NJ, 1970 | MR | MR | Zbl | Zbl

[14] A. I. Markushevich, Teoriya analiticheskikh funktsii, t. 2. Dalneishee postroenie teorii, 2-e izd., Nauka, M., 1968 | Zbl

[15] P. A. Borodin, “Estimates of the distances to direct lines and rays from the poles of simplest fractions bounded in the norm of $L_p$ on these sets”, Math. Notes, 82:5–6 (2007), 725–732 | DOI | MR | Zbl

[16] I. G. Petrovsky, Lectures on partial differential equations, Interscience Publ., New York–London, 1954 | MR | MR | Zbl