Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1957. Vol. 114, No. 4
MATHEMATICS
G. Ts. TUMARKIN
ON SIMULTANEOUS APPROXIMATION IN THE MEAN OF COMPLEX-VALUED FUNCTIONS GIVEN ON SEVERAL CONTOURS
(Presented by Academician M. A. Lavrent’ev on 11 XII 1956)
1. Let (\gamma) be a closed Jordan rectifiable curve. Denote by (s) the length of the arc of the curve (\gamma), measured from some point (\zeta_0 \in \gamma); (0 \le s \le l), where (l) is the length of (\gamma); let (\sigma(s)) be a nondecreasing function of bounded variation for (0 \le s \le l)*. Consider the space (L^p(d\sigma,\gamma)), (p>0), of complex-valued functions (f(\zeta)), defined on (\gamma), for which
[
\int_\gamma |f(\zeta)|^p\,d\sigma(s)<\infty^{**}.
]
In our note ((^2)) a theorem was given in which a characterization was supplied of the closure of the linear span of the system ({\zeta^m}), (m=0,1,2,\ldots), in (L^p(d\sigma,\gamma)), (p>0). In the present note we shall consider the approximation of functions given on several contours. Let (G) be an (n)-connected domain bounded by (n) closed rectifiable curves (\gamma_1,\ldots,\gamma_n). For definiteness we shall regard (G) as a finite domain, and (\gamma_1) as the outer contour. The complete boundary of (G) will be denoted by (\Gamma). For each of the curves (\gamma_i), (i=1,2,\ldots,n), we define the spaces (L^p(d\sigma_i,\gamma_i)) and shall henceforth denote by (L^p(d\sigma,\Gamma)) the totality of all complex-valued functions (f(\zeta)), defined on (\Gamma), belonging on each component (\gamma_i) of the boundary (\Gamma) to the corresponding space (L^p(d\sigma_i,\gamma_i)). The distance between two functions (f_1(\zeta)) and (f_2(\zeta)) belonging to (L^p(d\sigma,\Gamma)) will be defined by the formula
[
\rho(f_1,f_2)=\int_\Gamma |f_1-f_2|\,d\sigma
=\sum_{i=1}^n \int_{\gamma_i} |f_1-f_2|\,d\sigma_i.
]
2. We first consider the approximation of functions in the metric (L^p(d\sigma,\Gamma)) by sequences ({\Pi_k(\zeta)}) of polynomials in (\zeta). For this purpose we investigate the behavior, inside a simply connected domain (g) bounded by a curve (\gamma), of a sequence of polynomials ({\Pi_k(z)}) satisfying on (\gamma) the condition
[
\lim_{k\to\infty}\int_\gamma |f(\zeta)-\Pi_k(\zeta)|^p\,d\sigma=0,\qquad p>0.
\tag{1}
]
Theorem 1. If for (\sigma(s)) the condition
[
\int_\gamma \ln \sigma'(s)\,|\psi'(\zeta)\,d\zeta|>-\infty,
\tag{2}
]
is fulfilled, where (w=\psi(z)) maps (g) conformally onto (|w|<1), then the sequence ({\Pi_k(z)}), satisfying condition (1), will converge uniformly inside (g)
* (\sigma(s-0)=\sigma(s)).
** The integrals are understood in the Lebesgue–Stieltjes sense.
converge, and the function (f(z)=\lim\limits_{k\to\infty}\Pi_k(z)) will have on (\gamma) almost everywhere the angular boundary values (f(\zeta)) coinciding with the function appearing in equality (1).
Theorem 2. If condition (2) is not fulfilled for (\sigma(s)), then for any function (f(\zeta)\in L^p(d\sigma,\gamma)), (p>0), and an arbitrary function (F(z)) analytic in the domain (g), there exists a sequence ({\Pi_k(z)}) such that on (\gamma) (1) will hold, while inside (g) ({\Pi_k(z)}) will converge uniformly to (F(z)).
