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On an extremal problem in the theory of rational approximation
JOURNAL
OF APPROXIMATION
THEORY
50,
127-132 (1987)
On an Extremal Problem in the Theory Rational Approximation
of
D. KHAVINSON AND D. LUECKING* Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, -5J.S.A Communicated by Philip J. Davis
Received October 30, 1984 DEDICATED TO THE MEMORY OF &ZA
1.
FREUD
INTRODUCTION
Let G be an n-connected domain in C bounded by simple analytic curves Let I-= U;=, rj. Let R(G) be the uniform closure in G of the Yn. algebra of rational functions with poles outside of G. We recall that, for domains with analytic boundary, one can define a Smirnov class E,(G) as the collection of analytic functions in G whose boundary values belong to the closure of R(G) in L’(ds). Here, ds is Lebesgue measure on IY (Details concerning the general definition and properties of the classes E, can be found in [3, 5, 91.) In [6] the following concept of rational capacity I has been introduced:
YlY..,
A= l(G) ‘ZfBEi;fG, IIC- d(i)ll. Here, II.11denotes the uniform norm on G. The importance of J. can be seen from a simple observation that A = 0 if and only if G degenerates to the union of analytic arcs. Also, it turns out that 1 enjoys simple estimates in terms of elementary geometric characteristics of G: area and perimeter, i.e., a(G) > I > 24G) -/ /d-- 7l f’(G) ’
(*I
Here, a(G) = a and P(G) = P denote the area and perimeter ( = [r ds) of G, respectively. The first inequality in (*) was proved in [l] and the second in [6]. * Both authors were partially supported by the National Science Foundation.
127 @O21-9045/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
128
KHAVINSON
AND
LUWKING
Using a standard duality argument (see [3], for the case of the unit disk [4-81) one can easily show that
Also, it is not hard to see (via F. and M. Riesz and Banach-Alaoglu theorems) that there exist $* E H”(G) and f* EEL, the extremal functions for which the infimum and supremum are attained. Moreover, as we assume r to be analytic, i.e., [I,= S(c), where S(i) is the so-called Schwarz function analytic in a tubular neighborhood of r (see [2]), it follows from a result of S. Ya. Khavinson (see [7, 81) that both functions ,f* and #* can be analytically continued across I-. In this paper in Section 2 we obtain a simple expression for the area of the image 4*(G) in terms of L and the number of zeros N,. of the extremal funtionf*(z) (zeros on r are counted with a half-multiplicity). In Section 3 as a corollary of this theorem we obtain the classical isoperimetric inequality with sharp constants (Corollary 3). Also, we show (Corollary 2) that the area of the image of the best approximation d* to C in simply connected domains is always dominated by the “isoperimetric deficiency” 1 - &a/P2 of G. In particular this estimate gives another proof of the well-known fact that equality in the isoperimetric inequality occurs if and only if the domain is a disk.
2. THE MAIN THEOREM We keep the same notation as above. THEOREM. Let d* and f * be the extremal functions in the problem (1). Let N,, denote the number of zeros off * in G. (Zeros on r are counted with half-multiplicity.) Then
SIG I(q3*)‘j2dx dy = (urea
of 4*(G) with multiplicity)
(2)
= -a+LIm
ds - 742 - n + Nf,) A2.
A PROBLEM
IN RATIONAL
129
APPROXIMATION
Note. By continuity, we extend the function F/If*1 points where f * vanishes.
= ePrargf* to the
Proof. The following little argument is due to S. Ya. Khavinson (see [I7,81).
For any OEWG),fEEI(
Ilfll,G 1, we have
(3)
If f *, $* are extremal and if we assume without loss of generality that
then everywhere in (3) equalities hold. Therefore,
CC-~*(i,lf*(i)di=~lf*(i)l
ds
(4)
a.e. on r. Since r is analytic and c$*,f * are analytic near r, we conclude that (4) holds everywhere on r. Let us rewrite (4) in the form
(5) Set
and extend it by continuity to the points where f* vanishes. From (5), it follows that on r we have (6)
According to Stokes’ theorem
130
KHAVINSON
AND
LUECKING
Then, using (5), (6) and taking into account that on f ItI = I&/&( = 1, we obtain j jG ,((*).,'d*d~=~{j)idjl -Lj
j&T
r
ird(;)-ijrigd[
+A2 jrfdzf12
jr;d(z]}.
(7)
Applying Stokes’ theorem again, we obtain (d[=
(8)
-2ia.
Also, ~~~d(~~=j~~=id,arg(~~ = -27ci( 2 - n).
(9)
Integration by parts gives us
so
= 2i Im
Ir
zds.
(10)
Finally, since (71= 1 on r, jrFdi=jr
$=iil,argr = -id,
arg f * = -2niNf..
Combining formulas (7)-( 11) we obtain (2).
(11)
131
A PROBLEMIN RATIONAL APPROXIMATION
3. APPLICATIONS
We keep the same notations as in Sections 1 and 2. COROLLARY
1.
dY= (urea of 4*(G) ilG I(d*)‘I*dx
with multiplicity) (12)
Proof.
