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The Convergence of Some Non-Linear Processes of Approximation Author(s): Dunham Jackson Source: American Journal of Mathematics, Vol. 55, No. 1 (1933), pp. 515-524 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/2371148 . Accessed: 04/12/2014 20:35 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]. . The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. http://www.jstor.org This content downloaded from 128.83.63.20 on Thu, 4 Dec 2014 20:35:18 PM All use subject to JSTOR Terms and Conditions

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Page 1: The Convergence of Some Non-Linear Processes of Approximation

The Convergence of Some Non-Linear Processes of ApproximationAuthor(s): Dunham JacksonSource: American Journal of Mathematics, Vol. 55, No. 1 (1933), pp. 515-524Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371148 .

Accessed: 04/12/2014 20:35

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].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

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Page 2: The Convergence of Some Non-Linear Processes of Approximation

THE CONVERGENCE OF SOME NON-LINEAR PROCESSES OF APPROXIMATION.*

By DUNHAM JACKSON.

1. Introduction. Let f(x) be a positive continuous function of period 2'r, and let an approximation be sought for it in the form eTn(x), where Tl (x) is a trigonometric sum of the n-th order. If Tn (x) is any sum approximating log f(x), naturally eTn(,) gives some sort of approximation to f(x). This paper is concerned specifically with the properties of sums Tln(x) chosen so as to minimize the integral

7r (1) t (X) I f(x) - eTn(x) Int dx,

where p(x) is a given non-negative summable weight function and m a given positive exponent. A significant feature of the problem is the fact that the approximating function depends non-linearly on the fundamental functions cos kx and sin kx in terms of which it is expTessed, and the particular form chosen, in spite of its simplicity, introduces complications which were not encountered in an earlier paper by the writer with a similar title.t It will be shown nevertheless that under appropriate hypotheses the minimum prob- lem has a solution (which to be sure is not shown to be unique), and that with more restrictive hypotheses the approximating functions eTn(S) converge uniformly toward f(x) as n becomes infinite. (Under the conditions imposed convergence of eTn(x) toward f(x) and convergence of T. (x) toward log f(x) are equivalent; the point- is that the criterion defining Tn(x) in the first place is altogether different from that which would be set up by requiring that an integral in terms.of I log f(x)) - Ti2n(x) be a minimum.)

A concluding section will give a brief discussion of the corresponding problem of polynomial approximation.

2. Theorems of existence and convergence for function having a con- tinuous derivative. It will be assumed throughout this section that m > 1, and that the function f(x), continuous, of period 2-r, and everywhere positive, has a continuous derivative for all values of x. The hypotheses imply of course that f(x) has a positive minimum; let h > 0 be its minimum, M its

* Presented to the American Mathematical Society December 29, 1932. t " Some nQn-linear problems in approximation," Transactions of the American

Mathematical Society, Vol. 30 (1928), pp. 621-629.

515

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516 DUNHAM JACKSON.

maximum, and A an upper bound for the absolute value of its derivative. Since the function

+ (x) r- log f(x)

has everywhere a continuous derivative, there exist * trigonometric sums t, (x) such that lim Tn2 = 0, if en is for each m the maximum of ( (x) - t. (x) I. By the mean value theorem,

f (x) - etn(x) e() - etn(x) [p(X) -tn (X) ]etx),

where en(x) is intermediate in value between +(x) and tn(x). Since the sums t,n(x) uniformly approach (p(x) they are uniformly bounded, the func- tions 4n(x) and eWx() are uniformly bounded, and f (x) - etn(x) j does not exceed a constant multiple of en.

The functions f(x) and p (x) and the exponent m ?_ 1 being given, let yyn be the greatest lower bound of the integral (1) as T. (x) ranges over all trigonometric sums of the n-th order; it is not assumed as yet that this lower bound is a minimum actually attained. It is clear that yn_ - kEfnm, where E.1

has the meaning given to it in the preceding paragraph and k is independent of nr, and hence that (2) lim ',n ?0.

n-*oo

Let it be assumed throughout the rest of this section that the weight function p (x) has a positive lower bound: p (x) ? v > 0 for all values of x.

Let G > 0 be the larger of the numbers I logh |, I logM |, the trivial case f(x) = 1 being ruled out. For an arbitrary Tn(x), let g denote the value of the integral (1). It will be shown that for m sufficiently large all sums Tn(X) of the n-th order for which g _ 2yn are subject to the inequality Tn(x) I < 4G, for all values of x.

