A Problem in Minima

Annals of Mathematics

A Problem in MinimaAuthor(s): Dunham JacksonSource: Annals of Mathematics, Second Series, Vol. 28, No. 1/4 (1926 - 1927), pp. 587-592Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968400 .

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A PROBLEM IN MINIMA.*

BY DUAXA JACKSON.

If f(x) is a given function of period 2ir, the trigonometric sum T, (x), of order n, which minimizes the integral

[f (x) - T. (x)]2 dx,

in comparison with all other trigonometric sums of like order, is the partial sum of the Fourier series for f(x). In terms of the geometry of function space, if P is the point corresponding to f (x), and T the point representing T. (x), it is a question of minmizing the distance PT, when P is fixed and T ranges over the linear (2n + 1)-dimensional space of trigonometric sums of the nth order. A natural step in the direction of generalization is this. Let f (x) and So (x) be two given functions of period 27r, represented by points P and Q in function space: to determine the point T which minimizes the area of the triangle PTQ, or, in analytic languaget to find the trigonometric sum T. (x), of the ntlh order, which minimizes the Gramian of the functions f (x) - T. (x) and Sp (x) - T, (x). The existence of a solution of this minimum problem will be discussed below, together with the question of the convergence of the minimizing sums T, (x) as n becomes infinite.

Let it be supposed throughout, for convenience, that f (x) and sp (x) are continuous. If f and p themselves are trigonometric sums of the nth order, any T. (x) of the form cf (x) + (1 -c) 9' (x) will reduce the Gramian to zero; this case will be left out of consideration henceforth. There may still be a single value of c, though there can not be more than one, for which cf+ (1 - c) 9' is of the form T. (x), and so gives a trivial solution of the problem. This will happen whenever there is a linear combination clf+ c, a, with cl + c2 t 0, which is a trigonometric sum of the nth order, as may be seen be taking c = c1/(c, + ce). This special case also may be put aside.

Under all other circumstances, the points P, T, Q form a triangle of positive area, whatever trigonometric sum of the nth order is represented by T. The base PQ being fixed, the point T has to be found so that

* Presented to the American Mathematical Society, April 16, 1927; received April 28, 1927. t Cf., e. g., D. Jackson, The elementary geometry of function space, Amer. Math.

Monthlv. vol. 31 (1924), pp. 461471; p. 463. 587



588 D. JACKSON.

its perpendicular distance from the line PQ, constituting the altitude of the triangle, shall be as small as possible. If t() (x) = f(x)- g (x), an arbitrary point of the line PQ corresponds to a function of the form

co(X) = 9(X)+,4(x) = f(X)+(1-A)(X)-

The foot of the perpendicular to PQ from the nearest point T represents the function w (x) which can be most closely approximated by a trigonometric sum of the nth order, the approximation being measured by the integral of the square of the error.

One more special case remains to be considered, namely that in which W(x) is a trigonometric sum of the nth order, neither f nor T by itself

being such a sum. In this case all functions of the form co (x) can be equally well approximated, since the addition of a constant multiple of yt (x) simultaneously to the function to be approximated and to the approximating function leaves the integral of the square of the error unchanged. The minimum problem has a solution in the partial sum of the Fourier series for any function w(x), through terms of the nth order.

Let it be supposed from now on that no linear combination of f(x) and ip (x) is identically equal to a trigonometric sum of the nth order. This is equivalent to assuming that the 2n +3 functions Sp (x), V (x), 1, cos x, . * *, cos nx, sin , . * *, sinnx, are linearly independent. If T. (x) is represented by the notation

Tn(t) ao + a, cos x + ascos 2x + *+ a cos nx

+ bi sin X + bNsin 2x + * * * + bn sil nX,

the problem is to determine the 2n + 2 parameters A, as, al, ..., an, b1 . I be, so as to minimize the integral

[ nw(x) + IV (x)- Tn (x)]2 dx,

regarded as a function of the 2 n +2 variables mentioned. To shift the emphasis slightly, a least-square approximation is sought for fy (x), by means of a linear combination of the 2n + 2 linearly independent functions t(x), 1, cosx, .., cos nx, sin x, . . sin nx. It is wvell known that this problem has one and just one solution.*

For each value of n, let So (x) be the trigonometric sum which enters into the solution-the sum, then, which solves the problem originally

e Cf., e. g., D. Jackson, On functions of closest approximation, Trans. Amer. Math. Soc., vol. 22 (1921), pp. 117-128; Theorem I.



