Orthogonal Trigonometric Sums

Annals of Mathematics

Orthogonal Trigonometric SumsAuthor(s): Dunham JacksonSource: Annals of Mathematics, Second Series, Vol. 34, No. 4 (Oct., 1933), pp. 799-814Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968700 .

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ORTHOGONAL TRIGONOMETRIC SUMS.1

BY DUNHAM JACKSON.

1. Introduction. The purpose of this paper is to develop the beginnings of a theory analogous to that of systems of polynomials orthogonal with respect to a given weight function, for the corresponding case of trigonometric sums.2

It is well known that if a set of functions VI (x), f2(x), *. is given in an interval (a, b), each of the i/'s being summable over the interval together with its square, and any finite number of them being properly independent, in the sense that every linear combination with coefficients that do not all vanish is different from zero over a set of positive measure, then it is possible to construct a normalized orthogonal system " (x), SP2 (X), * ,

in which the general function Sp, (x) is a linear combination of Vi (x), * , Ale (x), so that

xb

f9m (X) !n(X) d x = 0 (m tn ), b

fPn n(x)]2 dx

=

1.

Let e(x) be a non-negative summable function of period 2ir, different from zero over a set of positive measure in any interval of length 2ir, and let Vl,(x), b2(x), *.* be the set of functions

[e(x)]"2, [e(x)]J2 cos x, [e(x)]1/2 sin x, [e(x)]"2 cos 2x, [Q(x)]1/2 sin 2x,

The corresponding Sp's for a period interval will be a sequence3

IQ (X)j112 UO(X), [e (X)]112 of(X), [e (X)]112 V,(x), [e (X)]1,2 tUa (x), [e (X)] 1,2 V2 () ***

in which u,,(x) and Vn(x) are trigonometric sums of the nth order, the former containing no term in sinn x, and the characteristic properties of the sequence are expressed by the relations

I Received February 21, 1933.-Presented to the American Mathematical Society, August 30, 1932, and April 15, 1933.

2 The treatment is to a considerable extent parallel to that in the writer's expository paper entitled Series of orthogonal polynomials, Annals of Mathematics, vol. 34 (1933), pp. 527-545.

3 See e. g. the writer's Theory of Approximation, American Mathematical Society Colloquium Publications, vol. XI, New York, 1930, pp. 89-91.

799



800 D. JACKSON.

JeF(X) u.m(X) Un(x) dx 0, f (X) V (X) Vn(X) d x = 0 (Mtn),

(1) f 2(x) um(x) v.(x) dx = 0 for all m and n,

fe%(Z) [un(X)]2 dx= 1, J (X) [vn(x)W dx = 1.

The content of the equations with right-hand member zero can be expressed by saying that any two functions of the sequence of u's and v's are orthogonal to each other with respect to e(x) as weight function. The following discussion is concerned with further properties of the u's and v's belonging to a given weight function &(x), and in particular with the convergence of series expansions in terms of them.

It is apparent from the manner of construction of the orthogonal system that, in the general formulation, So (x) as written in terms of the Vt's actually contains a term in qmn(x), with non-vanishing coefficient, for each value of n, and hence that each .m(x) is linearly expressible in terms of S (x), ..., ?Pn(x). Consequently any linear combination of i/l, , bn-, is a linear combination of Sol, *, wn- and !fn(x) is orthogonal to any such combination. In the special case under consideration, un(x) actually has a term in cos nx, and vn(x) actually has a term in sin nx. Any trigonometric sum of the nth order containing no term in sin n x is linearly expressible in terms of uo(x), ul(x), vi(x), *-, Un_-(X), vn-1(x), un(x), and any trigonometric sum of the nth order is expressible in terms of uo(x), * , u, (x), vn (). With respect to the weight function e (x), U. (x) and vn (x) are orthogonal to any trigonometric sum of order lower than the nth, and v. (x) is orthogonal also to any sum of the nth order lacking the term in sinnx.

2. Recursion formulas. The u's and v's are connected by recursion formulas analogous to that of Darboux' for orthogonal polynomials, though somewhat less simple in form. Let p& be any constant. (The essence of the method would be sufficiently illustrated by taking & = 0, but it is desirable for the applications to have the degree of generality conferred by the presence of the arbitrary constant.) The product cos(x-P&)Un(x) is a trigonometric sum of order n + 1, and as such can be expressed in terms of uo (x), . . ., Un+1 (x), Vn+1 (x):

(2) cos(x-p) un (x) = an Un+1 (x)+An un x)?An unf-l(x)+A4'unf-2(X)?+ ..

