Abstract
Full Text
UDC 517.537
MATHEMATICS
N. A. LEBEDEV
ON INVERSE THEOREMS OF UNIFORM APPROXIMATION
(Presented by Academician V. I. Smirnov on February 12, 1966)
This note is devoted to a certain generalization of the inverse theorems of V. K. Dzyadyk \((^{1,2})\).
Let \(\overline{B}\) be a bounded set of points \(z\) of type \(\mathfrak{M}_s^*\) (see \((^3)\), pp. 79–81) and let \(L\) be the boundary of \(\overline{B}\). The complement of \(L\) in the extended plane consists of domains \(B^j,\ j=1,2,\ldots,s\), not belonging to \(\overline{B}\), and of no more than a countable number of domains \(B^j,\ j=s+1,s+2,\ldots\), belonging to \(\overline{B}\). Further we assume that every point \(z\) of the boundary \(L^j\) of the domain \(B^j\) is accessible from \(B^j,\ j=1,2,\ldots\).
Let \(a_j\) be a fixed point in \(B^j,\ j=1,2,\ldots,s\). We shall consider approximation of a function \(f(z)\), given on \(L\), by rational functions (r.f.) of the form
\[ R(z)=\sum_{j=1}^{s}\sum_{\nu=1}^{n_j}\frac{A_{j,\nu}}{(z-a_j)^\nu}+A,\qquad A_{j,n_j}\ne 0 \tag{1} \]
(if some \(a_j=\infty\), then \(1/(z-a_j)^\nu\) in (1) must be replaced by \(z^\nu\)). The number \(n=\max_{j=1,2,\ldots,s} n_j\) will be called the degree of the r.f. (1).
Let: \(g_j(z,\zeta)\) be the Green function of the domain \(B^j,\ j=1,2,\ldots,s\);
\(L_\rho^j=\{z:\ g_j(z,a_j)=\ln\rho\},\ \rho>1;\quad L_\rho=\bigcup_{j=1}^{s}L_\rho^j;\quad d(z,u)\) be the distance from the point \(z\in L\) to \(L_{1+u},\ u>0\). Let \(A(z),\ z\in L\), be a continuous positive function. Introduce the function \(A^*(z)\), putting \(A^*(z)=A(z)\) for \(z\in L\) and \(A^*(z)=\exp G(z,B^j)\) for \(z\in B^j,\ j=1,2,\ldots\), where \(G(z,B^j)\) is a function harmonic in \(B^j\) and continuous on \(\overline{B}^j\) such that \(G(z,B^j)=\ln A(z)\) for \(z\in L^j\). The function \(A^*(z)\) is continuous on the extended plane \(Z\), and
\[ \sup_{z\in Z} A^*(z)=\sup_{z\in L} A(z). \]
Put
\[ A^*(z,y)=\sup_{|\zeta-z|=y} A^*(\zeta),\qquad z\in L,\quad y>0. \]
Lemma. Let the r.f. (1) of degree not exceeding \(n\) satisfy the condition \(|R(z)|\le A(z)\) for \(z\in L\). Then:
1) \(|R(z)|\le A^*(z)\) for \(z\in \overline{B}\) and \(|R(z)|\le A^*(z)\rho^n\) for \(z\in L_\rho,\ \rho>1\);
2) for any \(u>0\) and \(z\in L\) we have
\[ |R^{(\nu)}(z)|\le \nu! e^u A^*(z,d(z,u/n))\,d(z,u/n)^{-\nu},\qquad \nu=1,2,\ldots; \]
3) for any \(u>0,\ z_1\in L,\ z_2\in L,\ |z_1-z_2|<d(z_1,u/n)\), we have
\[ |R(z_1)-R(z_2)|\le \frac{2}{\pi}\,K(k)e^u A^*\!\left(z_1,d\!\left(z_1,\frac{u}{n}\right)\right)k,\qquad k=\frac{|z_1-z_2|}{d(z_1,u/n)}, \]
where \(K(k)\) is the complete elliptic integral of the first kind.
For the ideas of the proof see \((^1)\), pp. 730–732 (additionally see \((^3)\), pp. 83, 97).
Let \(\omega(x),\ x>0,\ \omega(0)=0\), be a nondecreasing continuous function. Put \(A(z)=A_u(z)=\omega(d(z,u)),\ z\in L,\ u>0\). If there exists a number \(M\le 1\) such that \(A^*(z,d(z,u))=A_u^*(z,d(z,u))\le M A_u(z)\) for every \(z\in L\) and every \(u>0\), then we write \(\omega(x)\in\{\overline{B},M\}\). If \(\omega_1(x)\in\)
\(\in \{\bar B,M_1\};\ \omega_2(x)\in\{\bar B,M_2\}\) and \(\alpha>0\) and \(C>0\) are constants, then \(C\omega_1(x)^\alpha\in\{\bar B,M_1^\alpha\}\) and \(\omega_1(x)\omega_2(x)\in\{\bar B,M_1M_2\}\).