Since condition (2) is necessary and sufficient for the non-closedness of ({\zeta^m}, m=0,1,\ldots,) in (L^p(d\sigma,\gamma))*, Theorems 1 and 2 show that, in the case of non-closedness of the system ({\zeta^m}, m=0,1,2,\ldots,) in (L^p(d\sigma,\gamma)), the convergence of ({\Pi_k(\zeta)}) on (\gamma) in the metric of (L^p(d\sigma,\gamma)) implies the uniform convergence of ({\Pi_k(z)}) inside (g), while in the case of closedness of the system under consideration the requirement that ({\Pi_k(\zeta)}) converge on (\gamma) in the metric of (L^p(d\sigma,\gamma)) “does not constrain” the behavior of ({\Pi_k(z)}) in closed subdomains lying inside (g). In the latter case, using Theorem 2, it is easy to construct examples of ({\Pi_k(z)}) satisfying condition (1) and converging uniformly inside (g) to (\infty), etc.
Remark. The dependence of the behavior inside the domain of polynomials whose norms in (L^p(d\sigma,\gamma)) are uniformly bounded on whether the system ({\zeta^m}, m=0,1,\ldots,) is closed or non-closed was also pointed out by S. N. Mergelyan, who established that in the case of closedness of ({\zeta^m}, m=0,1,2,\ldots,) in (L^p(d\sigma,\gamma)) the set of polynomials under consideration is noncompact inside (g), while in the case of non-closedness it is compact.
- With the aid of Theorems 1 and 2 and the earlier investigation of approximation by polynomials of functions defined on a closed rectifiable curve, one can completely study the question of approximation by sequences of polynomials of functions defined on a composite contour (\Gamma). Here we restrict ourselves only to formulating a theorem giving a sufficient condition for the closedness of the system ({\zeta^m}, m=0,1,2,\ldots).
Theorem 3. If the system ({\zeta^m}, m=0,1,2,\ldots,) is closed in each of the spaces (L^p(d\sigma_i,\gamma_i)), (i=1,\ldots,n), (p>0), then this system is closed also in the space (L^p(d\sigma,\Gamma)).
Remark. Theorem 3 would be a simple consequence of Runge’s theorem only in the case when the finite domains bounded by the curves (\gamma_i), (i=1,\ldots,n), had no common points. Recall that in our case (\gamma_2,\ldots,\gamma_n) lie inside (\gamma_1). Therefore the possibility of simultaneous approximation by one and the same sequence of polynomials of functions prescribed on (\gamma_1,\ldots,\gamma_n) requires special consideration. We also note that Theorem 3 will be valid for any system of pairwise nonintersecting contours (\gamma_i). The necessity of the conditions of Theorem 3 is obvious.
- Let us now consider the question of which functions (f(\zeta)), defined on (\Gamma), can be approximated arbitrarily well in the metric (L^p(d\sigma,\Gamma)) by sequences of boundary values of functions analytic in the closed domain (\overline{G}). For this it is necessary to investigate the closure of the linear span of the system of functions
[
\left{z^m,\ \frac{1}{(z-\alpha_i)^m}\right}, \qquad m=0,1,2,\ldots,\quad i=1,\ldots,n,
\tag{3}
]
where (\alpha_i) is a point chosen inside the domain complementary to (\overline{G}) and bounded by (\gamma_i). As the investigation shows, the resulting
* For (p\geqslant 1) this was first proved by Ya. L. Geronimus.
the results turn out to be analogous to those obtained in considering approximation by polynomials of functions defined on a closed Jordan rectifiable curve (1). In what follows we shall denote by (R(z)) a linear combination of the functions of the system (3).