+ fin Vn+1 (x)+Bn vn (x)+Bn vn-1 (x)+B" Vn-2 (x)+ * * *

I. Darboux, Sur l'approximation des fonctions de trWs-grands nombres ..., Journal de mathematiques pures et appliquees (3), vol. 4 (1878), pp. 377-416, see pp. 411-416.



ORTHOGONAL TRIGONOMETRIC SUMS. 801

The coefficients of course depend on Ak, but it is not necessary to indicate that in the notation. For any value of k from 0 to n, multiplication by e (X) n-k (x) and integration from - 7 to g, with attention to the relations (1), gives

(3) e () COS (x-P) Un (x) Un-k(x) dx = An.

If the trigonometric factors under the integral sign are regarded as forming the product of un (x) and cos (x- ) Un-k (x), the latter is a trigonometric sum of order lower than n if k has any value greater than 1, and the integral consequently vanishes: A(=) 0 0 for k ? 2. Similarly

(4) tf c(x) cos(x-P&) un (x) vn-k (x) dx =B

for k - 0,1, *, n- 1 (there is no vo (x)), and the integral vanishes for k > 2. Replacement of n by n-1 in (2) gives an identity of the form

(5) COS (X - P) Un-1 (X) - an-1 U (X) - n- v (X) ?

If this expression for cos (x -)un, (x) is substituted in (3), for k = 1, the value of the integral defining A' is seen to be an-1. So (2) may be rewritten in the form

COS (X - Un (X) == an Un+, (x) + An an (x) + an-1 Un-1 (x)

(6) ?+ AI Vn+1 (x) + Bn Vn (x) + Bn Vn-1 (x)

Further specifications with regard to the coefficients will be brought out later.

After the manner of (2) there may be written down a corresponding identity

COS (X -P) Vn ( = n Un+1 ()?+Cn un (x)+Cn ?ln-1 ()+Cn" Un-2(x)+ .

(7) + nd Vn+l (x)+Dnvn(x)+Dnvn-i(x)+Dl v--2(X)+. _

The coefficients are given by the integrals

e e(x) cos (x- )vn (x) un-k (x) d x C (8) -

J (X) COs (X--,) Vn (X) n-k (x)dx = D(3,

and it is seen that Ck) = )() 0 for k > 2. Substitution from the identity

(9) COS (X - ) Vn-i (X) W=- rnl-e (x) + dn-1 Vn (x) ? *



802 D. JACKSON.

gives Dn = , Also, (9) may be used in (4) and (5) in (8) to show that Bn = - and Cn, = fin- And it appears from comparison of (4) and (8) for k 0 that Bn = Cn. So the identities (6) and (7) take the form

COS (X -P) Un (X) an Un+l (x) + An Un (x) + an-1 Uni (x)

(10) + 8nVn+1 (x) +BnVn (x)?+ n-i vn-1 (x), COS (X P) Vn (X) = n Un+1 (x) + Biw tn (x) + an-1 Un-1 (X)

(11l) + 6n Vn+1 (x) + Dn vn (x) + an-1 vn-1 (X).

These hold for n ? 1, provided that when n = 1 it is understood that vo (x) is to be replaced by 0. For n = 0 no calculation is required to justify the formula

cos (x-its) uo (x) ao 1 (x) +?io v1 (x) + Ao uo (x).