Theorem. Let \(f(z)\) be a function defined on the boundary \(L\) of the set \(\bar B\); let \(\lambda\), \(\lambda>0\), be a fixed number, \(r\) a nonnegative integer; \(x^r\omega(x)\in\{\bar B,M\}\) and \(\omega(\lambda x)\le \lambda'\omega(x)\) for all \(x>0\) and \(\lambda'\ge 1\); suppose there exists a sequence of rational functions \(R_n(z)\) of degree not exceeding \(n\), \(n=m,m+1,\ldots\), of the form (1), such that
\[ |f(z)-R_n(z)|\le d(z,\lambda/n)^r\omega(d(z,\lambda/n)),\qquad z\in L . \tag{2} \]
Then:
1) if \(r=0\) and \(0<x\le e^{-1}\inf_{z\in L} d(z,\lambda/m)=x_0\), we have
\[ \omega(x,f)\le \omega(x,R_m)+C_0Mx\int_x^{l_m}\frac{\omega(t)}{t^2}\,dt, \tag{3} \]
where \(l_m=\sup_{z\in L} d(z,\lambda/m)\) and \(C_0\) is an absolute constant \((0<C_0<4.2e^{\lambda_0}\), \(\lambda_0=\max\{(m+1)\lambda/m,\lambda+2\})\) (\(\omega(x,\varphi)\) is the modulus of continuity of the function \(\varphi(z)\) on \(L\));
2) if \(r>0;\ \nu=1,2,\ldots,r\),
\[ |f^{(\nu)}(z)-R_n^{(\nu)}(z)|\le C_r^{(\nu)}M\Omega_{r-\nu}(d(z,\lambda/n)),\qquad z\in L,\quad n=m,m+1,\ldots, \tag{4} \]
\[ \Omega_p(x)=\int_0^x t^{p-1}\omega(t)\,dt,\qquad 0<C_r^{(\nu)}<\frac{4\nu!\,e^{\lambda_0+q}}{q(e^q-1)},\qquad q=r-\nu+1 . \]
Remark 1. For \(\nu=r\), from (4) we have
\[ |f^{(r)}(z)-R_n^{(r)}(z)|\le C_r^{(r)}M\Omega_0\left(d\left(z,\frac{\lambda_*}{n+m}\right)\right),\qquad z\in L,\quad n=m,m+1,\ldots, \]
where \(\lambda_*=\frac{m+r}{m}\lambda\). If \(\Omega_0(x)\in\{\bar B,M_*\}\), then the first assertion of the theorem can be applied to \(f^{(r)}(z)\).
Remark 2. In the first part of the theorem one can also give an estimate of \(\omega(x,f)\) for \(x>x_0\).
Proof (see additionally \((^2)\), pp. 371–374).
1) \(r=0\). Let \(0<x\le x_0\), \(z_1\in L\), \(z_2\in L\), and \(|z_1-z_2|=x\). We have
\[ \begin{aligned} f(z_2)-f(z_1) ={}&[f(z_2)-R_{n_1}(z_2)]-[f(z_1)-R_{n_1}(z_1)]\\ &+\sum_{j=1}^{k-1}\{[R_{n_j}(z_2)-R_{n_{j+1}}(z_2)]-[R_{n_j}(z_1)-R_{n_{j+1}}(z_1)]\}\\ &+[R_{n_k}(z_2)-R_{n_k}(z_1)], \end{aligned} \tag{5} \]
where \(n_j\), \(m=n_k<n_{k-1}<\cdots<n_1<\infty\), are natural numbers.
We turn to the choice of \(n_j\). Let \(n\) be the largest natural number such that
\[ \max\{d(z_1,\lambda/n),d(z_2,\lambda/n)\}\ge ex, \]
and suppose, for example, that \(d(z_1,\lambda/n)\ge d(z_2,\lambda/n)\). If \(d(z_1,\lambda/n)\le e^2x\), then put \(n_1=n\). If \(d(z_1,\lambda/n)>e^2x\), then put \(n_1=n+1\) and \(n_2=n\). If some \(n_j>m\), \(j\ge 1\), has been found, then we find the largest natural \(n\) such that \(d(z_1,\lambda/n)\ge ed(z_1,\lambda/n_j)\), and if either \(d(z_1,\lambda/n)\le e^2d(z_1,\lambda/n_j)\), or
\[ n\ge \frac{m}{m+1}n_j, \]
then put \(n_{j+1}=n\). In the contrary case put \(n_{j+1}=n+1\) and \(n_{j+2}=n\). We continue this process until we obtain (or are forced to take) some \(n_k=m\). For brevity denote
\[ d_0=x,\qquad d_j=d(z_1,\lambda/n_j). \]
There are three possible cases: a) \(ed_j\le d_{j+1}\le e^2d_j\); b) \(e^2d_j<d_{j+1}\), but \(n_j/n_{j+1}\le (m+1)/m\); c) \(d_{j+1}<ed_j\), but \(d_{j+2}>e^2d_j\) and \(n_{j+1}/n_{j+2}\le (m+1)/m\), or \(j+1=k\).