Since
(12) immediately follows from (2). COROLLARY 2.
If G is simply connected, then
SI
G I@*)‘,*dxdy<$
Proof. As n = 1, NY, 30 and since q(A) =1P-a-nJ2 maximum at lb= P/2n, (13) follows directly from (12). COROLLARY
(13)
attains its
3. (Isoperimetric inequality). 4na < P2
(14)
Moreover, equality occurs in (14) lf and only if G is a disk of radius 1. Proof For n = 1 (14) follows from (13). If n > 1, then (14) holds for the interior G, of yn (we assume that yn is an outer contour). But a = a(G) < a(G,) and P(G) > P(G,). So, (14) is verified for n > 1. If equality occurs in (14), then it is clear that n = 1. Then, according to (13) 47~2= P* if and only if (#*)’ -0, i.e., c#*E const. Therefore, I[- const) 1r = A. Thus, G is a disk of radius 1.
REFERENCES 1. H. ALEXANDER, On the area of the spectrum of an element of a uniform algebra, in “Complex Approximation” (B. Aupetit, Ed.), pp. 3-12, Birkhluser, Base1 1980. 2. P. J. DAVIS, The Schwarz function and its applications, Cams Math. Monographs 17 (1974).
132
KHAVINSON
AND
LLJECKING
3. P. DUKEN, “Theory of HP-Spaces,” Academic Press, New York/London, 1970. 4. S. FISHER, “Function Theory on Planar Domains,” Wiley, New York, 1983. 5. D. KHAVINSDN, Factorization theorems for different classes of analytic functions m multiply connected domains, Pucific J. Math. 108 (1983) 295-318. 6. D. KHAVINSON, Annihilating measures of the algebra R(X), J. Funct. Anal. 58 (1984). 175--l 93. 7. S. YA. KHAVINSON, Extremal problems for certain classes of analytic functions in finitely connected regions, Muf. Sb. 36 (78)(1955), 445478. [English transl.: Amer Mafh. Sm. Trans. 5 (2)( 1957), l-331 8. S. YA. KHAVINSON, “Foundations of the Theory of Extremal Problems for Bounded Analytic Functions and Their Various Generalizations.” Ministry of High and School Education, SSSR; Moscow Institute of Civil Engineering. Moscow, 1981. [In Russian; English transl. to appear] 9. S. YA. KHAVINWN AND G. C. TUMARKIN. On the definition of analytic functions of class E,, in multiply connected domains, Uspekhi Mar. Nuuk 13 (1958), 201-206. [In Russian]
OF APPROXIMATION
THEORY
50,
127-132 (1987)
On an Extremal Problem in the Theory Rational Approximation
of
D. KHAVINSON AND D. LUECKING* Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, -5J.S.A Communicated by Philip J. Davis
Received October 30, 1984 DEDICATED TO THE MEMORY OF &ZA
1.
FREUD
INTRODUCTION
Let G be an n-connected domain in C bounded by simple analytic curves Let I-= U;=, rj. Let R(G) be the uniform closure in G of the Yn. algebra of rational functions with poles outside of G. We recall that, for domains with analytic boundary, one can define a Smirnov class E,(G) as the collection of analytic functions in G whose boundary values belong to the closure of R(G) in L’(ds). Here, ds is Lebesgue measure on IY (Details concerning the general definition and properties of the classes E, can be found in [3, 5, 91.) In [6] the following concept of rational capacity I has been introduced:
YlY..,
A= l(G) ‘ZfBEi;fG, IIC- d(i)ll. Here, II.11denotes the uniform norm on G. The importance of J. can be seen from a simple observation that A = 0 if and only if G degenerates to the union of analytic arcs. Also, it turns out that 1 enjoys simple estimates in terms of elementary geometric characteristics of G: area and perimeter, i.e., a(G) > I > 24G) -/ /d-- 7l f’(G) ’
(*I
Here, a(G) = a and P(G) = P denote the area and perimeter ( = [r ds) of G, respectively. The first inequality in (*) was proved in [l] and the second in [6]. * Both authors were partially supported by the National Science Foundation.
127 @O21-9045/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
128
KHAVINSON
AND
LUWKING
Using a standard duality argument (see [3], for the case of the unit disk [4-81) one can easily show that
Also, it is not hard to see (via F. and M. Riesz and Banach-Alaoglu theorems) that there exist $* E H”(G) and f* EEL, the extremal functions for which the infimum and supremum are attained. Moreover, as we assume r to be analytic, i.e., [I,= S(c), where S(i) is the so-called Schwarz function analytic in a tubular neighborhood of r (see [2]), it follows from a result of S. Ya. Khavinson (see [7, 81) that both functions ,f* and #* can be analytically continued across I-. In this paper in Section 2 we obtain a simple expression for the area of the image 4*(G) in terms of L and the number of zeros N,. of the extremal funtionf*(z) (zeros on r are counted with a half-multiplicity). In Section 3 as a corollary of this theorem we obtain the classical isoperimetric inequality with sharp constants (Corollary 3). Also, we show (Corollary 2) that the area of the image of the best approximation d* to C in simply connected domains is always dominated by the “isoperimetric deficiency” 1 - &a/P2 of G. In particular this estimate gives another proof of the well-known fact that equality in the isoperimetric inequality occurs if and only if the domain is a disk.