3. Christoffel-Darboux formula. From the recursion formulas it is possible to obtain an analogue of the Christoffel-Darboux formula' in the theory of orthogonal polynomials. It is a question of an identity for the sum

(12) Kn (x 2 ) _ Uoo Wx Uo Wg + U1 WZ u1 (W +*** + Un(x) Un t + V1 (X V1 Wt + ***+ Vn (x) Vn(t)-

Let (10) and (11) be multiplied by un(t) and Vn(t) respectively, and let the corresponding formulas for cos (t -A) Un(x) Un(t) and cos (t- i) v,(x) v,,(t) be written down by interchange of t and x. Combination of the resulting identities by the indicated additions and subtractions yields

[COS (t - A) - cos (x - i)] [Un (X) Un n) ?+ V (X) WVn (t)]

an [Un(X) Un+l (t)- Un+l (x) Ut (t)]- an-1 [Un- (X) U n(t) - ut(X) UIn-1 (t)] (13) + fin [Un (x) Vn+i (t) -Vnl (x) un (t)] - n-i [un-l (x) v, (t) -n (X) un-i (t)]

+ rn [Vn (x) Un+1 (t) - + (x) WVn (t) - rn-I [vn-i (X) U (t) - U (X) Vn-i (t)] + dn [n (x) Vn+i (t) -V?i (x) Vn (0] - Jn-1 [Vn-i (X) Vn (t) - Vn (X) Vn-1 (t)].

These relations hold for n > 1, with the previous understanding as to vo. For n = 0,

(14) [cos (t -) cos (x - u)J u0(x) uO(t) So [U0 (N ) Ui (t) - U () Uo (t)] + i0 [t (x) Vi (t) W V (x) oU (t)]

The desired identity for Kn (x, t) is derived by writing (13) with n replaced by k for the values k = 1, 2, * , n successively, summation over these values of the index, and addition of the terms obtained from (14):

5 Darboux, loc. cit.




[COS (t- p) -COS (x -p)] Kn, (x, t) an, [Un (x) Un+ (t) Un+1 (x) Un ()] + An [Un (x) Vn+1 (t) - V+1 (X) Us (t)]

+ Yn [Vn (x) U-n+1 (1)- Un+1 (x) Vn (1)] + dn [n (x) V+1 (t) - Vn+1 (X) Vn ,

or, if the long right-hand member is denoted by 2Vn(x, t),

(15) Kn t)~ - N (Xi t) (15) -x [cos (t-p)-cos (x-IC)]

The constants an, any rn J. can be evaluated in terms of the leading coefficients in the expressions of Utn(x) and Vn(x) as combinations of sines and cosines. More important for the subsequent applications, however, is the fact that their absolute values are less than unity. Multiplication of (10) by Q(x)un+1(x) and integration from -7f to ns gives

=s feQ(X) COS (X-P)Un Wxun+?B(x)dx.

Hence

an (X) i cos (x-) un(x) Un+1 (x) d x.

Let Schwarz's inequality be applied to this integral, the integrand being regarded as the product of the factors [e (x)]J12 I cos (x -j) us (x) I and [e (X)1/2 | >n+1(X) |:

a2 _ J Q(X) COS2 (X-P) [un (x)]2 dx () [Un+1 (x)]2 dx

<fQ W x)[Un(x)]2dx f (x)[un+l(x)]2dx 1; the replacement of cos2(x -i) by unity actually increases the value of the integrand over a set of positive measure and increases the value of the integral, the alternative of equality being ruled out. A corresponding calculation gives a similar result for fin, Yn and dn (and incidentally for each of the coefficients denoted by An, Bn, Dn in the preceding section; in the case of An and Dn the fact can also be recognized directly from the form of the integrals defining them, without the use of Schwarz's inequality).

If the terms of the nth order in un (x) and va (x) are denoted by an COS nx, cX cos nx, and dn sin nx respectively, the explicit formulas for an, fangn yes dn are found to be as follows:

1 an ( C4+1 .

atn- 2 an?1 \COs h d sin

-8n ? an sin /h, 2 dn~l



804 D. JACKSON.

Y = 2a i [(Cn COS I,,d sin 1) Cn+1 (cn sin P + dnco sP )],

A 2d,?1 (cn sin?+ d cos p),

for n > 1, while

asO ao cos cl din A), To ao sin It.

These are set down merely as a matter of record, and will not be used further.

4. Formal expansion of an arbitrary function. An arbitrary functionf (x) of period 2 ir, subject to the condition that e (x)f(x) be summable over a period, can be formally expanded in a series

(16) ao uo (x) + a, Ul (x) + as us (x) +

+ b, v, (x) + b2 v2 (x) + by the procedure that is usual in connection with series of orthogonal functions; the meaning of an here of course has no connection with the use of the same symbol in the preceding paragraph. The general coefficients are

ak e(x)f(x) uk (x) dx, bk e(x)f(x) vk (x) dx.

