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Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence Author(s): J. L. Walsh Source: Proceedings of the National Academy of Sciences of the United States of America, Vol. 55, No. 1 (Jan. 15, 1966), pp. 20-25 Published by: National Academy of Sciences Stable URL: http://www.jstor.org/stable/57413 . Accessed: 07/05/2014 17:40 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the National Academy of Sciences of the United States of America. http://www.jstor.org This content downloaded from 169.229.32.136 on Wed, 7 May 2014 17:40:13 PM All use subject to JSTOR Terms and Conditions

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Page 1: Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence

Approximation by Polynomials: Uniform Convergence as Implied by Mean ConvergenceAuthor(s): J. L. WalshSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 55, No. 1 (Jan. 15, 1966), pp. 20-25Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/57413 .

Accessed: 07/05/2014 17:40

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

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Page 2: Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence

APPROXIMATION BY POLYNOMIALS: UNIFORM CONVERGENCE AS IMPLIED BY MEAN CONTVERGENCE*

BY J. L. WALSH

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Communicated November 24, 1965

There are very few places in the literature where degree of convergence in the qth power mean norm of a series of polynomials (complex domain) is used to deduce degree of convergence in the Tchebycheff (uniform) norm of that same series; the object of this note is to indicate that methods previously used for other purposes apply with appropriate modifications also for this purpose.

In one of the early papers on this subject, J3ackson2 considers approximation to a given function f(z) on the boundary C of a given Jordan region (satisfying suitable auxiliary conditions) where f(z) is analytic interior to C, continuous on C. He assumes as known the degree of Tchebycheff approximation to f(z) of an auxiliary sequence of polynomials Pn(z) of respective degrees n: ||f(z) - Pn(Z)|| <_ En and then shows for the polynomials Pn(Z) of respective degrees n of best approxi- mation to f(z) on C in the mean of order q(> 0) that we have ||f(z) - Pn(Z) I_I <

const n1/ En, provided nllIq En O* 0 as n -- co. Jackson's theorem is direct and easy to apply, but has two disadvantages: (i) it requires knowledge of degree of convergence in the Tchebycheff norm of an auxiliary sequence of polynomials Pn(z) to f(z) on C, and (ii) the conclusion applies only to sequences of polynomials of best approximation in the qth power norm. Nevertheless, Sewell's book3 contains numerous applications of Jackson's theorem.

The present results deduce degree of convergence on C in the Tchebycheff norm directly from degree of convergence in the qth power norm. The method of proof, of choosing various partial sums of the given series, of considering the difference of successive partial sums, of applying a convenient lemma concerning poly- nomials to estimate the derivatives or some other related expressions of these differences, and finally summing the newly obtained sequence, is due to de la Vall6e-Poussin, and has been used at various times by Walsh, Sewell, and Elliott;6 Elliott;' and Walsh and Russell.5 The last two of these papers are especially close to the present note in method and result, but do not make full use of the method.

A Jordan curve C is said to be of type B if it is rectifiable, and if there exists a fixed number bo(>O) such that through each point of C passes a circle y of radius bo whose closed interior lies in the closed interior of C. We denote by tv = (0(z) the function which maps the exterior of a Jordan curve J onto the region iwI > 1, with p(co) = co, and denote generically by JR(R > 1) the locus 1so(z) = R. If ey: lz -zo = bo denotes one of the circles Just described, then z-zo = &J? is the locus yR. By the monotonic property of the level loci of Green's function for a variable region, it follows for fixed R that CR contains in its closed interior each circle 'YR, since the closed interior of C contains each circle 'Y. Thus, the distance from C to CR is not less than bo(R - 1), the distance from 'y to yR. The following is a slight generalization of Theorem 6.1 of reference 1:

THEOREM 1. Let IF consist of a finite number of mutually exterior Jordan curues

20

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Page 3: Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence

VOL. 55, 1966 MATHEMATICS: J. L. WALSH 21

each of type B. Let P(z) be an arbitrary polynomnial of degree n (>0). Then we have for z in the closed interior of r

IP(z)lg < Lin,fr jP(#'ljdzj, q> O, 1

where the constant L, depends only on P. It is clearly sufficient to prove this theorem, as we do, in case r consists of a single

Jordan curve C. We set fc P(z) lads L", and set also (in the notation already introduced)

P(z) [1 - fo(ai) (Z) ](*. [1 - (ah) f(Z)

[k (z) ] [f (z) - 9(ai)j ...* [ f (z) 9p(ah)]

where m (< n) is the order of the pole (if any) of P(z) at infinity, and al, a2, ah are the zeros of P(z) exterior to C. On C we have 9(z)# = 1, whence fClQ(z)Iqds = L- . The function [Q(z)]q/l(z) is analytic exterior to C and zero at infinity, whence for z exterior to C

[Q(z) ]q 1 IQ(t) I qdt ~P(z) 27ri 3Jc s(t)(t -Z)

We choose now z on CR with R = 1 + 1/n, and note that It - zl > bo/n:

IQ(Z )I n- n > O IP(z)|2 ?

