Kolmogorov inequalities for functions in classes $W^rH^\omega$ with bounded $\mathbb L_p$-norm

S. K. Bagdasarov

Izvestiya. Mathematics , Tome 74 (2010) no. 2, pp. 219-279.

Résumé

We find the general solution and describe the structural properties of extremal functions of the Kolmogorov problem $\|f^{(m)}\|_{\mathbb L_\infty(\mathbb I)}\to\sup$, $f\in W^r\!H^\omega\!(\mathbb I)$, $\|f\|_{\mathbb L_p(\mathbb I)}\le B$, for all $r,m\in\mathbb Z$, $0\le m\le r$, all $p$, $1\le p\infty$, concave moduli of continuity $\omega$, all positive $B$ and $\mathbb I=\mathbb R$ or $\mathbb{I}=\mathbb R_+$, where $W^rH^\omega(\mathbb I)$ is the class of functions whose $r$th derivatives have modulus of continuity majorized by $\omega$. We also obtain sharp constants in the additive (and multiplicative in the case of Hölder classes) inequalities for the norms $\|f^{(m)}\|_{\mathbb L_\infty(\mathbb I)}$ of the derivatives of functions $f\in W^rH^\omega(\mathbb I)$ with finite norm $\|f^{(r)}\|_{\mathbb L_p(\mathbb I)}$. We also investigate some properties of extremal functions in the special case $r=1$ (such as the property of being compactly supported) and obtain inequalities between the knots of the corresponding $\omega$-splines. In the case of the Hölder moduli of continuity $\omega(t)=t^\alpha$, we find that the lengths of the intervals between the knots of extremal $\omega$-splines decrease in geometric progression while the graphs of the solutions exhibit the fractal property of self-similarity.

Détail

Keywords: Kolmogorov–Landau inequalities, moduli of continuity.

Bibliographie

[1] A. N. Kolmogorov, “O neravenstvakh mezhdu verkhnimi granyami posledovatelnykh proizvodnykh funktsii na beskonechnom intervale”, Uchenye zapiski MGU, 30 (1939), 3–16 | MR | Zbl

[2] A. N. Kolmogorov, Selected works of A. N. Kolmogorov. Vol. I. Mathematics and mechanics, Math. Appl. (Soviet Ser.), 25, Kluwer Acad. Publ., Dordrecht, 1991 | MR | MR | Zbl | Zbl

[3] A. S. Cavaretta, jr., “An elementary proof of Kolmogorov's theorem”, Amer. Math. Monthly, 81:5 (1974), 480–486 | DOI | MR | Zbl

[4] I. J. Schoenberg, A. Cavaretta, “Solution of Landau's problem concerning higher derivatives on the halfline”, Constructive theory of functions, Proc. of the Intern. Conf. (Varna, 1970), 1, Izdat. Bolgar. Akad. Nauk, Sofia, 1972, 297–308 | MR | Zbl

[5] S. Karlin, “Some one-sided numerical differentiation formulae and applications”, Studies in spline functions and approximation theory, Academic Press, New York, 1976, 485–500 | MR | Zbl

[6] V. M. Tihomirov, “Best methods of approximation and interpolation of differentiable functions in the space $C_{[-1,1]}$”, Math. USSR-Sb., 9:2 (1969), 275–289 | DOI | MR | Zbl

[7] V. M. Tikhomirov, Nekotorye problemy teorii priblizhenii, Izd-vo MGU, M., 1976 | MR

[8] V. N. Malozemov, A. B. Pevnyi, Polinomialnye splainy, Izd-vo LGU, L., 1986 | MR | Zbl

[9] G. G. Magaril-Il'yaev, “Inequalities for derivatives and duality”, Proc. Steklov Inst. Math., 161 (1984), 199–212 | MR | Zbl

[10] M. K. Kwong, A. Zettl, Norm inequalities for derivatives and differences, Lecture Notes in Math., 1536, Springer-Verlag, Berlin, 1992 | DOI | MR | Zbl

[11] D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Inequalities involving functions and their integrals and derivatives, Math. Appl. (East European Ser.), 53, Kluwer, Dordrecht, 1991 | MR | Zbl

[12] V. F. Babenko, N. P. Korneichuk, V. A. Kofanov, S. A. Pichugov, Neravenstva dlya proizvodnykh i ikh prilozheniya, Naukova dumka, Kiev, 2003

[13] V. V. Arestov, V. N. Gabushin, “Best approximation of unbounded operators by bounded ones”, Russian Math. (Iz. VUZ), 39:11 (1996), 38–63 | MR | Zbl

[14] V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126 | DOI | MR | Zbl

[15] D. Jackson, Über die Genauigkett des Annäherung stetigen Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrischen Summen gegebener Ordnung, Diss., Göttingen, 1911

[16] N. P. Korneichuk, Ekstremalnye zadachi teorii priblizhenii, Nauka, M., 1976 | MR

[17] N. P. Korneichuk, “S. M. Nikol'skii and the development of research on approximation theory in the USSR”, Russian Math. Surveys, 40:5 (1985), 83–156 | DOI | MR | Zbl | Zbl