The partial sum of the series through terms of the nth order is

(17) Sn(X) W J (t)f(t)Kn(x, t)dt,

where Kn (x, t) is defined as above. In particular,

1-J e(t)Kn(x, t)dt;

multiplication of this identity byf(x), which is independent of the variable of integration, and subtraction from (17) gives

(18) s (x) -f(x) e Q (t) [f(t) -f(x)] Kn (x, t) d t.

5. Analogue of the Riemann-Lebesgue theorem. If e(x) [f(x)]2 as well as e (x) f (x) is summable, further calculation which is well known as applied to orthogonal functions in general yields the relation

f e(x) f(x)-sn(x)]r dx je(x) If(x)]2 dx-a -(ak+bk). e -71 ~~~~~k==1

As the magnitude of the sum on the right can never exceed that of the integral on the right as n increases, by reason of the non-negative character




of the left-hand member, the corresponding infinite series is convergent, and the general term approaches zero:

lim ak-0, lim bk =0 k-aoo k-aoo

If the sums Uk (x), Vk (x) are uniformly bounded, as will be true at least in particular cases, this conclusion can be made more general by omission of the requirement that efJI be summable, the hypothesis of summability being imposed merely on e and ef. Suppose I Uk(x) I < H, I vk (x)I < H, for all k and all x. For arbitrary positive N, let fN(x) f(x) when If(x) I ? N, fN(x) = 0 when f(x) I >N. If e is any positive number,

fne(x)lf(x)-fN(x)Idx< '

when N is sufficiently large, and hence 71 ~ ~ ~ ~ 7

f Q (x)f(x) Uk(x) dx - (x) fN(x) k (x) dx < -2

and similarly with uk (x) replaced by vk (x, for all values of k. Let N be given a fixed value so large that these inequalities hold. Since efN is summable,

| e W( ~fN~z)%k~z~d~l< 2 AU1 Q(x)fN(x)vk(x)dxd < 2 e

for all values of k from a certain point on, and when this is true

|akj HE, jbkI < -,

if ak and bk are the coefficients defined for the original function f (x). It will be desirable subsequently to have the conclusion formulated for

a function other than the one whose series expansion is under investigation, in the form of a

LEMMA. If e (t) so (t) and e(t) [So (t)]' are summable, or if e (t) So (t) is summable and the functions Uk (t) Vk(t) are uniformly bounded, the integrals

J Q(t) So (t) uk (t) d ty Q Wt SP Wt Vk (t) d t

approach zero as k becomes infinite. (It is understood throughout the paper that the weight function itself

is summable.) 6. Special uniformly bounded systems. As there has been occasion

above to suppose the functions Uk (x), Vk (x) uniformly bounded, and as this hypothesis will be fundamental in a large part of what follows, it is worth while, pending investigation of general conditions under which



806 D. JACKSON.

the property in question is realized, to point out that in a particular class of cases its validity can be recognized immediately. In other cases it can be seen that the u's and v's are bounded for specified values of x, or uniformly bounded over a part of a period, and this also is significant for the theory of convergence.

Let e(x) be any weight function for which the u's and v's are bounded for a particular value of x as k becomes infinite, or uniformly bounded over a designated range of values of x:

(19) Uk (X) < H, I vk (x) I ? H,

for values of x belonging to a point set E. Certainly this condition is fulfilled everywhere in the special case e (x) = 1, the functions of the normalized system for k ? 1 being (1/irll2) cos kx and (1/4r/2) sin kx. Let Uk(x), Vk (x) be the sums of the kth order in the normalized orthogonal system corresponding to the weight function e(x) i(x), where i(x) is any non-negative trigonometric sum. It will appear that Uk (x), Vk (x) are bounded at any point of E where r(x)tO, and uniformly bounded over any part of E where i(x) has a positive lower bound.

Let p be the order of the sum i (x). Since i (x) U. (x) is a trigonometric sum of order n +p it can be expressed in the form

i (x) Un (x) = Ao uo (x) + Au1 l (x) + + An+p U?t+p (

+ Bi vi (x) + Bn+p vn+p(x) ,

the coefficients depending on n, to be sure, as well as on the index which appears explicitly. (The present A's and B's have no relation to those used previously). By the property of orthogonality of the sums Uk (x), Vk (x),

Ak fe()T ( U.n (x) Uk(x) d x.

