A CLOSED SET OF NORMAL ORTHOGONAL FUNCTIONS*

Transcription

1 A CLOSED SET OF NORMAL ORTHOGONAL FUNCTIONS* BY J. L. WALSH Itroductio. A set of ormal orthogoal fuctios χ} for the iterval x has bee costructed by Haar each fuctio takig merely oe costat value i each of a fiite umber of sub-itervals ito which the etire iterval ( ) is divided. Haar s set is however merely oe of a ifiity of sets which ca be costructed of fuctios of this same character. It is the object of the preset paper to study a certai ew closed set of fuctios ϕ} ormal ad orthogoal o the iterval ( ); each fuctio ϕ has this same property of beig costat over each of a fiite umber of sub-itervals ito which the iterval ( ) is divided. I fact each fuctio ϕ takes oly the values + ad except at a fiite umber of poits of discotiuity where it takes the value zero. The chief iterest of the set ϕ lies i its similarity to the usual (e.g. sie cosie Sturm-Liouville Legedre) set of orthogoal fuctios while the chief iterest of the set χ lies i its dissimilarity to these ordiary sets. The set ϕ shares with the familiar sets the followig properties oe of which is possessed by the set χ: the th fuctio has zeroes (or better sig-chages) iterior to the iterval cosidered each fuctio is either odd or eve with respect to the mid-poit of the iterval o fuctio vaishes idetically o ay sub-iterval of the origial iterval ad the etire set is uiformly bouded. Each fuctio χ ca be expressed as a liear combiatio of a fiite umber of fuctios ϕ so the paper illustrates the chages i properties which may arise from a simple orthogoal trasformatio of a set of fuctios. I we defie the set χ ad give some of its pricipal properties. I we defie the set ϕ ad compare it with the set χ. I 3 ad 4 we develop some of the properties of the set ϕ ad prove i particular that every cotiuous fuctio of bouded variatio ca be expaded i terms of the ϕ s ad that every cotiuous fuctio ca be so developed i the sese ot of covergece of the series but of summability by the first Cesàro mea. I 5 it is proved that there exists a cotiuous fuctio which caot be Preseted to the America Mathematical Society Feb Mathematische Aale Vol. 69 (9) pp ; especially pp

2 J. L. Walsh: Normal Orthogoal Fuctios. 6 expaded i a coverget series of the fuctios ϕ. I 6 there is studied the ature of the approach of the approximatig fuctios to the sum fuctio at a poit of discotiuity ad i 7 there is cosidered the uiqueess of the developmet of a fuctio.. Haar s Set χ. Cosider the followig set of fuctios: f (x) x f () x < (x) < x f () (x) < x x < i f (i) k < x < i k i = 3 k k x < i k or i k = 3 ; k < x these fuctios may be defied at a poit of discotiuity to have the average of the limits approached o the two sides of the discotiuity. If we have at our disposal all the fuctios f (i) k it is clear that we ca approximate to ay cotiuous fuctio i the iterval x as closely as desired ad hece that we ca expad ay cotiuous fuctio i a uiformly coverget series of fuctios f (i) k. For a cotiuous fuctio F (x) is uiformly cotiuous i the iterval ( ) ad thus uiformly i that etire iterval ca be approximated as closely as desired by a liear combiatio of the fuctios f (i) k where k is chose sufficietly large but fixed. The approximatio ca be made better ad better ad thus will lead to a uiformly coverget series of fuctios f (i) k. Haar s set χ may be foud by ormalizig ad orthogoalizig the set f (i) k those fuctios to be ordered with icreasig k ad for each k with icreasig i. The set χ cosists of the followig fuctios: x < χ (x) x χ (x) < x χ () (x) = χ () = x < 4 = = 4 < x < = = < x < 3 4 = = < x..... L. c. p. 36.

3 7 J. L. Walsh: Normal Orthogoal Fuctios. χ (k) = = k k < x < k = 3 k < x < k = 3 = < x < k or k < x <. The same covetio as to the value of χ (k) at a poit of discotiuity is made as for the f (k) ad χ (k) () ad χ (k) () are defied as the limits of χ (k) as x approaches ad. For ay particular value of N all the fuctios f (k) < N ca be expressed liearly i terms of the fuctios χ (k) < N ad coversely. Let F (x) be ay fuctio itegrable ad with a itegrable square i the iterval ( ); its formal developmet i terms of the fuctios χ is () F (x) χ (x) + χ (k) (x) F (y)χ (y)dy + χ (x) F (y)χ (k) (y)dy +. F (y)χ (y)dy + This series () is formed with coefficiets determied formally as for the Fourier expasios ad it is well kow that S m (x) the sum of the first m terms of this series is that liear combiatio F m (x) of the first m of the fuctios χ which reders a miimum the itegral (F (x) F m (x)) dx. That is S m (x) is i the sese of least squares the best approximatio to F (x) which ca be formed from a liear combiatio of the first m fuctios χ; it is likewise true that S m (x) is the best approximatio to F (x) which ca be formed from a liear combiatio of those fuctios f (k) that are depedet o the first m fuctios χ. Let F (x) be cotiuous i the closed iterval ( ). If ɛ is ay positive umber there exists a correspodig umber such that F (x ) F (x ) < ɛ wheever x x <. We iterpret S (x) as a liear combiatio of the fuctios f (k). The multiplier of the fuctio f (k) which appears ( i S (x) is chose so as to k furish the best approximatio i the iterval k ) to the fuctio F (x) so it is evidet that S (x) approximates to F (x) uiformly i the