IQ(z)#q(1 + 1/n)n < L? Lqn. R 2wr8o

This inequality is valid for z on CR, hence for z on C, and (1) follows. THEOREM 2. Let P consist of a finite number of mutually exterior Jordan curves

each of type B, and let there be given on F a function f (z) and a sequence of polynomials pN(z) of respective degrees n (>0) such that we have for the qth power norm on r

|f(z) - pn(Z)llIq - En, q > 0, En ?4O (2)

where En is monotonic nonincreasing for n sufficiently large. Then a sufficient condition for the uniform convergence of pn(z) to f(z) on P is the existence and boundedness (n -- co) of

2m/qEn + 2(m+l)/I.2. + 2:(`+2)/qC'(.+1) + *3

n lIEn (3)

where 2?-1 ? n < 2m. If this condition is satisfied, we have for the Tchebycheff norm on P

||f(Z) - Pn(Z)11 ? Anl/ Es. (4) Here and below, A represents a constant independent of n and z, which may change from one inequality to another.

It is sufficient to establish Theorem 1 when P consists of a single curve. We write, for the moment with purely formal significance, for z on r

If - Pnl - IP,m - PnI + P2-+ P, + Ip,m+2 - Pm+I + . . . < A [2ml Ip2n - Pnfl I + 2(m+1)Iql p 2in- p,l 2 +11 . . ], (5)

by Theorem 1. With q _ 1, we may write by the triangle inequality

11P2- Pn| I a _ 1I If P p2m | M+ f - Pn| Iq; (6)

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22 MATHEMATICS: J. L. WALSH PRoc. N. A. S.

with q < 1, we use the standard inequalities Jx1 + X2 q lXllq + JX21q2 lX1 + X211lq < 21/q%1l1N[ /q + |X2 l'], to deduce p2- Pn||, < (f V - p2-q + f If -Pn I Iq ? 2'l-'[(f If - M d)1 + (f If - pnl2)l/l] < 2'lq-[ [|f -

p211 q+ f 1 q] From (5) we now have formally for z on F

If- Pn| < Aj[2"/qEn + 2(m+l)Iqe2 2 + 2(m+2) Iq '2 + ..

if (3) has a mneaning and is bounded, we deduce (4), not muerely formally but effec- tively. Of course, f(z) is defined merely almost everywhere on r by (2); the function f(z) is defined everywhere on F by (4).

COROLLARY 1. If en = A/n'3 where f3> 1/q, then (3) is bounded and (4) is satisfted. Here the expression (3) is not greater than

2mlq 2(m+1) Iq *2-m3 + 2(m+2)/q.2-(m+1)'3 +

n lq + 2(m-1) /I2. -m'3

_ 2Ilq + 22Iq/(l - 21 lq-I3)

so the condition of Theorem 2 is satisfied, and (4) is established. Corollary 1 is a contribution to the study of Problem a in approximiiation by

polynomials, namely, approximation on a set P to a function whose properties on P are known, as we indicate in more detail below.

There exists a corresponding result5 for Problem f, where the function f(z) is analytic throughout the interior of rP , and for p > 1 of class H(k, a, p) or Z(k, p) on Fp, assumed to have no multiple points; inequality (2) is satisfied with en= A/n'3pn, n > 1; here k may be negative:

COROLLARY 2. If en - A/n 0pn, p > 1, then (3) is bounded and (4) is satisfied; the exponent d mnay be negative.

The expression (3) is not greater than (we set X = l/q - 3)

2mIq 2(m+l)/q-mP-2m + 2(m+2)/q-(m+l)i3P-2m+ +

n +ll 2(m-1) q-m p-2?