But UV (x) is orthogonal to any trigonometric sum of lower order with respect to the weight function e (x) r (x) for which the U's and V's are constructed, and hence Ak 0 for k < n. Similarly, Bk = 0 for k < n. So

Un (x)- [An un (x)+ +An+p un rp (x)+Bn vn (x) + *+ Bn+p vn+p (x

the number of terms in the bracket being 2p + 2, and remaining finite as n increases. In the same way Vn (x) has the form

Vn (X) X) [Cn Un (x) + + Cn+p un+p (x) + Dn vn (x) + * + Dn vn+p (x)] .




For x in E, if r(x)+0,

|Un(x)l - TH() (a i+ +An-+-p+ Bn + I Bnip

whence, as any one of the absolute values in the parenthesis is certainly not greater than the square root of the sum of the squares of all of them,

(20) Un(x)I < H (2p+2) (A2+ +A2+ B2+ +B 2 1/2

(By Schwarz's inequality the last relation holds in fact with (2p+2) replaced by (2p+2)1/2). If Mo is the maximum of T(x),

2 + 2 = e (x) [T(x) Un(x)]2dx

? Mi fe(x) r (x) [Un (x)]I d x = Mo X

the last equality resulting from the normalization of the U's with respect to e(x) i(x) as weight function. So the whole right-hand member of (20) has an upper bound independent of n. Similar reasoning applies to V,(x). So the assertion with regard to the boundedness of the U's and V's is justified. In particular, if (19) holds for all values of x and if j(x) has a positive minimum the U's and V's are uniformly bounded everywhere.

7. First convergence theorem. Let e (x) be a non-negative summable weight function of period 2i for which the sums Uk(x), vk(x) are everywhere uniformly bounded. Let f(x) be a function of period 2 ir such that e(x)f(x) is summable. Let sn(x), as above, be the partial sum of the series (16) through terms of the nth order. This section is concerned with the setting up of conditions for the convergence of sn(x) toward f(x) for an arbitrarily specified value of x, to be held fast throughout the discussion. The question of uniformity of convergence will be reserved for treatment by a different method in the next section.

The value of x being given, let V, (t) and V2(t) be functions of period 2 ir, defined in the interval x- (in2) _ x<x+(3 f!2) as follows:

i1L'(t) = f(t)-f(x) for I t-x x i 2 p1(t) = O for It-x[> j

2 M) = 0 for I t-xIj l_ 2 (t)=f (t)-f (x) for Xt-xl>-2 .

Then t1 () + 2 (t) f(t) f(x) everywhere, and by (18)

sn) W x) et)+1 (t) K.(x t) d t2+ e .2 W Kn(X t) d t -7 -l7I2



808 D. JACKSON.

It will be recalled that in the expression (15) for KR(x, t) the constant A is arbitrary, the coefficients a., 8nn, rn, J. which enter into the numerator depending on pA, but being always less than 1 in absolute value. For substitution of (15) in the integral I, let , = x+(7r/2). Then

Cos(X-A) = Cos(- -) = ?, cos(t-I) = cos (t-x- j-) = sin(t-x),

cos (t -) cos(x-) =sin (t -x). So I, is the sum of

(21) an W - t sin (t - x) [un Wx U+1 t- Un+1 (x u1B (t)] d t

and three other integrals of similar form. Let - f(t)-f(x)

t-X

and let it be supposed that the function f(t) is such that there is an interval of values of t about the point x over which e(t) sD(t) is summable. Then e(t) D(t) is summable over any finite interval, being the product of the summable function e(t) [f(t) -f(x)] by the continuous function 1(t -x) over any interval which does not approach the point t = x. If

(t) - 'sin

(t-X)

e(t) p1(t) is summaJble over the interval x- (in2) ?t<x+(7r/2), being the product of e(t) Md (t) by the continuous function (t -x)/sin (t -x); it is summable from x+((i/2) to x+(37r/2), being identically zero except at the point t = x +? i where it is not defined; and therefore it is summable over the entire period from x- (7r/2) to x+(3nr/2), or equally well over any other period interval, in particular that from -a to 7r. The expression (21) is equal to

an (x) e (t)5p1(t) un+1(t)dt -Un+1(x) e(t) Soi(t) un(t) dt.