4 J. L. Walsh: Normal Orthogoal Fuctios. 8 etire iterval ( ) with a approximatio better tha ɛ. The fuctio S +(x) caot differ from F (x) by more tha ɛ at ay poit of the iterval ( ) ad so for all fuctios S +l(x). Thus we have Theorem I. If F (x) is cotiuous i the iterval ( ) series () coverges uiformly to the value F (x) if the terms are grouped so that each group cotais all the terms of a set χ (k) k = 3. Haar proves that the series actually coverges uiformly to F (x) without the groupig of terms ad establishes may other results for expasios i terms of the set χ; to some of these results we shall retur later.. The Set ϕ. The set ϕ which it is the mai purpose of this paper to study cosists of the followig fuctios: x < ϕ (x) x ϕ (x) < x ϕ () x < (x) < x 4 < x < 3 4 ϕ () x < (x) 4 < x < < x < <. x..... ϕ (k ) ϕ (k) (x) x < + (x) ( ) k+ ϕ (k) (x ) () < x ϕ (k) + (x) ϕ (k) (x) x < ( ) k ϕ (k) (x ) < x k = 3 = 3. I geeral the fuctio ϕ () > is to be used with the horizotal scale reduced to oe half ad the vertical scale uchaged to form the fuctios ϕ () + ad ϕ() + i each of the halves ( ) ( ) of the origial iterval; the fuctio ϕ () + to the poit x = fuctios ϕ (k ) + ad ϕ (k) (x) is to be eve ad the fuctio ϕ() +. Similarly the fuctio ϕ(k) + odd with respect is to be used to form the the former of which is eve ad the latter odd with respect to the poit x =. All the fuctios ϕ(k) ( are to be take positive i the iterval ). The fuctio ϕ (k) is to be defied at poits of discotiuity as were the fuctios f ad χ ad at x = to have the L. c. p. 368.

5 9 J. L. Walsh: Normal Orthogoal Fuctios. value ad at x = to have the value ( ) k+. The fuctio ϕ (k) is odd or eve with respect to the poit x = accordig as k is eve or odd. The fuctios ϕ ϕ ϕ () ϕ() have 3 zeroes (i.e. sig-chages) respectively iterior to the iterval ( ). The fuctio ϕ (k ) + (x) has twice as may zeroes as the fuctio ϕ (k) ; ad ϕ (k) + (x) has oe more zero amely at x = tha has ϕ(k ) + (x). Thus the fuctio ϕ (k) has +k zeroes. This formula holds for = ad follows for the geeral case by iductio. Hece each fuctio ϕ (k) has oe more zero tha the precedig; the zeroes of these fuctios icrease i umber precisely as do the zeroes of the classical sets of fuctios sie cosie Sturm-Liouville Legedre etc. We shall at times fid it coveiet to use the otatio ϕ ϕ ϕ for the fuctios ϕ (k) ; the subscript deotes the umber of zeroes. The orthogoality of the system ϕ is easily established. Ay two fuctios are orthogoal if < 3 as may be foud by actually testig the various pairs of fuctios. Let us assume this fact to hold for = 3 N ; we shall prove that it holds for = N. By the method of costructio of the fuctios ϕ each of the itegrals ϕ (k) ϕ (k) N (x)ϕ(i) m (x)dx is the same except possibly for sig as a itegral ϕ (k) N (x)ϕ(i) m (x)dx m N ϕ (j) N (y)ϕ(l) m (y)dy after the chage of variable y = x or y = x. Each of these two itegrals [i fact they are the same itegral] whose variable is y has the value zero so we have the orthogoality of ϕ (k) (x) ad ϕ(i) m (x): N ϕ (k) N (x)ϕ(i) m (x)dx =. This proof breaks dow if the two fuctios ϕ (j) N (y) ϕ(l) m (y) are the same but i that case either ϕ (k) N (x) ad ϕ(i) m (x) are the same ad we do ot wish to prove their orthogoality or oe of the fuctios ϕ (k) N (x) ϕ(i) m (x) is odd ad the other eve so the two are orthogoal. Each of the fuctios ϕ (k) (x) is ormal for we have ϕ (k) (x) If it is desired to develop periodic fuctios by meas of the set ϕ [or the similar sets f ad χ] simultaeously i all itervals ( ) ( ) ( ) ( ) it will be wise to chage these defiitios at x = ad x = so that always the value of ϕ (k) (x) is the arithmetic mea of the limits approached at these poits to the right ad to the left.

