Abstract
Full Text
UDC 517.52
MATHEMATICS
P. K. SUETIN
POLYNOMIALS ORTHOGONAL OVER AREA AND BIEBERBACH POLYNOMIALS
(Presented by Academician M. A. Lavrent'ev on 9 I 1969)
If in a simply connected domain \(G\), lying in the finite part of the plane and bounded by a Jordan curve \(\Gamma\), there is defined a nonnegative function \(h(z)\), summable over the area of the domain \(G\) and differing from zero almost everywhere, then the system of polynomials \(\{K_n(z)\}\), having positive leading coefficient and orthonormal over the area of the domain \(G\) with weight \(h(z)\), is uniquely determined. The simplest properties of these polynomials in the case of an analytic contour and unit weight were first considered by T. Carleman \((^{1})\), while other cases and further properties were studied in the works \((^{3,5,6,8})\). In the present note we set forth asymptotic and approximation properties of the polynomials \(\{K_n(z)\}\) under various smoothness conditions on the contour and the weight, as well as estimates for the rate of convergence of Bieberbach polynomials to the mapping function in a closed domain, since Bieberbach polynomials are represented in terms of polynomials orthogonal over area.
Let the function \(w=\Phi(z)\) map conformally and one-to-one the exterior \(D\) of the contour \(\Gamma\) onto the domain \(|w|>1\) under the conditions \(\Phi(\infty)=\infty\) and \(\Phi'(\infty)=\gamma>0\). We shall say that a rectifiable Jordan curve \(\Gamma\) belongs to the class \(C(p,\alpha)\), where \(p\) is a natural number and \(0<\alpha<1\), if in the equation of the curve \(z=\lambda(s)\), where \(s\) is arc length, the periodic function \(\lambda(s)\) is continuously differentiable \(p\) times, and moreover \(\lambda^{(p)}(s)\in \mathrm{Lip}\,\alpha\).
§ 1. Let us first consider the case of unit weight.
Theorem 1. If \(\Gamma\in C(p+1,\alpha)\), where \(p\geq 1\), then for the polynomials \(\{K_n(z)\}\), orthonormal over the area of the domain \(G\) with weight \(h(z)\equiv 1\), the formula holds
\[
K_n(z)=\sqrt{\frac{n+1}{\pi}\,\Phi'(z)\Phi^n(z)}
\left[1+O\!\left(\frac{\ln n}{n^{p+\alpha}}\right)\right],
\qquad z\in \overline{D}.
\tag{1}
\]
The estimate of the remainder term in formula (1) cannot be improved in order if one considers the set of curves of the class \(C(p+1,\alpha)\) with a fixed constant in the Lipschitz condition for the last derivative. For curves of the class \(C(1,\alpha)\), where \(\alpha>1/2\), the remainder term decreases at the rate \(n^{1-2\alpha}\ln^2 n\). The estimate of the remainder term is improved on a closed set \(F\), located inside the domain \(D\). Namely, if \(\Gamma\in C(p+1,\alpha)\), where \(p+\alpha>1/2\), then for \(z\in F\subset D\) the remainder term in formula (1) will decrease at the rate \(n^{-p-\alpha-1}\). Under the same conditions \(\Gamma\in C(p+1,\alpha)\), where \(p+\alpha>1/2\), for the leading coefficient \(\lambda_n\) of the orthonormal polynomial \(K_n(z)\) the formula holds
\[
\frac{\gamma^{2n+2}}{\lambda_n^2}
=
\frac{\pi}{n+1}
\left[1+O\!\left(\frac{1}{n^{2p+2\alpha}}\right)\right],
\qquad
\gamma=\Phi'(\infty).
\]
Let the function \(f(z)\) be analytic in the domain \(G\), continuous in the closed domain \(\overline{G}\); let \(\{a_n\}\) be its Fourier coefficients with respect to the system \(\{K_n(z)\}\), and let \(E_N(f,G)\) be the best uniform approximation of this function by polynomials of degree not exceeding \(N\) in the closed domain \(\overline{G}\).