Here each integral approaches zero as n becomes infinite, by the Lemma of Section 5; un(x) and u.+,(x) are bounded by hypothesis; and <1anI<. So the whole expression approaches 0. Similar reasoning, with a similar conclusion, is applicable to each of the other three parts of II. Hence lim 1 ?0.

Fpf1oo

For the application of (15) ill Is, let , x,




and let S92(t) = i(t)/[cos (t -x) 1]. With new values of a,, **, E

the integral I2 is the sum of

all [Un (X) r e (t) 92 (t) Un,1 (t) d t - un+1 (x) t 7 (t) 2 (t) un (t) dt]

and three other expressions of like character. The product e(t) !P2(t) is summable from x- (7n2) to x+(nI 2), where it is identically zero except for t = x, and from x+@(/2) to x+(3 n/2), where I cos (t -x)- I I >1, and so is summable over any period interval. By use of the Lemma again it is seen that lim 12 = 0. The fact that I1 and 12 both approach zero

nc-oo means that

liM Sn(X) =f (x).

The proof of convergence goes equally well without the requirement that the u's and v's be uniformly bounded, if it is assumed that they are bounded at the point x and that e (t) [f(t)]2 and e (t) [0(t)]2 are summable.

These results are expressed in THEOREM I. The series (16) will converge to the value f (x) for a specified

value of x if the functions Uk (t) and Vk (t) are uniformly bounded, if e (t) f (t) is summable, and if there is an interval about the point x over which e (t) d) (t) is summable, where '9 (t) denotes the difference quotient [ f (t) -f (x) ] /(t -x); or if uk(x) and Vk(x) are bounded for the particular value of x in question, if e (t) [f(t)]2 is summable, and if e (t) [0(t)]2 is summable over an interval about the point x.

The conditions on 0(t) can of course be interpreted in more specific terms6 with reference to properties of f(t). They will be more than satisfied, for example, if f(t) satisfies a Lipschitz condition in the neighborhood of t = x.

More particularly still, the condition on 0D(t) will be satisfied if f(t) vanishes identically throughout an interval about the point x. If fi (t) and f2(t) are two functions which are equal throughout such an interval, the series for their difference, which is the difference of their respective series, will converge toward zero at the point x; if the series for fi converges, the series for fs will converge likewise, regardless of any difference between the functions outside the interval designated, provided efi and ef2 are summable, and, in case of need, e fZ and eJ2 also. The convergence of the series for a function f (x) at a specified point depends only on the behavior of the function in the neighborhood of the point, if the u's and v's are uniformly bounded and ef is summable, or if the u's and v's are bounded at the point in question and ef3 is summable.

6 Cf. e. g. H. Lebesgue, Lefons 8ur les series trigonometriques, Paris, 1906, Chapter III.



810 D. JACKSON.

8. Second convergence theorem. Let Tn(x) be any trigonometric sum of the nth order. It can be expressed as a linear combination of u0 (x), u (x), v1 (x), .., un (x), Vn (x), and this linear expression is at the same time the series development of Tn(x) in terms of the u's and v's. The partial sum of the series through terms of order n is identical with T,1(x itself. This means that

75n (xW t Tn (t) K. (x, t) d t.

Let f(x)- Tn(x) rn(x). Then f(x)= T,,(x)+r,?,(x), and if sn(x) is the partial sum of the series for f (x),

be(x) = f (t)f (t) Kn (x, t) d t

Tt (X t)r n X )d = Jn (t) 7,4 (t) K, (x t) dt+ F e(t) rn(t)Kn((x t) d t

=T(x)? W n(t)rn(t)Kn(x, t)dt,

f (x)-so, (x) =(x rn (x) (tr(t) Kn (x t) d t .