6 J. L. Walsh: Normal Orthogoal Fuctios. except at a fiite umber of poits. Each of the fuctios χ χ χ () χ() ) χ( + ca be expressed liearly i terms of the fuctios ϕ ϕ ϕ () ϕ() ) ϕ( +. Thus for = we have χ = ϕ χ = ϕ χ () = (ϕ () + ϕ () ) χ() = () ( ϕ + ϕ () ). It is true geerally that except for a costat ormalizig factor the fuctio χ (k) + k is the same liear combiatio of the fuctios [ϕ(k ) + +ϕ (k) + ] as is χ(k) of the fuctios ϕ (k) is the same liear combiatio of the fuctios ( )k+ [ϕ (k ) ad the fuctios χ (k) + k > + ϕ (k) + ] as is ) χ(k of the fuctios ϕ (k). It is similarly true that all the fuctios ϕ ϕ ϕ ( ) + ca be expressed liearly i terms of the fuctios χ χ χ ( ) +. Thus we have for = ϕ = χ ϕ = χ ϕ () = () (χ χ () ) ϕ() = () (χ + χ () ). The geeral fact appears by iductio from the very defiitio of the fuctios ϕ. The set χ is kow to be closed ; it follows from the expressio of the χ i terms of the ϕ that the set ϕ is also closed. The defiitio of the fuctios ϕ (k) Let us set i biary otatio eables us to give a formula for ϕ (k) (x). x = a + a + a a i = or. If x is a biary irratioal or if i the biary expasio of x there exists a i i > the followig formulas hold for ϕ (k) : (3) ϕ = ϕ = ( ) a ϕ () = ( ) a +a ϕ () = ( ) a ϕ () 3 = ( ) a +a 3 ϕ () 3 = ( ) a +a +a 3 ϕ (3) 3 = ( ) a +a 3 ϕ (4) 3 = ( ) a 3 ϕ () 4 = ( ) a 3+a 4 ϕ () 4 = ( ) a +a 3 +a 4 ϕ (3) 4 = ( ) a +a +a 3 +a 4 ϕ (4) 4 = ( ) a +a 3 +a 4 ϕ (5) 4 = ( ) a +a 4 ϕ (6) 4 = ( ) a +a +a 4 ϕ (7) 4 = ( ) a +a 4 ϕ (8) 4 = ( ) a That is there exists o o-ull Lebesgue-itegrable fuctio o the iterval ( ) which is orthogoal to all fuctios of the set; l. c. p. 36.

7 J. L. Walsh: Normal Orthogoal Fuctios. The geeral law appears from these relatios; always we have (4) ϕ () = ( ) a +a ϕ (k) = ϕ k ϕ (). A geeral expressio for ϕ (k) (x) whe x is a biary ratioal ca readily be computed from formulas (3) for we have expressios for the values of ϕ (k) for eighborig larger ad smaller values of the argumet tha x. 3. Expasios i Terms of the Set ϕ}. The followig theorem results from Theorem I by virtue of the remark that all fuctios ϕ (k) ca be expressed i terms of the fuctios χ (i) ad coversely ad from the least squares iterpretatio of a partial sum of a series of orthogoal fuctios: (5) Theorem II. If F (x) is cotiuous i the iterval () the series F (x) ϕ (x) + ϕ (j) i (x) F (y)ϕ (y)dy + ϕ (x) F (y)ϕ (j) i (y)dy + F (y)ϕ (y)dy coverges uiformly to the value F (x) if the terms are grouped so that each group cotais all the terms of a set ϕ (k) k = 3. Series (5) after the groupig of terms is precisely the same as series () after the groupig of terms. Theorem II ca be exteded to iclude eve discotiuous fuctios F (x); we suppose F (x) to be itegrable i the sese of Lebesgue. Let us itroduce the otatio F (a + ) = lim F (a + ɛ) F (a ) = lim F (a ɛ) ɛ > ɛ= ɛ= ad suppose that these limits exist for a particular poit x = a. We itroduce the fuctios F (x) x < a F (a + ) x a (6) F (x) = F F (a ) x a (x) = F (x) x > a The least squares iterpretatio of the partial sums S (x) of the series () or (5) as expressed i terms of the f (j) i gives the result that if h < F (x) < h i ay iterval the also h < S (x) < h i ay completely iterior iterval if is sufficietly large. It follows that F (x) is closely approximated at x = a by its partial sum S if is sufficietly large ad that this approximatio is uiform i ay iterval about the poit x = a i which F (x) is cotiuous. A similar result holds for F (x).