Theorem 2. If \(\Gamma \in C(2,\alpha)\) and \(h(z) \equiv 1\), then the inequality
\[ \left| f(z)-\sum_{n=0}^{N} a_n K_n(z)\right| \le c_1 E_N(f,\overline{G})\ln N,\qquad z\in \overline{G}, \tag{2} \]
holds, where the constant \(c_1\) depends only on \(\Gamma\).
If, however, \(\Gamma \in C(1,\alpha)\), where \(\alpha>1/2\), then in the analogue of inequality (2), instead of \(\ln N\) there will be the product \(N^{1-\alpha}\ln N\).
§ 2. Let now the weight function have the form \(h(z)=|\gamma(z)|^2\), where the function \(\gamma(z)\) is analytic in the domain \(G\), continuous and nonzero in the closed domain \(\overline{G}\). Denote by \(H_2(h,G)\) the set of functions analytic in the domain \(G\) whose modulus squared is integrable over the area of the domain \(G\) with weight \(h(z)\), and let \(E_N(f,H_2)\) be the best approximation of the function \(f(z)\) by polynomials in the metric of the space \(H_2(h,G)\). Further, we shall say that a function \(f(z)\), analytic in the domain \(G\), belongs to the class \(W^{(p)}H^{(\alpha)}(\overline{G})\) if it is continuously differentiable \(p\) times in the closed domain \(\overline{G}\), and \(f^{(p)}(z)\in \operatorname{Lip}\alpha\). For \(p=0\) the corresponding class will be denoted by \(H_A^{(\alpha)}(\overline{G})\).
Theorem 3. If \(\Gamma\in C(p+1,\alpha)\), where \(p\ge 0\), and \(\gamma(z)\in W^{(p)}H^{(\alpha)}(\overline{G})\), then for every function \(f(z)\in H_2(h,G)\) the inequality
\[ \left| f(z)-\sum_{n=0}^{N} a_n K_n(z)\right| \le c_2(F)E_N(f,H_2)/N^{p+\alpha},\qquad z\in F\subset G, \]
holds uniformly inside the domain \(G\), where the constant \(c_2(F)\) contains in the denominator the distance from the closed set \(F\) to the contour \(\Gamma\) to the power \(p+2\).
Theorem 4. If \(\Gamma\in C(1,\alpha)\) and \(\gamma(z)\in H_A^{(\alpha)}(\overline{G})\), then the estimate
\[ |K_n(z)|\le c_3\sqrt{n+1},\qquad z\in \overline{G}. \]
holds.
Theorem 5. If \(\Gamma\in C(1,\alpha)\) and \(\gamma(z)\in H_A^{(\alpha)}(\overline{G})\), then for every function \(f(z)\in W^{(p)}H^{(\alpha)}(\overline{G})\) the inequality
\[ \left| f(z)-\sum_{n=0}^{N} a_n K_n(z)\right| \le c_4\ln N/N^{p+\beta-1/2},\qquad z\in \overline{G}. \]
holds.
Theorem 6. If \(\Gamma\in C(p+1,\alpha)\), where \(p+\alpha>3/2\), and \(\gamma(z)\in W^{(p)}H^{(\alpha)}(\overline{G})\), then every function \(f(z)\) analytic in the domain \(G\), whose primitive of order \(p\) is uniformly bounded in the domain \(G\), is expandable in a series in the orthogonal polynomials \(\{K_n(z)\}\), converging uniformly inside the domain \(G\).
§ 3. In the more general case of a nonanalytic weight, the following results hold.
Theorem 7. If \(\Gamma\in C(1,\alpha)\), and the weight function is merely bounded away from zero, i.e. satisfies the condition \(h(z)\ge c_5>0\), then the estimates
\[ |K_n(z)|\le c_6(n+1),\qquad z\in \overline{G}, \]
\[ \int_{\Gamma}|K_n(z)|^2|dz|\le c_7(n+1). \]
hold.