2. THE MAIN THEOREM We keep the same notation as above. THEOREM. Let d* and f * be the extremal functions in the problem (1). Let N,, denote the number of zeros off * in G. (Zeros on r are counted with half-multiplicity.) Then
SIG I(q3*)‘j2dx dy = (urea
of 4*(G) with multiplicity)
(2)
= -a+LIm
ds - 742 - n + Nf,) A2.
A PROBLEM
IN RATIONAL
129
APPROXIMATION
Note. By continuity, we extend the function F/If*1 points where f * vanishes.
= ePrargf* to the
Proof. The following little argument is due to S. Ya. Khavinson (see [I7,81).
For any OEWG),fEEI(
Ilfll,G 1, we have
(3)
If f *, $* are extremal and if we assume without loss of generality that
then everywhere in (3) equalities hold. Therefore,
CC-~*(i,lf*(i)di=~lf*(i)l
ds
(4)
a.e. on r. Since r is analytic and c$*,f * are analytic near r, we conclude that (4) holds everywhere on r. Let us rewrite (4) in the form
(5) Set
and extend it by continuity to the points where f* vanishes. From (5), it follows that on r we have (6)
According to Stokes’ theorem
130
KHAVINSON
AND
LUECKING
Then, using (5), (6) and taking into account that on f ItI = I&/&( = 1, we obtain j jG ,((*).,'d*d~=~{j)idjl -Lj
j&T
r
ird(;)-ijrigd[
+A2 jrfdzf12
jr;d(z]}.
(7)
Applying Stokes’ theorem again, we obtain (d[=
(8)
-2ia.
Also, ~~~d(~~=j~~=id,arg(~~ = -27ci( 2 - n).
(9)
Integration by parts gives us
so
= 2i Im
Ir
zds.
(10)
Finally, since (71= 1 on r, jrFdi=jr
$=iil,argr = -id,
arg f * = -2niNf..
Combining formulas (7)-( 11) we obtain (2).
(11)
131
A PROBLEMIN RATIONAL APPROXIMATION
3. APPLICATIONS
We keep the same notations as in Sections 1 and 2. COROLLARY
1.
dY= (urea of 4*(G) ilG I(d*)‘I*dx
with multiplicity) (12)
Proof.
Since
(12) immediately follows from (2). COROLLARY 2.
If G is simply connected, then
SI
G I@*)‘,*dxdy<$
Proof. As n = 1, NY, 30 and since q(A) =1P-a-nJ2 maximum at lb= P/2n, (13) follows directly from (12). COROLLARY
(13)
attains its
3. (Isoperimetric inequality). 4na < P2
(14)
Moreover, equality occurs in (14) lf and only if G is a disk of radius 1. Proof For n = 1 (14) follows from (13). If n > 1, then (14) holds for the interior G, of yn (we assume that yn is an outer contour). But a = a(G) < a(G,) and P(G) > P(G,). So, (14) is verified for n > 1. If equality occurs in (14), then it is clear that n = 1. Then, according to (13) 47~2= P* if and only if (#*)’ -0, i.e., c#*E const. Therefore, I[- const) 1r = A. Thus, G is a disk of radius 1.
REFERENCES 1. H. ALEXANDER, On the area of the spectrum of an element of a uniform algebra, in “Complex Approximation” (B. Aupetit, Ed.), pp. 3-12, Birkhluser, Base1 1980. 2. P. J. DAVIS, The Schwarz function and its applications, Cams Math. Monographs 17 (1974).
132
KHAVINSON
AND
LLJECKING
3. P. DUKEN, “Theory of HP-Spaces,” Academic Press, New York/London, 1970. 4. S. FISHER, “Function Theory on Planar Domains,” Wiley, New York, 1983. 5. D. KHAVINSDN, Factorization theorems for different classes of analytic functions m multiply connected domains, Pucific J. Math. 108 (1983) 295-318. 6. D. KHAVINSON, Annihilating measures of the algebra R(X), J. Funct. Anal. 58 (1984). 175--l 93. 7. S. YA. KHAVINSON, Extremal problems for certain classes of analytic functions in finitely connected regions, Muf. Sb. 36 (78)(1955), 445478. [English transl.: Amer Mafh. Sm. Trans. 5 (2)( 1957), l-331 8. S. YA. KHAVINSON, “Foundations of the Theory of Extremal Problems for Bounded Analytic Functions and Their Various Generalizations.” Ministry of High and School Education, SSSR; Moscow Institute of Civil Engineering. Moscow, 1981. [In Russian; English transl. to appear] 9. S. YA. KHAVINWN AND G. C. TUMARKIN. On the definition of analytic functions of class E,, in multiply connected domains, Uspekhi Mar. Nuuk 13 (1958), 201-206. [In Russian]