_22/ + 22/q + 2 31q-,l0 p-2m + 24/q-21 p-2m+2 +2 +t

2' 22X 2x = 22/q + 22/q 2+ + p2 + +2 + 2~ ~ 3.2"n 7.2m .

p p* p* 22/q

< +2lq-

1 - 2X/p2m

which is bounded; this completes the proof of (4). Nowhere in Theorem 2 or one of its corollaries do we assume that the sequence

Pn(Z) is extremal. It is of considerable interest to study the aiialogues of Theorems 1 and 2 with

the line integrals replaced by surface integrals. THEOREM 3. Let C be an analytic Jor-dan curve and let J'(z) be an arbitrary poly-

nomial of degr-ee n (>O). Then we have for z in the closed interior D of C

JP(z) I _ L2n 2LO, Ljq = f fDIP(z)qdS, q > 0, (7)

tvhere the constant L2 depend-s only on C.

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Page 5: Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence

VOL. 55, 1966 MATHEMATICS: J. L. WALSH 23

Contrary to our usual custom, we denote by CR the level locus I |Z) = R for the function w = p(z) which maps the exterior of C onto lwl > 1 with s0( o) = , even when R is less than but sufficiently near unity; then CR remains an analytic Jordan curve. The shortest distance to C from an arbitrary point of CR, R < 1, is greater than some constant Q times (1 - R) when R is sufficiently near unity, because w = sp(z) transforms conformally and one-to-one the closed annulus R ? iwl ? 1, so that the difference quotient Sw/Az is bounded in that closed alinulus

Q' > ?Aw/Az|, lAzl > IAwI/Qj > (1 - R)/Q1 = Q( -R),

where the vector Az goes from a point of CR to a point of C. It follows by a well-known lemma (?5.3, Lemma II, of ref. 4) that if z lies on

CR, R = n/(n + 1), n > 0, we have

______<__L _ _ (n + 1) 2L1

7rQ2(j - R)f2 - rQ2

The generalized Bernstein lemma (?4.6 of ref. 4) now yields for z on [CRl(n?1)In = C

IP(z)Iq < (n + 1) ( + 2)n 4n2eLq

which is of form (7). The formal difference between (1) and (7) is that the respective exponents of

n are unity and two. Theorem 2 thus has an analogue that refers to the surface integral over D rather than the line integral over C, and we may use r rather than a single curve C.

THEOREM 4. Let F consist of a finite number of mutually exterior analytic Jordan curves, the union of whose closed interiors is denoted by E, and let there be given on E a function f(z) and a sequence of polynomials Pn(Z) of respective degrees n (>0) such that we have for the qth power norm on E

|\f(z) - Pn(Z)| fq >= E,n q > 0, E n> O, (8)

where En is monotonically nonincreasing for n sufficiently large. Then a sufficient condition for the uniform convergence of Pn(Z) to f(z) on E is the existence and bounded- ness (n --- co) of

22m/Me + +2(m?l)I + 2(m+2)/q1 +.

n2/2en n (9)

where 2`1 < n < 2m. If this condition is satisfied, we have for the Tchebycheff norm on E

llf(Z) - Pn(Z) <An2I qC-. (10) COROLLARY 1. If en = A/no where : > 2/q, then (9) is bounded and (10) is

satisfied. COROLLARY 2. If en = A /n pn, p > 1 , then (9) is bounded and (10) is satisfied;

the exponent : may be negative. Each of these three propositions may be proved from the analogous proof (of

Theorem 2 or one of its corollaries) by-replacing the previous q by q/2.

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Page 6: Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence

24 MATHEMATICS: J. L. WALSH PRoc. N. A. S.

Jackson2 considers also surface integrals instead of line integrals in defining the norm as in (8), and his corresponding condition for uniform convergence of the extremal polynomials is n2qe1 _* 0. Clearly, the results of the present note tend to supplement, rather than replace, the results of Jackson. Introduction of positive continuous weight functions presents no difficulties.

Theorems 1-4 and those of Jackson refer to approximation by polynonmials, but precisely similar theorems, including the corollaries, apply to approximation by sequences of rational functions of respective types (n, ') for constant v, with some free poles. The writer plans to publish elsewhere8 these analogues of the theorems of Jackson; the analogues for rational approximation of the present theorems can be similarly treated by the present methods.

The results established above catn be generalized to include approxim-iation by bounded analytic functions. We indicate an application of the preceding results.