If Irn(x)l ? en,

and consequently | f(X)-S (X)l | En [l+ An (X)]2

if

n ( fQ(t) K . (x, t) I d t. -7t

So the convergence of s. (x) toward f (x) can be discussed in terms of the order of magnitude of en and the order of magnitude of In (x). Information with regard to the former, when the sums Ti (x) are appropriately chosen, is given by known theorems on degree of approximation. The latter will be examined here.7

Let it be supposed that e (x) is such that Uk (x), Vk (x) are uniformly bounded, and futrthermore, for simplicity at least, that e(x) itself is bounded. Let e(x) < G, Iuk(x)? < H, IVk (x) I _ H. In view of the periodicity of the functions involved, the integral defining n (x), instead of being taken from -a to Ar, may be extended over any other interval of length 27r,

7For a corresponding treatment of the problem of polynomial approximation see J. Shohat, On the development of continuous functions in series of Tchebycheff polynomials, Trans- actions of the American Mathematical Society, vol. 27 (1925), pp. 537-550.




say that from x- (r/2) to x+(37r/2). Let the integrals from x-(7r/2) to x+(7 '/2) and from x + (zr/2) to x + (37r/2) be represented by J1 and J2 respectively, and let J1 be further subdivided into the integrals from x-(nf/2) to x-(1/n), from x-(1/n) to x+(1/n), and from x+(1/n) to x+(ir/2), denoted respectively by JA? Ji', and Ji", so that

in (x) J1 + 12 -=J1+ J1 + J1= + J2

the integrand each time being Q (t) I K,, (x, t) . In J1' and Jj" let K.(x, t) be represented by (15) with a - x=+(n/2),

whereby the denominator takes the form

cos (t-I) -cos (x-t) sin (t-x).

Since t x I<r/2 over the range in question, |sin (t -x) ?(2/ ) It-x-. In the first of the four parts of the numerator. the relations

aI?,Il<l, Un(X) ? H, I?1n+i(t)I ? H. In+,U (x) I ? H, Iu(t)n < H

give I a, [U,, (xI)t/n? (t! - un+l (X Un (t] ? 2 H2. As corresponding inequalities hold for the other three parts, and as e(t) ? 6G,

2(t) 2 n(x t)! -' -2 tGz 2 It--xI By explicit integration.

t-_d t O X( 2)2 J-x-0vt2) - J x?(ll ) + 1t l g lg

and J1 and Ji"' have each an upper bound of the order of log n. In Jf' it suffices to read off from the definition of K, (x, t) in (12), together with the hypothesis on the W's and v's, that I Kn (x, t) < (2 n + 1) Ha; the length of the interval being 2/n, it is seen that J1" ? 2[2+(1/n)16Y'H < 6CGH2. Consequently the whole of J1 does not exceed a constant multiple of log n.

In J2 the formula (15) is to be used with a = x. As t-x ranges from n/2 to 3 7/2,

1cos(t- U)-cos (x--U)l = cos(t-x)-1I > 1,

and e(t) IKdn(x t)| < 8GH2, J2 < 87(GH2. By combination of these results it is recognized that

,('x) < C log n

for n -? 1, where C is independent of n and x. The conclusions with regard to convergence and degree of convergence

are expressed in



812 D. JACKSON.

THEOREM II. If e (x) is bounded, if uk (x) and vk (x) are uniformly bounded, and if trigonometric sums Tn(x) of the nth order exist' so that

If(x) -Tn(x) I < En then

fX) - Sn(x) I _ C? En 1og n

for n> 1, where G1 is independent of n and x; in particular,9 the series converges uniformly to the value f(x) for all values of x if f(x) has a modulus of continuity w (d) such that limb,0o (w) log d = 0.

9. General weight functions. Some information as to the order of magnitude of the u's and v's, when they are not known to remain bounded, is readily accessible in the case of weight functions of considerable generality.

Suppose that e (x) is summable and has a positive lower bound for all values of x: (22) e(x) ? v>O.

Let Tn (x) be any trigonometric sum of the nth order (not necessarily one of those in the orthogonal sequence) satisfying the condition that

(23) f2(x, [T~z(x)]2 dx 1 .

In consequence of (22),

(24) JTn(x)]2 dx< ? .

