Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence, IIAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 55, No. 6 (Jun. 15, 1966), pp. 1405-1407Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/57210 .

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APPROXIMATION BY POLYNOMIALS: UNIFORM CONVEIRGENCE AS IMPLIED BY MEAN CONVERGENCE, II*

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, UUNIVERSITY OF MARYLAND

Communicated April 22, 1966

I published recently in these PROCEEDINGS a short note' with this same title (except for the numeral), in which I deduced for functions of a complex variable z analytic in a region results on degree of uniform convergence in the closed region of sequences of polynomials, from hypotheses of mean convergence of those sequences on the boundary of the region. The present note has the purpose of (1) establishing similar results for approximation by polynomials in z and I/z on a smooth Jordan curve, and (2) deriving results on mean approximation of order p from hypotheses of mean convergenice of order q, 0 < q < p. This note is to be considered a continu- ation of the preceding one; notation and numbering of theorems and formulas are in accord with this interpretation.

We say that a Jordan curve C is of type B' if it is rectifiable, and if there exists a fixed number 6' (>O) such that through eac:h point of C passes some circle y of radius 6' whose closed interior lies in the closed exterior of C. If the origin 0 lies interior to C, it follows that the image of C under the transformation w = I/z is of type B.

Theorem I for a siiigle Jordan curve (due to H. E\'t. Elliott) admits a generaliza- tion:

THEOREM 6. Let C be a Jor-dan curve of types B and B' containing in its interior the origin z = 0, and let a be greater than zero. Then there exists a constant L1 depend- ing only on C and q such that for an arbitrary polynomial P(z, i/z) in z and I/z of degree n (>0) we have for z on C

I P(z, llz) I, < Lin X p(Z, I/Z) I qIdzl. (17)

As in the proof of Theorem 1, we set fC P(z, I/z) I q dz = Lq, and set also (in the notation preceding Theorem 1)

P(z, l/z) [I - -(ai) 9(z)] ... [I - (k) 9(z)

[9(z)] [9(z) - (oi)] * * (z) - (k)

where in is the order of the pole (if any) of iP(z, l/z) at infinity, and ai,, a2,. . ak

are the zeros of P(z, l/z) exterior to C. C)n C we have I 5P(z)I = 1, whence f'cI Q(z) I ads = L. The function [Q(z) ] /sp(z) is analytic exterior to C and zero at infinity, so for z exterior to C we have

[Q(z)] __ 1 [Q (t) ] dt. (P(z) 2 7ri J c p(t) (t -z)

We choose now z on C, with R = 1 + I/n, and note that I t- z > o/n:

Q(z)Iq < LQn IP(Z, I qZ)I < IQ(z) I(1 + 1/n)nq < L20LIn. (18)

In (18) we can choose L'1 = 2e, where L2 depeinds only on C. 1405

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1406 MATHEMA TICS: J. L. WALSII P)RoC. N. A. S.

Let the functioin W = 1/z map the interior of C onito the exterior of some Jordaln curve r in the W-plane which then is rectifiable and of type B. We have

f P(z, l/z) Iq dW If P(z, 1/z) I dWI dzI < L3fIP'(z; I/z) q dz| (19)

wlhere L3 = maxI dW/dz| = max| z 21 for z on C. Just as the properties of C and P(z, l/z) imply (18), so do the properties of r and P(z, l/z) imply with some L4

IP(z, I/z)I a < L4nq (20)

for z on the loc us |b(1/z) I R = 1 + I/n interior to C, where W141 = (W) miaps the exterior of r onto I W1 > 1 with 4( C) = Co. This locus anid CR together bound ani annular region which contains C and whose closure conitains rno siingularity of P(z, l/z), so (17) follows from (18) and (20).

The method of proof of Theorem 2 now yields THEOREM 7. Let C be a Jordan curve of types B and B' containing in its interior

the origin z = 0, and let there be given on C a function f(z) and a sequence of poly- nomials pn(Z, l/z) in z and I/z of respective degrees n (>O) such that we have (2) for the qth power norm on C, wher-e pn(Z) is replaced by pn(Z, l/z), and where En is mono- tone nonincreasing for n sufficiently large and tends to zero (n -a Co). Then a sufficient condition for the uniform convergence of pn(z, l/z) to f(z) on C is the existence and boundedness (n co) of (3) where 2m-1 < n < 2m. If this condition is satisfted, we have (4) for the Tchebycheff norm on C.

