Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence, III

Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence, IIIAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 56, No. 5 (Nov. 15, 1966), pp. 1406-1408Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/57752 .

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APPROXIMATION BY POLYNOMIALS: UNIFORM CONVERGENCE AS IMPLIED BY MEAN CONVERGENCE, II1*

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MARYLAND

Communicated September 6, 1966

The present note is the continuation of two notes" 2 with the same title that I have recently published. For polynomials in the complex variable, these notes use degree of convergence in the mean to establish results on degree of uniform convergence and also (Theorems 8 and 9) to establish results on degree of mean convergence of one order from degree of mean convergence of a lower order. The purpose of the present note is to strengthen the previous results on the latter topic, by replacing an inequality in the original hypothesis by a weaker and more natural inequality, notably by replacing a > l/q (which is independent of p) in Theorem 8 by a > l/q - l/p; the latter inequality is essential for the convergence of a certain comparison series. Notation and numbering of theorems and equations here agree with those of references 1 and 2.

The method of (11), (14), (15), and (16), which is also the method of (21), (22), and (23) concerning approximation, can be used to strengthen Theorem 1 concerning a single polynomial. We proceed to prove

THEOREM 10. With the hypothesis of Theorem 1 and 0 < q < p < oo, we have for the norms on r

IIP(Z)IIP < L2n,,q-l/pll()lX(7

where the constant L, is independent of n and z. As before, we write |P(Z) lq = L, whence by (1)

|IP(z)11- :! L1nl,qL,

fr IP(Z)lPdzI < fr |P(z)jI|dz| *lp(z)jp_ < L,P-qnP-l L

which is essentially (27). The same method yields the COROLLARY. With the hypothesis of Theorem 6 and 0 < q < p < ca, we have for

the norms on r (or on C)

||P(z,11z)j|p _ L3nl/q-l/pl jP(zj1Z)ll q1 (28)

where the constant L3 is independent of n and z. This corollary applies to an arbitrary trigonometric sum in 0 of order n, if r

is chosen as the unit circle, with z = etG; it is related to a lemma involving general- ized derivations due to Nikolskogo and used by Ogieveckii (ref. 2 of paper II), both of whom require q _ 1.

THEOREM 11. With the hypothesis of Theorem 2, including (2) and with 0 < q < p _ co , a sufficient condition for the convergence in the mean of order p of the sequence pn(z) to f(z) on r is nre n 0 plus the existence and boundedness of (2m1 _ n < 2')

2mrEC + 2(m +1)r 2m + 2(m+2)rM+E + 1, (29)

1406



VOL. 56, 1966 MATHEMATICS: J. L. WALSH 1407

(2`m)PrenP + (2mn+l)Pv-2mP + (2m+2)Pre2m lP + ... < 1, (30)

where r = l/q - 1/p. If that condition is satisfied, we have (23) for the norm on r:

lif(Z) -pn(z)I|p ? An"en. (31) In particular if En _ A/no, where ,B > r, then (29) or (30) is satisfied, as is nrEn 0;

we have (31) with the second member equal to A/n~r. The conclusion of Theorem 2 deals with p = o, thus p > 1, but the usual dis-

tinction in treatment of the classes p ? 1 and p < 1 is to be made in the proof of Theorem 11. We write formally (p _ 1)

Iif - Pnl P ? IIPAm - PnlJP + IlP2m+1 - P2mIIp + lIP2m+2 - P2m+cI p +

and by (2) and (27) the second member is dominated by A multiplied by

2mrEn + 2(m+l)rE2m + 2(m+2)rE2m+l + *. . ,

so by the existence and boundedness of (29) we deduce effectively (31), which shows convergence of pn(z) to f(z) on r in the mean of order p.

If p < 1, we write formally from the general inequality

Ix1 + X21P < Ix11P + 1x21P,

l-f pn|| P < IlP2n -

Pn|I|P + IlP2rn+1 -

P2mlIpP + IP2m+2 - P2mlIPI + * X

whose second member is dominated by A multiplied by

(2m)prE nP + (2m+l)Pr f2m p + (2m+2)prE2m+lp + *. ,

from (27), so by the existence and boundedness of (30) we deduce effectively

IIf - pnl lP < AnprEnp,

which is equivalent to (31). The remaining part of Theorem 11 follows at once. The method of proof of Theorem 11, by use now of the Corollary to Theorem 10

instead of Theorem 10 itself, gives now THEOREM 12. With the hypothesis of Theorem 7, including

jjf(Z) - pn(z,1/z)flq _ E, q > 0 ,n ? 0 (32) and nrE- 0, a sufficient candition for the convergence in the mean of order p of the sequence pn(z,l/z) tof(z) on r is the existence and boundedness of (29) or (30). If that condition is satisfied, we have (23) for the norm on r:

Ilf(z) - pn(Z,1/Z)lIp _ AnrEn (33)

In particular if en ? A/no where , > r, then (29) or (30) exists and is bounded. We have nrEn -- O and hence (33) with the second member replaced by A/n'r.

The preceding results of the present note extend easily to the use of double integrals as norms. An analogue of Theorem 3, whose proof is similar to that of Theo- rem 10, is

THEOREM 13. With the hypothesis of Theorem 3, if we denote with primes the norms as measured by double integrals over D, we have (0 < q < p < co)



1408 MATHEMATICS: J. L. WALSH PROC. N. A. S.

IIP(Z)IIP' ? L3n91IP(z)Iqf', s = 2/q - 2/p. (34)

THEOREM 14. Let the hypothesis of Theorem 4 be satisfied, including (O < q < p < co~

IIf(z) - Pn(Z) IIq' a< En (8)

and also n"en 0. Then a sufficient condition for the convergence in the mean of order p of pn(z) to f(z) on D is the existence and boundedness of (29) or (30) with r replaced by s. If that condition is satisfied, we have for the norm on D

flf(Z) - Pn(Z)|l|p < An' En. (35) In particular, if en ? A/n, where : > s, then, with r replaced by s, (29) or (30)

exists and is bounded. We have n8en -O 0, and hence (35) with the second member equal to A/n-8.

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

1 Walsh, J. L., these PROCEEDINGS, 55, 20 (1966). 2 Ibid., p. 1405.



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Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence, III