Let ,u be the maximum of j Tn(x) |, and let it be supposed that p ? 4G. Let xo be a value of x such that I T,, (xo) ,u. By Bernstein's theorem, T'n (x) ? - n everywhere, and

I Tn(x)- Tn(xo) _nf t I x-xo|

For I xxo l 1/(2n),

Tn(X) -Tn(xo)f?_ U, Tn(x) | s_2G.

Throughout this interval, then, eT-(x) ? e2G, or else eTn(x) ? e_2G, one or the

* See e. g. D. Jackson, "The theory of approximation," American Mathematical Society Colloquium Publications, New York, Vol. 11 (1930), p. 12, Theorem IV, Corollary.

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Page 4: The Convergence of Some Non-Linear Processes of Approximation

CONVERGENCE OF NON-LINEAR PROCESSES OF APPROXIMATION. 517

other of the indicated relations holding throughout the whole interval, according as Tn (x0) = , or -,u. On the other haind,

e-Ga h?f(x) ?M? eG

everywhere. Consequently, for x x- ? 1/(2n), and so throughout an interval of length 1/n,

f(x) - eTn(x) I _ e2G eG,

or else (3) f f()-eTn(x) - 2G

as e2G - eG e3G(e-G - e-2G) > e-G e-2G, it may be asserted without distinction of alternatives that (3) is satisfied. Hence

7r

g = fp(x) I f(x) - eT)m dx ? (V/n) (e-_ e-2G)m

which is inconsistent with (2) and the supposition that g _? 2y,, for all values of n from a certain point on.

Inasmuch as the 2n + 1 coefficients of any trigonometric sum Tn (x) of specified order n for which max I T,n (x) I < 4G belong to a certain bounded domain in (2n + 1) -dimensional space, and as it is now seen that if a sequence of snms T, (x) is constructed for which the value of (1) approaches y. the condition max I Tn (x) I < 4G must be satisfied by all sums in the sequence from a certain point on, provided n is sufficiently large, it follows that there must be at least one limiting set of coefficients for which the integral (1) is actually equal to yn; the minimum problem proposed at the outset has a solution. (For completeness the possibility that yn == 0 requires separate notice, but the conclusion for this special case is justified with equal facility.) Furthermore (still under the supposition that n is sufficiently large) any minimizing sum T. (x) satisfies the condition max I T4(x) I < 4G. There is no assertion that the minimizing sum is uniquely determined, and no assumption to this effect will be needed in the subsequent work. The question of the existence of a minimizing sum for all values of n from the beginning, and for more general functions f (x), will be considered in the next section; for the problem of convergence as n becomes infinite, with which this section is primarily concerned, any finite number of values of n may be left out of account.

In further preparation for the convergence proof an extension of Bernstein's theorem is needed, going beyond one that was presented in an earlier paper to which reference has been made. Let f(x) be subject to the hypotheses already imposed, and let Tn (x) be an arbitrary trigonometric sum of the n-th order; let L, h', M' be positive numbers such that

4

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518 DUNHAM JACKSON.

f (x) - eTn(x) ?'-C L, log h' _ Tn(x) _ log M',

for all values of x. (The value L 0 would be admissible but trivial.) Let ho be the smaller of h and h', and let log f(x) be denoted once more by +(x). By the law of the mean,

f (x) - eTn(x) eO' eTn() [+(x) -T4 (x)]e(x),

where t (x) has a value intermediate between +) (x) and T" (x). As log ho is a lower bound both for +(x) and for T(x), it is a lower bound for (x), and

eE()_ ho, I p(x) - T(x) I I f(x) - eTn(X) I e ? L/ho.

Also, as it has been assumed that I f'(x) I ? A,

I+'(x) | I I f(x)/fQX) I'? A/ho. It is possible then to apply the extension of Bernstein's theorem given in the earlier passage referred to,* with the conclusion that

I T'(x) I < nL/ho + CAG/ho, where C is an absolute constant. (More specifically, the statement is true with C 4, though the numerical value is not needed for present purposes.) Hence

(d/dx)eTn(x) - T'_ (x) eTx) _ (Ml/ho) (nL + CA)P I(dldx) [f (x) el(x) |_ + (MW/ho) (nL + CA).

