$L_p$ extensions of Gonchar's inequality for rational functions

A. L. Levin   ; E. B. Saff

Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 199-210

Résumé

Given a condenser $~(E,\, F)$ in the complex plane, let $~C(E,\, F)$ denote its capacity and let $~\mu^*=\mu_E^*-\mu_F^*$ be the (signed) equilibrium distribution for $~(E,\, F)$. Given a finite positive measure $\mu$ on $E\cup F$, let $$ G(\mu_E')=\exp\biggl(\,\int\log(d\mu/d\mu_E^*)\,d\mu_E^*\biggr),\quad G(\mu_F')=\exp\biggr(\,\int\log(d\mu/d\mu_F^*)\,d\mu_F^*\biggr). $$ We show that for $0<p,q<\infty$ and for any rational function $r_n$ of order $n$ \begin{equation} \|r_n\|_{L_p(d\mu,E)}\|1/r_n\|_{L_q(d\mu,F)}\geqslant e^{-n/C(E,F)}G^{1/p}(\mu_E') G^{1/q}(\mu_E'), \tag{1} \end{equation} which extends a classical result due to A. A. Gonchar. For a symmetric condenser we also obtain a sharp lower bound for $\|r_n-\lambda\|_{L_p(d\mu,\,E\cup F)}$, where $\lambda=\lambda(z)$ is equal to $0$ on $E$ and $1$ on $F$. The question of exactness of (1) and the relation to certain $n$-widths are also discussed.

Bibliographie

[1] Gonchar A. A., “Otsenki rosta ratsionalnykh funktsii i nekotorye ikh primeneniya”, Matem. Sb., 72:3 (1967), 489–503 | MR | Zbl

[2] Gonchar A. A., “Ob obobschennom analiticheskom prodolzhenii”, Matem. Sb., 76:1 (1968), 135–146 | MR | Zbl

[3] Gonchar A. A., “O zadachakh Zolotareva, svyazannykh s ratsionalnymi funktsiyami”, Matem. Sb., 78:4 (1969), 640–654 | MR | Zbl

[4] Widom H., “An inequality for rational functions”, Proc. Amer. Math. Soc., 24 (1970), 415–416 | DOI | MR | Zbl

[5] Saff E. B., Totic V., Ljgarithmic Potentials with External Fields, Springer-Verlag (to appear)

[6] Ganelius T. H., “Some extremal functions and approximations”, Fourier Analisys and Approximation Theory, Proc. conf. (Budapest, 1976), North-Holland, Amsterdam–Oxford–New York, 1978, 371–381 | MR

[7] Vyacheslavov N. S., “O naimenshem uklonenii funktsii $\text {\rm sign}x$ i ee pervoobraznykh ot ratsionalnykh funktsii v metrikakh $L_p$, $0

\infty $”, Matem. Sb., 103:1 (1977), 24–36 | MR | Zbl

[8] Fisher S. D., Micchelli C. A., “Fhe $n$-widths of sets of analityc functions”, Duke Math. J., 47 (1980), 789–801 | DOI | MR | Zbl

[9] Parfenov O. G., “Poperechniki odnogo klassa analiticheskikh funktsii”, Matem. Sb., 117:2 (1982), 279–285 | MR | Zbl

[10] Parfenov O. G., “Singulyarnye chisla operatorov vlozhenii nekotorykh klassov analiticheskikh i garmonicheskikh funktsii”, Lineinye i nelineinye uravneniya v chastnykh proizvodnykh. Spektralnye asimptotiki, Izd-vo LGU, L., 1984, 56–66 | MR

[11] Bagby T., “The modulus of the plane condenser”, J. Math. and Mech., 17 (1967), 315–329 | MR | Zbl

[12] Hille E., Analytic Function Theory, V. 2, Ginn and Company, Boston, 1962 | MR | Zbl

[13] Levin A. L., Saff E. B., “Szegö type asymptotics for minimal Blaschke products”, American-Russian Advances in Approximation Theory, Springer-Verlag (to appear) | MR

[14] Newman D. J., “Rational approximation to $|x|$”, Michigan Math. J., 11 (1964), 11–14 | DOI | MR | Zbl

[15] Ganelius T. H., “Rational approximation in the complex plane and on the line”, Ann. Acad. Sci. Fennicae Ser. Math., 2 (1976), 129–145 | MR | Zbl

[16] Widom H., “Rational approximation and $n$-dimensional diameter”, J. Appr. Theory, 5 (1972), 343–361 | DOI | MR | Zbl