Let us estimate the right-hand side in (5). For \(\nu=1,2\) we have
\(|f(z_\nu)-R_{n_1}(z_\nu)|\leqslant \omega(d(z_\nu,\lambda/n_1))\leqslant \omega(d_1)\). If \(d_1<ex\), then \(\omega(d_1)\leqslant \omega(ex)\), and if \(ex\leqslant d_1\leqslant e^2x\), then \(\omega(d_1)\leqslant \dfrac{d_1}{ex}\omega(ex)\leqslant e\omega(ex)\), and, consequently,
\[ \bigl|[f(z_2)-R_{n_1}(z_2)]-[f(z_1)-R_{n_1}(z_1)]\bigr|\leqslant 2e\omega(ex). \]
\[ |R_{n_j}(z_1)-R_{n_{j+1}}(z_1)| \leqslant |f(z_1)-R_{n_j}(z_1)|+|f(z_1)-R_{n_{j+1}}(z_1)| \leqslant 2\omega(d_{j+1}), \]
and, by virtue of the lemma (in our case \(k\leqslant e^{-1}\) and \(\dfrac{2}{\pi}K(k)<1.05\)), we have
\[ I_j=\bigl|[R_{n_j}(z_2)-R_{n_{j+1}}(z_2)]-[R_{n_j}(z_1)-R_{n_{j+1}}(z_1)]\bigr| \leqslant \]
\[ \leqslant 2.1\,Me^{\lambda u}\omega(d_{j+1})d(z_1,\lambda u/n_j)^{-1}x \]
for \(0<u\leqslant n_j/n_{j+1}\). If \(n_j/n_{j+1}\leqslant (m+1)/m\), then put \(u=n_j/n_{j+1}\). In the contrary case put \(u=1\) and use the fact that \(d_{j+1}\leqslant e^2d_j\). In both cases we obtain the estimate
\(I_j\leqslant 2.1e^{\lambda_0}M\omega(d_{j+1})d_{j+1}^{-1}x\).
From what has been stated and from (5) we have
\[ |f(z_2)-f(z_1)|\leqslant 2e^2\frac{\omega(ex)}{ex}x +2.1e^{\lambda_0}Mx\sum_{j=2}^{k}\frac{\omega(d_j)}{d_j} +\omega(x,R_m). \tag{6} \]
Let us estimate \(\omega(d_j)/d_j\). If \(d_j\geqslant ed_{j-1}\), then
\[ \int_{d_{j-1}}^{d_j}\frac{\omega(t)}{t^2}\,dt \geqslant \frac{\omega(d_j)}{d_j}\int_{d_{j-1}}^{d_j}\frac{dt}{t} \geqslant \frac{\omega(d_j)}{d_j}. \]
If \(d_j<ed_{j-1}\), then \(d_{j-1}\geqslant ed_{j-2}\), and therefore
\[ \int_{d_{j-2}}^{d_j}\frac{\omega(t)}{t^2}\,dt \geqslant \frac{\omega(d_j)}{d_j}\int_{d_{j-2}}^{d_{j-1}}\frac{dt}{t} \geqslant \frac{\omega(d_j)}{d_j}. \]
\[ \int_x^{ex}\frac{\omega(t)}{t^2}\,dt\geqslant \frac{\omega(ex)}{ex}. \]
Now from (6) it follows that
\[ |f(z_2)-f(z_1)|\leqslant \omega(x,R_m)+4.2e^{\lambda_0}Mx\int_x^{d_k}\frac{\omega(t)}{t^2}\,dt, \]
and, since \(d_k\leqslant l_m\), the case \(r=0\) has been considered.
2) \(r>0\). Choose a natural number \(n_0\geqslant m\) and represent \(f^{(\nu)}(z)\) as
\[ f^{(\nu)}(z)-R_{n_0}^{(\nu)}(z) = \sum_{j=1}^{\infty}\bigl[R_{n_j}^{(\nu)}(z)-R_{n_{j-1}}^{(\nu)}(z)\bigr], \quad z\in L, \tag{7} \]
where \(m\leqslant n_0<n_1<n_2<\cdots\) are natural numbers chosen analogously to what was set out in the first part of the proof. Further, relying on the lemma, we estimate the right-hand side in (7) and obtain (4).
For the functions \(\omega(x)\) and sets \(\overline{B}\) considered by V. K. Dzyadyk, it was in fact proved by him that \(\omega(x)\in\{\overline{B},M\}\). It is not difficult to indicate broader classes of sets \(\overline{B}\) for which \(\omega(x)\in\{\overline{B},M\}\) (\(\omega(0)=0\), \(\omega(\lambda x)\leqslant \lambda\omega(x)\) for all \(x>0\) and \(\lambda\geqslant 1\)), but a complete analysis could not be carried out.
The foregoing can easily be somewhat generalized.
Received
10 II 1966
CITED LITERATURE
- V. K. Dzyadyk, Izv. AN SSSR, Ser. Mat., 23, 697 (1959).
- V. K. Dzyadyk, Ukr. Mat. Zh., 15, No. 4 (1963).
- V. I. Smirnov, N. A. Lebedev, Constructive Theory of Functions of a Complex Variable, “Nauka,” 1964.