The result of this calculation may be recorded in

LEMMA I. If f(x) is a positive continuous function of period 2ir having a continuous first derivative subject everywhere to the condition I f'(x) | A and if Tn (x) is a trigonometric surm of the n-th order such that

log h' ? Tn (x) < log M' and f (x) -eTn (m) L

for all values of x, then

I (d/dx)[f(x)-eT(] C ClnL + C2,

where Ci and C2 depend on f(x) and on h' and M', but not on any other specification with regard to T; (x).

Throughout the rest of this section let Tn(x), for each n, denote spe- cifically a trigonometric sum of the n-th order for which the integral (1) has its minimum value yn, at least when n is large enough so that the existence of a minimizing sum is assured by the previous reasoning. It has been seen

* See Transactions, loc. cit., p. 622.

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CONVERGENCE OF NON-LINEAR PROCESSES OF APPROXIMIATION. 519

that for values of n from a certain point on I Tln(x) I < 4G, anid it will be understood that n is large enough so that this condition also is satisfied. It is to be shown that eTn(*) converges uniformly toward f (x) as n becomes infinite.

Let R (x) rf(x) -eTn() let n be the maximum of I R4E(x) 1, and let x1 be a value of x such that I Rn (xi) = n. By application of the Lemma,

| R'(X) Cln + C2.

The bounds log h' 4G, log M' 4G, on which the determination of Ci and C2 depends, are independent of n, and C1 and C2 therefore are independent of n also.

Let it be supposed temporarily that C2 ? nm,pi; the contrary case will be considered separately. Then

I R"n (X) lC(Cl + 1)?AnPn

R. R(x) -Rn (xi) | (Cl + 1), n/-n x xil

For I x - xil < 1/[2 (C + 1) n],

R.(x) -Rn(xi) I - lIn I R.n(x) | 1kn n

Inasmuch as p (x) ? v > 0 everywhere,

'yn ((7 + I)n (2 )' ( 2 VI

The supposition previously rejected, that C2 > nu,, would mean directly that P% < C2/n. In either case,

/, ? 2[(Cl + 1)/V]1/M(n-yn)l/m + C2/n.

Since m ? 1 it follows from (2) that lim nfyn = 0, lim n= 0, and the n--> n-->o

uniform convergence of Rn(x) toward zero is established. The conclusion is

THEOREM I. If f (x) is a positive continuous function of period 27r having a continuous derivative everywhere, if p (x) has a positive lower bound, and if m> 1, the sums Tn(x) minimizing the integral (1) will be such that eTn(x) converges uniformly toward f(x) as n becomes in/Inite.

3. More general existence theorem. In this section the existence of a trigonometric sum minimizing (1) is to be proved under hypotheses coIn- siderably more general than those previously admitted. The function f (x), of period 27r, is assumed to be bounded and measurable, with a positive lower bound; the weight function p (x) is of period 2vr, summable, non-negative everywhere, and positive over a set of positive measure in a period; the

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520 DUNHAM JACKSON.

exponent m may have any value > 0; the order n is any positive integer or zero. The problem of uniqueness of the minimizing sum will still be left untouched.

A preliminary stage of the reasoning may be summarized in

LEMIMA II. If n is a given integer 0, there is for every positive E a positive 8 such that if T'n(x) is any trigonometric susm of the n-th order having 1 as the maximumr of its absolute value, there is a set of measure at least 27r - E in a period throughout which Tn (x) I _8

Suppose this were not true. Then, for some positive E, there would exist a sequence of positive numbers 81,'82, . * *, k, * * *, approaching zero, such that for each k there is a trigonometric sum T. (x), of the n-th order, having 1 as the maximum of its absolute value, and satisfying the inequality I Ta(x) < 8i throughout a set ek of measure greater than e in the period interval (- r, r). As the coefficients in the sums T (x) are bounded by the restriction I T,(x) I< 1, the various sets of coefficients, regarded as coordinates of points in (2n + 1) -dimensional space, must have a limit point, and there is a sum T (x) of the n-th order,* still having 1 as maximum of its absolute value, uniformly approached by a sequence of the sums Tk. Let tn1 (x), t%2 (x), *, be such a sequence, the other sums Tn; being dis- missed from further consideration. Let e be the set of points which are common to infinitely many of the corresponding sets e>. The measure t of e is at least E. If x is any point of e, lim tln; (x), which exists and is equal

k-*oo to r (x), must be zero, by reason of the approach of the 8's to zero. So the sum i-, (x) is required to vanish throughout a set of positive measure and to take on an extreme value + 1, which is impossible.