We add the analogues of Corollaries I aricd 2 to Thleorem 2: COROLLARY. If En = A/na wher-e t > l/q, or- En = A/na pfn, p > 1, then (3) is

bounded and (4) is satisfied. The method of (11), (14), (15), and (16) to study a norm of order unity can also

be used to study the inorm of order p (>q) whein (2) is given. With the hypothesis of Theorem 2 or of Theorem 7, iincludiing existence aind boundedness of (3), we write (2) in the form

I f - Pn I "dzI 'E (21(2 1 )

wh1ere Pn may indicate pn(z) of Theorem 2 or pn(z, i/z) of Theorem 7. With Co > p > q > 0, we write (4) as

If - pn < An'lIfn (22)

fo:r z on C. If we take the (p - q)th power of bothi nemnbers of (22) ani id multiply inito (21), umder -the initegral sigin in the first miieinber, there results

ff -IPn pI dzI < AnP?q-1fnP)

II- pn p ? AnI/lIv-/P6n (23)

an inequality valid also for p = co. In particular, if en - A an-a, a > 1/q, the second member of (23) becomes A2nl1/1/Pa. We have proved

THEOREM 8. With the hypothesis of Theorem 2 or of Theorem 7, including the ex- istence and boundedness of (3), then (23) is valid, 0 < q < p < oo. In particular, if en -Ain-, a > l/q, the second member of (23) can be replaced by A2nlI-lIva

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VOL. 55, 1966 BOTANY: P. F. SCHOLA2NDER? 1407

In Theorems 7 and 8 we may choose C as the unit circle. Then by the Euler equations, any polynomial pn(z, 1/z) of degree n with z = -e is a trigoiioinetric sum in 0 on C of order n. We have

THEOREAMI 9. If F(O) is a function with period 2r, and if there exist trigonomnetric sumns tn(0) of order n satisfying for the qth power (q > 0) normn on the interval [0, 2X]

||F(O) - tn(o) 1 (24)

where en is mnonotone nonincreasing for n sufficiently large, then a sufficient condition for the uniforn convergence of tn (0) to F(0) for all values of 0 is the existence and bounded- ness (n - oo) of (3) wher-e 2m-1 < n < 2mn. If this condition is satisfted, we have for the Tchebycheff normn on [0, 2r]

I|F(0) - tn(0)OJ< AnlI An . (25)

More generally, we have (O < q < p < co)

||F(0) - tn(O)|| < A? I/q-1/Pen, (26)

and if En = An-a, a > 1/q, the second mnember of (26) becomnes An1q--11P-a. For q > 1 there is in the real domain an intimate connection between degree of

convergence in the qth power mean of trigonometric polynomials on the one hand, and integrated Lipschitz conditions on the other, first inidicated by Hardy and Littlewood, then developed in detail by Quade, and later by Zygmund. Instead of using integrated Lipschitz conditions, one may use fractional derivatives, inltroduced into the theory of approximation by trigonom-etric polynomials by M\'ontel (1918). Interrelationis among these properties were developed by Ogieveckil,2 who seems, however, not to have proved the properties expressed in Theorem 9 relating ap- proximation of one and the same function as measured by means of two different orders. Ogieveckil does not treat the cases p < 1 or q < 1.

* Research sponsored (in part) by U.S. Air Force Office of Scientific Research, Air Rtesearch and Development Command.

1 Walsh, J. L., these PROCEEDINGS, 55, 20 (1966). 2 Ogieveckil, I. I., in Investigations on Contemporary Problems of Constructive Theory of Functions

(Itussian), ed. V. I. Smirnov (Moscow, 1961), pp. 159-164.

THE ROLE OF SOLVENT PRESSURE IN OSMIOTIC SYSTEMJS*

BY P. F. SCHOLANDER

SCRIPPS INSTITUTION OF OCEANOGRAPHY, UNIVERSITY OF CALIFORNIA, S_AN DIEGO

Communicated March 22, 1966

The cohesive force of water has been given variously, on theoretical grounds,1 as between 10,000 and 17,000 atmospheres, and Briggs obtained negative pressures in spinning glass capillaries of some -260 aLm at room temperature.2 Since the establishment of the cohesion theory for sap rising in plants,3 most plant physiol- ogists have predicted that negative pressure -must prevail in the xylem. Negative hydrostatic pressure in a capillary or microporous system depends, of course, upon its rigidity; that is, at equilibrium the expansion force of the matrix is balanced by

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