Theorem 8. If the positive function \(h(z)\) satisfies in the closed domain \(\overline{G}\) a Lipschitz condition of order \(\alpha<1\), and the contour \(\Gamma\) is a regular analytic curve, then the relations
\[ |K_n(z)|\le c_8 n\left(\frac{\ln n}{n}\right)^{\alpha/2},\qquad z\in \overline{G}, \]
\[ K_n(z)=\sqrt{\frac{n+1}{\pi}}\,g(z)\Phi^n(z) \left\{1+O\left[\left(\frac{\ln n}{n}\right)^{\alpha/2}\right]\right\},\qquad z\in F\subset D, \]
hold.
where \(F\) is a closed subset of the domain \(D\), and the function \(g(z)\), analytic in the domain \(D\), is determined by the boundary values of the function \(h(z)\).
Theorem 9. If the weight function \(h(z)\) is uniformly separated from zero, then the inequality
\[
\left| f(z)-\sum_{n=0}^{N} a_n K_n(z)\right|\leq c_9 N E_N(f,\overline{G}), \qquad z\in \overline{G}.
\]
Theorem 10. If the function \(f(z)\) is analytic in the domain \(G_R\), bounded by the level line \(\Gamma_R\), on which \(|\Phi(z)|=R>1\), and is continuous in the closed domain \(\overline{G}_R\), then, under the conditions of Theorem 8, the inequality
\[
\left| f(z)-\sum_{n=0}^{N} a_n K_n(z)\right|\leq c_{10}(R)\sqrt{N}E_N(f,\overline{G}_R), \qquad z\in \overline{G}_R.
\]
§ 4. Let us now consider Bieberbach polynomials. Let the function \(w=\varphi(z)\) map the domain \(G\) conformally and one-to-one onto the disk \(|w|<r_0\), under the conditions \(\varphi(z_0)=0\) and \(\varphi'(z_0)=1\), where \(z_0\) is a fixed point of the domain \(G\), and \(r_0\) depends on \(z_0\). The Bieberbach polynomial \(\pi_n(z)\), by definition, minimizes the integral
\[
J(F_n)=\iint_G |F_n'(z)|^2\,dx\,dy
\]
on the set of polynomials \(F_n(z)\) of degree \(n\) satisfying the same normalization conditions \(F_n(z_0)=0\) and \(F_n'(z_0)=1\).
The conditions and rate of convergence of Bieberbach polynomials in the closed domain \(\overline{G}\) were first considered by M. V. Keldysh \((^2)\) and S. N. Mergelyan \((^4)\). In the present note we consider various cases of smoothness of the contour \(\Gamma\).
Theorem 11. If \(\Gamma\in C(1,\alpha)\), where \(\alpha>3/4\), then the inequality
\[
|\varphi(z)-\pi_n(z)|\leq c_{11}E_n(\varphi,\overline{G})\ln n, \qquad z\in \overline{G}.
\]
Theorem 12. If \(\Gamma\in C(1,\alpha)\), where \(\alpha>1/2\), then the estimate
\[
|\varphi(z)-\pi_n(z)|\leq c_{12}\ln n/n^{2\alpha}, \qquad z\in \overline{G}.
\]
Theorem 13. If \(\Gamma\in C(1,\alpha)\), then the estimate
\[
|\varphi(z)-\pi_n(z)|\leq c_{13}E_n(\varphi,\overline{G})\sqrt{n}\ln n, \qquad z\in \overline{G}.
\]
Theorem 14. If the Jordan curve \(\Gamma\) has bounded curvature, then the inequality
\[
|\varphi(z)-\pi_n(z)|\leq c_{14}\ln^2 n/n^2, \qquad z\in \overline{G}.
\]
Theorem 15. If \(\Gamma\in C(p+1,\alpha)\), where \(p\geq 1\), then for \(0\leq m\leq p\) the estimate
\[
|\varphi^{(m)}(z)-\pi_n^{(m)}(z)|\leq c_{15}n^{1/2+m}/n^{p+\alpha}, \qquad z\in \overline{G}.
\]
The estimates of Theorems 11–14 strengthen the results of our note \((^7)\).
Sverdlovsk Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
25 XII 1968
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