An immnediate corollary is that when 8 has been determined for a given E,

in accordance with the terms of the Lemama, then if T,n (x) is any trigono- metric surm of the n-th order whatever, and if I Tn (x) attains a value as large as H, whatever the value of H may be, then I Tn (x) ? H_ throughout a set of measure at least 2ir - E in a period.

To return to the problem of minimizing the integral (1), let h > 0 be a lower bound and M an upper bound for f(x). Let Tn(x) be an arbitrary trigonometric sum of the n-th order, let

=1 f- (x) If (x) - eT(wv) Im ix,

* Whenever reference is made to sums of the n-th order, the words are understood to mean of the n-th order at most.

t See e. g. de la Vanlie iPoussin, Inftgrales de Lebesgque, Paris, 1916, pp. 8-9, 26-27.

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CONVERGENCE OF NON-LINEAR PROCESSES OF APPROXIMATION. 521

and let 92 f (x) [f (x) - h]tm dx;

the quantity in brackets in the last integral is non-negative, and the constant h elog h may be regarded as a function of the form eTt(x) for any value of

n ? 0. Let h1 be a positive number less than h: 0 < h1 < h. Let E1 be the

set of points in (- 7r, 7r) (if any) at which eTtt() ? h - hl, let E2 be the

set where eTt(x) > 2M -h + h1, let E E1 + E2, and let CE be the set

complementary to E, on which

h -h < eT7t() < 2M -h + hl. Let r1(x) f(x) eTn(x), r2(x) =f(x) -hIo. Any point of E1,

ri(x) -r2(x) = r(x) -r2(x) =h - eTt7() > hl. At any point of E2, as f(x) M,

I r1 (x) > - h + hl,

and as r2 (x) -? M -- h everywhere,

I r1(x) I r2(X) ? h1,

f:or x in E2 as well as for x in E1. If m > 1, inasmuch as r,2 > the last

inequality implies that

I ri Im -2m ? (r2 + hl)m-r2in h 1m

the difference (r2 + h1)' ?-- r2m being smaller for r2 0 than for any positive

value of r2; if m < 1, inasmuch as r2 M - h, the corresponding inference

is that

r1 Im r2m (r2 + h1)fl -r2m_ (M -h + hl)m - (M1 -h)m,

the difference (r2 -+- h,)"I - r2m lnow being less for r2 M -h than for

any smaller non-negative value of r2. If D1 denotes hll' when m > 1,

(M h + h,)m (M h),n when m < 1, and the common value h1 to

which both expressions reduce when m = 1, then ri Im r2m> D1

throughout E, and

1 , p (x) (I ri rm -mr2m)dx > Di p (x) dx.

With regard to points of CE it can be said that

I r1 lm- r2 >- r2n >- (Al -h)m

and if (M h)m is denoted by D2,

J2 p(x) (I ri Im r2m))dx > D2 p(x)dx.

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022 DUNHAM JACKSON.

So

91 -g2 J1 +J2 DlfD p(x)dx- D2CEP(x)dx.

By the absolute continuity of fp(x) dx, the integral over CE can be

brought arbitrarily near to zero, and the integral over E arbitrarily near to the integral over an entire period, if the measure of CE, denoted by mCE, can be made sufficiently small. In particular, if

7r

I = J p(x)dx,

and if r is a positive number less than D11/(Di + D2), there will be a positive e such that

f Cp(x)dx < CE

if mCE < ,E and then it will follow further that

J p(x)dx > I-'q,

91 -92 > D1(I -) D2q > ?

Let 8 be the quantity associated with this e by Lemma II. Let H1 be the larger of the numbers log (h- h) 1, log (2M - h + h,). If Tn (x) H i H1 it will follow that

Tn(x) j H1?i log (h -h1) j3log (h- h), eTn(X) < h-hl, or else

Tn(x) ? H log (2M -h + h), eTn(X) > 2M h + hi.

That is to say, any x for which I Ten(x) ? Hi belongs to the set E. By the Lemma, as interpreted through its corollary, if Tn (x) I attains anywhere a value as large as H18, the measure of E will be at least 2ir - e, the measure of CE will be not more than E, and according to the preceding paragraph 1- g2 will be positive.