THEOREM 5. Let the point set E with boundary P consist of the closed interiors of a finite number of mutually exterior analytic Jordan curves, and let rP like I consist also of a finite number of mutually exterior Jordan curves. Let the function f(z) be analytic interior to rp and continuous on Ip, and let there exist a sequence of poly- nomials p, (z) of respective degrees n such that (for the boundary values of f(z))

frplf(z) - p.(z)jPldzl ? A/nOp, 0 < p < 1. (11)

Then there exists a sequence of polynomials Pn(z) of 7respective degrees n such that

If(z) - P.(z)1 < A/np+]?-l/Ppfn z on E, (12)

if + 1 - i/p > 0. The method previously used (ref. 7, Theorems 7.8 and 10.5; ref. 5, Theorem 3)

in case p ? 1 is the formula

f(z) - P (z) 1 , ()

U(t) - pn(t) ]dt z On E, (13) 27ri con t) (t -z) zn, (3

where Pn(z) coincides with f(z) in- n + 1 points equally distributed on sonme rP, r < 1, roots of the polynomial wOn(Z) zn+' + .... But that method uses H6lder's inequality (or its equivalent if p 1) and does not apply in the present case. However, Corollary 1 to Theorem 2 yields from (11)

lf(z) - Pn(Z)I < A/n0-11P, z on EX, (14)

if ll-i/p > 0; equation (13) holds, and we deduce (12) but with the first exponient replaced by d - i/p.

A more powerful method is to use again (11) and (14), to choose q 1 - p(> 0), to write

|f(z) - p.(z)|8 < A/nq(-11P), z on E., (15) as a consequence of (14), then to combine (11) and (15):

frplf(z) - pn(z)]?+qidz < A/n(P+q)#- P, p + q = 1. (16)

The conclusion (12) follows from (16) and (13). The choice q 1 - p gives the smallest exponent in (16) to which the usual

reasoning (Holder's inequality or a substitute) applies.

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Page 7: Approximation by Polynomials: Uniform Convergence as Implied by Mean Convergence

Voif. 55, 1966 MATHEMATICS: L. W. SMALL 25

This last method, use of (11) with a power (15) of (14), seems very convenient in the application of (13).

* Research sponsored (in part) by Air Force Office of Scientific Research. 1 Elliott, H. M., "On approximation to functions satisfying a generalized continuity condition,"

Trans. Am. Math. Soc., 71, 1 (1951). 2Jackson, D., "On certain problems of approximation in the complex domain," Bull. Am.

Math. Soc., 36, 851 (1930). 3Sewell, W. E., "Degree of approximation by polynomials in the complex domain," Annals of

Mathematics Studies (Princeton, 1942), no 9. 4Walsh, J. L., "Interpolation and approximation," in Coil. Pubs. Am. Math. Soc. (1935), vol. 20. 5 Walsh, J. L., and H. G. Russell, "Integrated continuity conditions and degree of approxima-

tion by polynomials or by bounded analytic functions," Trans. Am. Math. Soc., 92, 355 (1959). 6 Walsh, J. L., W. E. Sewell, and H. M. Elliott, "On the degree of polynomial approximation

to harmonic and analytic functions," Trans. Am. Math. Soc., 67, 381 (1949). 7 Walsh, J. L., and W. E. Sewell, Trans. Am. Math. Soc., 49, 229 (1941). 8 Walsh, J. L., "The convergence of sequences of rational functions of best approximation, III,"

in preparation.

HEREDITARY RINGS

BY LANCE W. SMALL

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY

Communicated by S. S. Chern, November 29, 1965

Kaplansky2 and the author' have given examples of right hereditary rings which were not left hereditary. In both instances the left global dimension was two. Indeed, in every known case of a ring with differing global dimensions, the difference was one. In this note we present an example of a right hereditary ring with left global dimension equal to three. This ring has the additional property of not being left semihereditary; thus, we sharpen an example of Chase.'

In the second section of this note we extend certain results of Chase to Noetherian hereditary rings. In particular, many theorems about Artinian hereditary rings, with minor modifications, are seen to be true for Noetherian hereditary rings.

1. An Example in Hereditary Rings.-Kaplansky constructed a ring A which has the following properties: (i) A is an algebra over a field F, (ii) A is von Neu- mann regular, (iii) the right ideals of A are countably generated, and (iv) there is a left ideal L such that the homological dimension (hdA) of L is 1.

The left A-module B = A/L may also be considered as a (right) vector space over F. Let T be the ring of all two-by-two "matrices" of the form:

(a b

I a E Al b EB, f C F

where addition is componentwise aind multiplication is given by:

(a b)(a' b') (aa' ab' + bf' T is a fas -oitv ffr w u

T is an associative ring with unit.

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