Theorem 4. For the system (3) to be closed in (L^p(d\sigma,\Gamma)), (p>0), it is necessary and sufficient that
[
\int_{\Gamma}\ln \sigma'(s)\,|\Psi'(\zeta)\,d\zeta|=-\infty,
\tag{4}
]
where (w=\Psi(z)) maps the domain (G) onto the (n)-connected circular canonical domain (K).*
Definition. An analytic function (F(z)) in the domain (G) belongs to the class (D) if the subharmonic functions
(\ln^+\left|\dfrac{F(z)}{M}\right|) have in the domain (G) least harmonic majorants (u^M(z)) satisfying the condition
(\lim_{M\to\infty}u^M(z_0)=0), where (z_0\in G). It follows from this definition that the class (D) is a subclass of the functions of bounded characteristic in (G). In the case when the domain (G) is the unit disk, it is not difficult to see that the class (D) introduced above coincides with the class of functions satisfying the condition of P. Ya. Polubarinova-Kochina ((1), Ch. II, Sec. 6.5).
Theorem 5. If condition (4) is not satisfied, then in order that the function (f(\zeta)\in L^p(d\sigma,\Gamma)), (p>0), belong to the closure of the linear span of the system (3) in the space under consideration, it is necessary and sufficient that there exist an analytic function (f(z)) of class (D) in (G), whose angular boundary values would coincide almost everywhere on (\Gamma) with (f(\zeta)).
Remark. It can be proved that condition (4) is equivalent to the following: among the integrals
[
\int_{\gamma_i}\ln \sigma_i'(s)\,|\psi'(\zeta)\,d\zeta|,
\quad i=1,2,\ldots,n,
]
where (w=\psi_i(z)), (i=1,\ldots,n), maps conformally the domain (G_i), bounded by (\gamma_i), (G_i\supset G), onto (|w|<1) for (i=1) and onto (|w|>1) for (i\ne1), at least one is equal to (-\infty).
- For multiply connected domains there will hold theorems analogous to Theorems 2 and 3.
Theorem 6. If ({R_k(z)}) converges in the metric (L^p(d\sigma,\Gamma)), (p>0), where (\sigma(s)) does not satisfy condition (4), then inside (G) the sequence ({R_k(z)}) will converge uniformly to that analytic function (f(z)) to whose boundary values ({R_k(\zeta)}) converged.
Theorem 7. If for (\sigma(s)) condition (4) is satisfied, then for any preassigned functions (f(\zeta)\in L^p(d\sigma,\Gamma)), (p>0), and (\Phi(z)), analytic inside (G), there exists ({R_k(z)}) such that on the boundary (\Gamma) the sequence ({R_k(\zeta)}) converges in the metric (L^p(d\sigma,\Gamma)) to (f(\zeta)), while inside (G) it converges uniformly to (\Phi(z)).
- We give a theorem that provides a sufficient condition for an analytic function (f(z)) in the domain (G) to belong to the class (D).
Theorem 8. If for (f(z)) there exists a sequence of analytic functions ({f_k(z)}), bounded in (G), converging uniformly to (f(z)) inside (G), and such that
[
\int_{\Gamma}|f_k(\zeta)|^p\rho(\zeta)\,|d\zeta|\leq\infty,
\quad k=1,2,\ldots,
]
where (\rho(\zeta)\geq0) satisfies the condition
[
\int_{\Gamma}\ln\rho(\zeta)\,|\psi'(\zeta)\,d\zeta|>-\infty,
]
then (f(z)\in D), and the boundary values (f(\zeta)) are summable on (\Gamma) to the power (p) with weight (\rho(\zeta)).
* Instead of a mapping onto the circular domain (K), one may take mappings onto any (n)-connected domain bounded by analytic curves.
From Theorems 8, 5, and 6 it follows:
Corollary. Under the hypotheses of Theorem 8 there exists ({R_k(z)}), converging uniformly inside (G) to (f(z)), and such that
[
\lim_{k\to\infty}\int_\Gamma |f(\zeta)-R_k(\zeta)|^p\rho(\zeta)|d\zeta|=0 .