A PROBLEM IN MINIMA. 589

proposed-and let 4. be the corresponding value of 2. One proceeds to inquire as to the convergence of these quantities when n becomes infinite. It is to be borne in mind that S1 (x) is the partial sum of the Fourier series for the function S (x) + 4n yi (x).

Let ak, bk be the coefficients of cos kx and sin kx respectively in the Fourier series for so(x), and let ak,@ k be the corresponding coefficients i the case of V(x). Let it be supposed that i'(x) is not a trigonometric sum of any order -in other words, that infinitely many of its Fourier coefficients are different from zero. The integral, extended over a period, of the square of the difference between Sp+2. and the partial sum of its Fourier series through terms of the nth order is*

xrZ [(ak +A ?ak)' + (bk + I6k) ]

-X = (a2+ bj) + 2A1(a, ak+ b,,by + A2 V(a. + 8P

the summation being extended in each case from n +1 to x, and 4n is merely the value of A which minimies this expression. By simple differentiation it is found that

4 -~(akak+ bkfik) (akak+ blk)

The next question at issue is whether 4 approaches a limit as n becomes infinite. It is recognized by the construction of examples that no simple general answer is possible. For convenience, let bk fik = 0 in each case, for all values of k, so that the Fourier series in question are series of cosines only. As a first example, let ak = 1/k3, ak = 1/k. The corresponding cosine series are uniformly convergent, and represent continuous functions. The quantity zea ak ak is of the order of magnitude of 1/n4, while 7n 1 ak is of the order of 1/n8, and limo=> 2n 0. On the other hand, if the definitions of ak and ak are interchanged, lim,=e , =oa> Thirdly, if ak = 0 when k is even, and ak =O when k is odd, then , = 0 constantly, regardless of the relative orders of magnitude of the

coefficients. Finally, let ak =1/2k/2 for all values of k, and let ak =1/2k/2 or 0 according as k is odd or even; then A, is alternately equal to 2/3 and to 1/3, and remains bounded without approaching a limit.

It may be pointed out at once that if 2A, does.approach a limit 4, and if, as will be supposed henceforth, f(x) and 9' (x), and consequently i (x), are such that their respective Fourier series are uniformly convergent,

* Cf., e. g., BMcher, Introduction to the theory of Fourier's series. Annals of Math., (2), vol. 7 (1906), pp. 81-152; pp. 85, 107.



590 D. JACKSON.

then S,, (x) converges uniformly toward So (x) + A1 q (x). For the remainder in the series for So + A yP is merely the sum of the corresponding remainders for So and y. multiplied by 1 and A respectively, and the series for S + A tp converges uniformly with regard to both . and x, as long as A remains bounded; and the difference between S,, (x) and 5p (x) + A if (x) is the sum of the quantity (A - An) if and the remainder in the series for S + X,,t i. Similarly, if a sub-sequence of the numbers 1, approaches a limit, the corresponding sub-sequence of sums S,, (x) uniformly approaches the combination ?P + 2 f formed with the limiting value of A in question. So the convergence of )n may be regarded as the main problem.

It may be noted also that the problem is concerned from the beginning with the family of functions Sp + A = If + (1 - I) Sp, rather than with f and so individually. The conditions would be essentially unchanged if f and 4y were replaced by two other functions of the family, and the value

= 0, which appears to have a peculiar significance m some of the examples, would be replaced by a different value.