Let co be the constant term in the trigonometric sum [T. (x)]2, and let Sn(x) = [Tn(X)]2_ co, so that Sn(x) is a trigonometric sum of order 2n without constant term. The integral (24) has the value 2 -rco, and hence Icol < 1/(2irv). For any value of x between -7c and a,

n ~~~~~1 T,, [n(X]2 dx < ' [Tn(x)]2 dx < -

and

fSn(x) dx < X[Tn(X)]2 dx I co I dx < -

8 For the requisite theorems on degree of approximation see e. g. the writer's Colloquium cited above, Chapter I.

9 For the case of Fourier series, and for the special theorem on degree of approximation needed here, see also Lebesgue, Sur la representation trigonometrique approchee desfonctions satisfaisant d une condition de Lipschitz, Bulletin de la SociWt6 Math6matique de France, vol. 38 (1910), pp. 184-210; pp. 201-202, 209.




The integral between bars in the left-hand member is itself a trigonometric sum of order 2n, having Sn(x) for its derivative. Hence, by Bernstein's theorem,

JSn(x) < n V

It follows that [Tn (x)]= co + Sn (x) has an upper bound of the order of n, and T, (x) has an upper bound of the order of n112. In particular, Iu,, (x)| and v, (x) can not exceed a constant multiple of n'2.

Now let the requirement that e(x) have a positive lower bound be replaced by a less restrictive one: let it be assumed merely that [e(x)]-f as well as e(x) is summable over a period. Let Tn (x) again be a trigonometric sum of the nth order satisfying (23). Let Schwarz's inequality be applied to the integral of the product of the factors [e (x)]-112 and [Q (A)]2 I Tn (x) 1: ri T 1(x) ] d]? [e(x)]-1 d fx (x) [T. (x)]2dx f (x)]-1 dx.

The square root of the value of the last integral being denoted by I,

Tn (x)Idx ?1.

Let co this time be the constant term in T. (x), and Sn (x) the sum of the remaining terms. It is seen that

2zr1col Tn(x)dx I, _ I<

Furthermore,

fIj Sn(x) I dx Tn (x) l dx + co ldx ? 21

Sn W dx-J Sn (x)J Idx < fSn(x)ldx? 2I. -n -7-

Since fSn(x) dx is a trigonometric sum of the nth order, Bernstein's theorem gives for I Sn (x) , and consequently for I Tn (x) , an upper bound of the order of n. Specifically, I (x) | and Vn(x) I can not exceed a constant multiple of n.

These bounds for the u's and v's, however, do not seem to lead immediately to any better theorems on convergence than can be obtained by direct use of the least-square property of the series.10 Consider for example

10 See e. g. the writer's Colloquium, already cited, Chapter III; D. Jackson, A generalized problem in weighted approximation, Transactions of the American Mathematical Society, vol. 26 (1924), pp. 133-154; J. Shohat, On the polynomial and trigonometric approximation of measurable bounded functions on a finite interval, Mathematische Annalen, vol.102 (1930), pp. 157-175.



814 D. JACKSON.

the case of the last paragraph, under which e (x) and [Q (x)]-' are summable. It follows from well-known properties of orthogonal functions in general that the integral

rn. = J 8(x) [f (x) - sn (x)]2 dx

has a smaller value than the corresponding integral with sn (x) replaced by any other trigonometric sum t, (x) of the nth order. Suppose sums t,, (x) exist so that

If(x)- tn (X) l ? En

for all values of x. Then it is certain that rn does not exceed a constant multiple of 22 say 7,, < h2 2. Let Schwarz's inequality be applied with e (x)]-12 and [e (x)]1/2 11(x) - Sn (x) I as factors under the integral sign:

[ If (x) -s (x)j dx? < I2 'n

I having the same meaning as before. It follows that

(25) f f(x)- sndx)dx ? I hen.

If this integral is denoted by gn it is easily shown" that

If (x)-Sn (x) ? _ 4 ngn+ 5 et,,

which means, in consequence of (25), that snI (x) converges uniformly toward f (x) if lim,. n e -= 0. And it will be possible to construct sums t. (x) so that this condition is satisfied, if f(x) has a continuous derivative.'2 The proof of convergence thus obtained applies even at points where e (x) vanishes.

" See Colloquium, pp. 87-88. It is to be observed that the function denoted by e (x) in the passage cited is not the e (x) of the present problem, but is to be taken as 1, while the sum T. (x) there is to be taken as the present sn (x), and m = 1.

12 See Colloquium, p. 12, Theorem IV, Corollary.

THE UNIVERSITY OF MINNESOTA, MINNEAPOLIS.



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