This means that all sums Ttn(X) for which g1 E g92 are such that Tn(X) I K H1/8 for all values of x, and the coefficients are thereby required

to belong to a bounded domain. If the greatest lower bound yn of the integral (1) is equal to g2, the constant h - elog h is itself a minimizing sum.; if not, then yn < g2, any sequence of sums 1Tn(x) for which the value of the integral approaches yyn will have coefficients belonging to the bounded domain just mentioned from a certain point on, and there will necessarily be a limiting set of coefficients and a correspondiing sum Tn(X) for which the value yn is attained. Thus the existence of a minimizing sum is assured:

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CONVERGENCE OF NON-LINEAR PROCESSES OF APPROXIMIATION. 523

THEOREM II. Under the conditions stated at the beginning of the sec- tion, there will be for each n ? 0 at least one surne T, (x) for which the value of the integral (1) is a minimum.

4. Polynomial approximation. A corresponding problem of polynomial approximation over a finite interval, which may without loss of generality be taken as that from - 1 to 1, relates to the minimizing of the integral

(4) p (x) I f (x) - ePn(x) mdx,

in which P,.(x) is a polynomial of the n-th degree (at most). The existence proof of the preceding section can be adapted to this case

without difficulty; under hypotheses similar in generality to those formulated at the beginning of Section 3 there exists for each value of n, a minimizing polynomial, which may or may not be uniquely determined.

The circumstances of the proof of convergence are materially changed, though not to the extent of obliterating the analogy, by the fact that Bernstein's theorem and its generalization are less simple for polynomials than for trigonometric sums.

The hypothesis with regard to f(x) (in addition to the requirement that it take on only positive values) will be once more that it have a continuous derivative, in the present case for - 1 < x ? 1. The exponent mn, however, will be restricted to values ? 2. It will be supposed again that p (x) has a positive lower bound. It can be shown by appropriate modification of the previous argument that the minimizing polynomials are uniformly bounded for all values of n. The question which then calls for special attention is the adaptation of Bernstein's theorem.

Let f(x) be continuous and positive for -1 <`x < 1, having h > 0 and M as its minimum and maximum values, and let f'(x) be defined and continuous throughout the interval, with X as an upper bound for its absolute value. Let Pn (x) be an arbitrary polynomial of the n-th degree, and let

I f (x) ePn(x) |_ L log h' P,(x) ? log ',

for - 1 x < 1. Let x = cos G. Then f (x) = f(cos 0) F(0) is a periodic function of 6,

defined and continuous for all real values of the variable. Its minimum and maximum values are those of f (x), and it has furthermore a continuous derivative PF(B) = -f(x) sin 0, subject to the inequality I '(0) I _ X. Also, Pt, (x) - P,, (cos 0) is a trigonometric sum of the n-th order in 0, which may be represented by T. (0). The bounds of Tn (0) are those of Pn (x), and

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524 DUNHAM JACKSON.

I F(0) - eTn(0) I I f (x) ep(x) L

for all values of 6. Lemma I is therefore directly applicable, with 0 as independent variable, to the effect that

I (d/d0) [F(0) - eT()] | CnL + C2.

For differentiation with respect to x, as

(d/dx) [f (x) eP- (x)] (d/dx) [F(0) - eTn()] (d/d0) [F(0) eTn(O)] (dO/dx),

this means that

I (d/dx) [f (x) ePn()] ( c n + Cl/2

In formal statement:

LEIM7MA III. If f (x) is a positive continuous function for -I x < 1, having throughout the interval a continuouws first derivative subject to the condition I f'(x) I-C A, and if P,(x) is a polynomial of the n-th degree such that logh"? P,(x) ?logM' and I f(x) ePn() ?-T) L for 1 1 then for - 1 < x < 1,

I (d/dx) [f (x) - eP- ( < C1nL + C2

where C0 and C2 depend on f(x) and on h' and M', but not onz any other specification with regard to Pl (x).

Repetition of an argument used elsewhere * in connection with the ordinary form of Bernstein's theorem leads directly to the

COROLLARY. If xl and X2 are any two numbers of the closed interval (-1, 1) differing by not more than utnity, and if f (x) - ePn(w), Rn (x),

I R,n (X2)- Rn(x1) I ? 2(CinL + C2) I X2-XI. 1/2-

With obvious readjustments, and in particular with replacement of an interval whose length is of the order of I/n by an interval whose length is of the order of 1/n2, the reasoning which gave a proof of Theorem I now shows that if Pn (x) is for each value of n a polynomial minimizing the integral (4), and if m > 2, eP-(w) converges uniformly toward f(x) for - 1C? x C I as n becomes infinite.

THE UNIVERSITY OF MINNESOTA,

MINNEAPOLIS.

* See the writer's Cottoquiutm, previously cited, pp. 93-94.

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