It has been observed that in some cases the behavior of the sequence t~l depends on the relative orders of magnitude of the Fourier coefficients for 9' and ip. More generally, by Schwarz's inequality,

IZ(akak+bk fk)| < |Z aI'kI|+|ZbkdfkI

< kz2)1/2 (,f )1# + /:2l2 (Z fl2)1/2

< [f (ak + bk)]"2 [Z (ak + f]1A2

the last member is justified by noting that if a, b, c, d are any four non- negative quantities,

(a b)1"2 + (c )1/2 < (a + c)1/2 (b + d)/2. Consequently

{ ? (at + b)] / +1 (a2 + fi2)]t'2

and it can be stated definitely that if the Fourier series for 9' (x) converges more rapidly than the series for if (x), in the sense that the integral of the spqare of the error is an infinitesimal of higher order in the former case then in the latter, then 4 approaches 0 as n becomes infinite, and S, (x) converges uniformly toward p (x). It should be pointed out that the replacement of f and T by two other functions of the family If + (1 -A) go does not affect the order of convergence of the Fourier seriers for their difference, since that difference is replaced merely by a constant multiple of itself. But the convergence might be more rapid for some other function



A PROBLEM IN MINIMA. 591

that would serve in place of g, then it is for ao. The essential thing, then, in the formulation of a sufficient condition for the convergence of 4 to some definite limit, is not that there should be a peculiarity in the function that happens to have been denoted by a, but that there should be in the family some function whose Fourier series converges with ex- ceptional rapidity. If the family of functions Iff+ (1- A) p is such that the Fourier series for an individual function of the family converges at least as rapidly as the series for the difference of two of them., the values of An will be bounded, and will converge toward one or more limits.

For an example, not without some generality, of the conditions insuring convergence of 4n to zero, let p(x) have a continuous second derivative everywhere, and let f (x) have a continuous second derivative, except that for a single point in each period, say for x = 0 and congruent points; the right-hand and left-hand first derivatives are different. Then V(x) and all the functions of the family If+ (1 - 1) s, except g itself, have the properties assigned to f (x). It is well known, and readily verified on the basis of successive integrations by parts, like those to be performed a few lines below, that

lim k'ak = lim k'bk = 0. k=o -k= a

Hence it follows that

lim n~' (a2++ b) = 0. n=QQ n

Considered merely for 0 ? x < 2 r, without regard to periodicity, the function i (x) has a continuous second derivative throughout the closed interval, but if (O +) 4 ! (2ir). By integration by parts,

Pt ~~~~1 C2 r ak - Jo i(x) coskxdx =-- J /(x) sinkx dx 0 ~~~~~k s

= +-[i" (x) coskx]0 -k- j f" (x) cos kx dx.

Since the last integral approaches zero,* r al, differs from [i' (2 i -) -a' (O +)I/k' by an infinitesimal of higher order, lim infk k' I ak I > 0, and

lim inf n;' :n+ k k) > ?. no=c

So it, is certain that lim n=c An = 0. The conditions discussed in the last paragraphs, however, are merely

sufficient conditions. One case has already been remarked in which the * Cf. B5cher, loc. cit., p. 86.



592 D. JACKSON.

precise order of magnitude of the coefficients was irrelevant. If ip (x) and t (x) are so related that every non-vanishing coefficient in the Fourier series for one corresponds to a vanishing coefficient in the series for the other, all the numbers 4 are zero. More generally, let g, and V.'~ be the remainders in the Fourier series for So and t respectively. In an abbreviated notation that will be readily understood, the integrals being extended over a period in each case,

Sn 9'n -

_ _ _ _ _5 At5=- [f]S]1.1/2 12

Since the last fraction on the right represents the cosine of the angle between the vectors Tn and Vn in function space, it may be said that 4n will approach zero if the series for 9' and V converge with the same order of rapidity, according to the interpretation that has been placed upon this mode of expression, and if at the same time 9n and i/n approach ortho- gonality with each other, so that the angle between them approaches 90?.

THE UNTVEsICY OF MESOTA, MINNEPOLIS, i.



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A Problem in Minima