]
- With the aid of the theorems given above one can investigate the question of the possibility of approximating the boundary values (f(\zeta)) of functions analytic in the domain (G) and belonging to the classes (E_\delta), (\delta>0). (The definition of the classes (E_\delta) in multiply connected domains, analogous to the definition of these classes for simply connected domains, is given in ((^3)).) The results obtained in this way turn out to be similar to those obtained in considering approximation of the boundary values of the classes (E_\delta) in simply connected domains by sequences of polynomials. (Formulations of the results for simply connected domains are given in our note ((^2)).) We shall confine ourselves here to considering only the most important case, when the domain (G) belongs to the class (S) of domains satisfying V. I. Smirnov’s condition. The definition of the class (S) in the case of multiply connected domains, which is given in ((^{3,5})), reduces to the case of a simply connected domain: a domain (G\in S) if each of the domains (G_i,\ i=1,\ldots,n), belongs to the class (S). (For the definition of the class (S) for simply connected domains, see, for example, ((^1)).) It can be proved that this definition is equivalent to the following one: the harmonic function (\ln |\varphi'(w)|), where (z=\varphi(w)) maps the disk (K) onto (G), is representable in (K) by Green’s formula. It is not difficult to see that if (f(z)\in E_\delta) in the domain (G\in S), then (f(z)\in D). Then Theorems 4 and 5 immediately imply Theorem 9.
Theorem 9. If (f(z)\in E_\delta) in the domain (G\in S), and the boundary values of this function (f(\zeta)\in L^p(d\sigma,\Gamma)), (p>0), then there exists a sequence ({R_k(\zeta)}) converging to (f(\zeta)) in the metric (L^p(d\sigma,\Gamma)).
From Theorems 6 and 9 follows Theorem 10.
Theorem 10. If (G\in S), then for the existence of ({\Pi_k(\zeta)}) satisfying the condition
[
\lim_{k\to\infty}\int_\Gamma |f(\zeta)-\Pi_k(\zeta)|^p|d\zeta|=0,
]
it is necessary and sufficient that (f(\zeta)) coincide on (\Gamma) with the boundary values of a function (f(z)), analytic in (G), of the class (E_p).
For (p>1) Theorem 10 was proved in ((^5)).
- In ((^{4,6})) the classes (H_\delta) in multiply connected domains were studied. In any domain (G), (H_\delta\subset D) for any (\delta>0). Therefore Theorems 4 and 5 imply Theorem 11.
Theorem 11. If (f(z)\in H_\delta), (\delta>0), in the domain (G) and (f(\zeta)\in L^p(d\sigma,\Gamma)), (p>0), then there exists ({R_k(\zeta)}), converging on (\Gamma) in the metric (L^p(d\sigma,\Gamma)) to (f(\zeta))—the boundary values of (f(z)).
For (\delta=p) and (\sigma(s)\equiv s) this was proved by Rudin ((^6)), moreover it was assumed that (\Gamma) is an analytic curve.
- Results similar to those given above are obtained if instead of the space (L^p(d\sigma,\Gamma)) one considers the space (C(\rho,\Gamma)), consisting of functions (f(\zeta)) continuous on (\Gamma) with norm (|f|=\max_\Gamma{|f(\zeta)|\rho(\zeta)}), where (\rho(\zeta)\geqslant0) is a function continuous on (\Gamma).
Moscow Geological Prospecting Institute
named after S. Ordzhonikidze
Received
5 V 1956
CITED LITERATURE
- I. I. Privalov, Boundary Properties of Analytic Functions, 2nd ed., 1950.
- G. Ts. Tumarkin, DAN, 84, No. 1, 21 (1952).
- S. Ya. Khavinson, Mat. sbornik, 36 (78), issue 3, 445 (1955).
- M. Parreau, Ann. Inst. Fourier, 3 (1951).
- J. Penez, Proc. Nat. Acad. Sci. USA, 40, No. 4 (1954).
- W. Rudin, Trans. Am. Math. Soc., 78, No. 1 (1955).