[18] N. Korneǐchuk, Exact constants in approximation theory, Encyclopedia Math. Appl., 38, Cambridge Univ. Press, Cambridge, 1991 | MR | MR | Zbl | Zbl

[19] C. M. Nikolskii, “Ryad Fure funktsii s dannym modulem nepreryvnosti”, Dokl. AN SSSR, 52:3 (1946), 191–194 | MR | Zbl

[20] S. K. Bagdasarov, “Zolotarev $\omega$-polynomials in $W^rH^\omega[0,1]$”, J. Approx. Theory, 90:3 (1997), 340–378 | DOI | MR | Zbl

[21] N. I. Achieser, Theory of approximation, Ungar Publ., New York, 1956 | MR | MR | Zbl | Zbl

[22] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, Selected works. Vol. 4. The mathematical theory of optimal processes, Classics Soviet Math., Gordon Breach, New York, 1986 | MR | MR | Zbl | Zbl

[23] S. Karlin, “Oscillatory perfect splines and related extremal problems”, Studies in spline functions and approximation theory, Academic Press, New York, 1976, 371–460 | MR | Zbl

[24] S. Karlin, Total positivity, Stanford University Press, Stanford, CA, 1968 | MR | Zbl

[25] S. K. Bagdasarov, “Maximization of functionals in $H^\omega [a,b]$”, Sb. Math., 189:2 (1998), 159–226 | DOI | MR | Zbl

[26] S. K. Bagdasarov, “Extremal functions of integral functionals in $H^\omega[a,b]$”, Izv. Math., 63:3 (1999), 425–480 | DOI | MR | Zbl

[27] S. K. Bagdasarov, “Generalizations of the time optimal problem and Lyapunov theorem on the range of vector measures”, J. Approx. Theory, 147:1 (2007), 81–111 | DOI | MR | Zbl

[28] S. K. Bagdasarov, Chebyshev splines and Kolmogorov inequalities, Oper. Theory Adv. Appl., 105, Birkhäuser, Basel, 1998 | MR | Zbl

[29] S. K. Bagdasarov, “General construction of Chebyshev $\omega$-splines with given norm”, St. Petersburg Math. J., 10:6 (1999), 965–996 | MR | Zbl

[30] S. K. Bagdasarov, “Kolmogorov problem in $W^rH^\omega[0,1]$ and extremal Zolotarev $\omega$-splines”, Dissertationes Math. (Rozprawy Mat.), 379 (1998), 1–81 | MR | Zbl

[31] F. H. Clarke, Optimization and nonsmooth analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley, New York, 1983 | MR | MR | Zbl | Zbl

[32] K. Borsuk, “Drei Sätze über die $n$-dimensionale euklidische Sphäre”, Fund. Math., 20 (1933), 177–190 | Zbl

[33] R. A. DeVore, H. Kierstead, G. G. Lorentz, “A proof of Borsuk's theorem”, Functional analysis (Austin, TX, 1986–87), Lecture Notes in Math., 1332, Springer-Verlag, Berlin, 1988, 195–202 | DOI | MR | Zbl

[34] S. B. Stechkin, “Ungleichungen zwischen den Normen der Ableitungen einer beliebigen Funktion”, Acta Sci. Math. (Szeged), 26 (1965), 225–230, Szeged | MR | Zbl

[35] G. G. Magaril-Il'yaev, “On Kolmogorov inequalities on a half-line”, Moscow Univ. Math. Bull., 31:5–6 (1976), 25–32 | MR | Zbl | Zbl

[36] A. T. Fuller, “Optimization of non-linear control systems with transient inputs”, J. Electronics Control (1), 8:6 (1960), 465–479 | MR | Zbl

[37] V. N. Gabushin, “Inequalities for the norms of a function and its derivatives in metric $L_p$”, Math. Notes, 1:3 (1967), 194–198 | DOI | MR | Zbl | Zbl

[38] G. A. Kalyabin, “Effective formulas for constants in the Stechkin–Gabushin problem”, Proc. Steklov Inst. Math., 248:1 (2005), 118–124 | MR | Zbl

[39] N. P. Korneǐčuk, “Extreme values of functionals and best approximation on classes of periodic functions”, Math. USSR-Izv., 5:1 (1971), 97–129 | DOI | MR | Zbl | Zbl

[40] S. K. Bagdasarov, “Properties of $\omega$-rearrangements”, Funct. Anal. Appl., 33:3 (1999), 161–177 | DOI | MR | Zbl

[41] R. J. McCann, “Exact solutions to the transportation problem on the line”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455:1984 (1999), 1341–1380 | DOI | MR | Zbl

[42] C. Villani, Topics in optimal transportation, Grad. Stud. Math., 58, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[43] V. L. Levin, “General Monge–Kantorovich problem and its applications in measure theory and mathematical economics”, Functional analysis, optimization, and mathematical economics, Oxford Univ. Press, New York, 1990, 141–176 | MR | Zbl