8 J. L. Walsh: Normal Orthogoal Fuctios. The fuctio F (x) + F (x) differs from the origial fuctio F (x) merely by the fuctio F (a + ) x < a G(x) = F (a ) x > a. The represetatio of such fuctios by sequeces of the kid we are cosiderig will be studied i more detail later ( 6) but it is fairly obvious that such a fuctio is represeted uiformly except i the eighborhood of the poit a. If F (x) is cotiuous at ad i the eighborhood of a or if a is dyadically ratioal the approximatio to G(x) is uiform at the poit a as well. Thus we have Theorem III. If F (x) is ay itegrable fuctio ad if lim F (x) exists for x=a a poit a the whe the terms of the series (5) are grouped as described i Theorem II the series so obtaied coverges for x = a to the value lim F (x). x=a If F (x) is cotiuous at ad i the eighborhood of a the this covergece is uiform i a eighborhood of a. If F (x) is ay itegrable fuctio ad if the limits F (a ) ad F (a + ) exist for a dyadically ratioal poit x = a the the series with the terms grouped coverges for x = a to the value [F (a + ) + F (a )]; this covergece is uiform i the eighborhood of the poit x = a if F (x) is cotiuous o two itervals extedig from a oe i each directio. It is ow time to study the covergece of series (5) whe the terms are ot grouped as i Theorems II ad III. We shall establish Theorem IV. Let the fuctio F (x) be of limited variatio i the iterval x. The the series (5) coverges to the value F (x) at every poit at which F (a + ) = F (a ) ad at every poit at which x = a is dyadically ratioal. This covergece is uiform i the eighborhood of x = a i each of these cases if F (x) is cotiuous i two itervals extedig from a oe i each directio. Sice F (x) is of limited variatio F (a + ) ad F (a ) exist at every poit a. Theorem IV tacitly assumes F (x) to be defied at every poit of discotiuity a so that F (a) = [F (a + ) + F (a )]. Ay such fuctio F (x) ca be cosidered as the differece of two mootoically icreasig fuctios so the theorem will be proved if it is proved merely for a mootoically icreasig fuctio. We shall assume that F (x) is such a fuctio ad positive. We are to evaluate the limit of K (k) F (y)k (k) (x y)dy (x y) = ϕ (x)ϕ (y) + ϕ (x)ϕ (y) + + ϕ (k) (x)ϕ (k) (y).

9 3 J. L. Walsh: Normal Orthogoal Fuctios. We have already evaluated this limit for the sequece k = so it remais merely to prove that (7) lim = F (y)q (k) (x y)dy = Q (k) (x y) = ϕ () (x)ϕ () (y) + ϕ () (x)ϕ () (y) + + ϕ (k) (x)ϕ (k) (y) whatever may be the value of k. We shall cosider the fuctio F (x) merely at a poit x = a of cotiuity; that is we study essetially the ew fuctios F ad F defied by equatios (6). I the sequel we suppose a to be dyadically irratioal; the ecessary modificatios for a ratioal ca be made by the reader. The followig formulas are easily foud by the defiitio of the Q (k) ; both x ad y are supposed dyadically irratioal: Q () (x y) = ± Q () if x < (x y) = y > or if x > y < ± if x < y < or if x > y > Q () (x y) = ± Q () (x y) = if x < y > or if x > y < Q () (x y) if x < y < Q () (x y ) if x > y > Q (k) (x y) = Q (k+) (x y) = if x < y > or if x > y < Q (k) (x y) if x < y < Q (k) (x y ) if x > y > ± if x < y > or x > y < Q (k) + Q (k+) if x < y < or if x > y >. The itegral i (7) for x = a is to be divided ito three parts. Cosider a iterval bouded by two poits of the form x = ρ ν x = ρ + ν where ρ ad ν are itegers ad such that ρ ν < a < ρ + ν. The we have (8) F (y)q (k) (a y)dy = + (ρ+)/ ν ρ/ ν ρ/ ν F (y)q (k) (a y)dy + F (y)q (k) (a y)dy F (y)q (k) (ρ+)/ ν (a y)dy.

10 J. L. Walsh: Normal Orthogoal Fuctios. 4 These itegrals o the right eed separate cosideratio. Let us set ρ ν = µ + µ + µ µ ν ν µ i = or. The first itegral i the right-had member of (8) ca be writte (9) µ / + (µ / )+(µ / ) µ / + ρ/ ν (ρ/ ν ) (µ ν/ ν ) F (y)q (k) (a y)dy. Each of the itegrals is readily treated. Thus o the iterval y µ Q (k) ±ϕ (k) (a y) takes oly the values ± or is if k is eve ad has the value (y) if k is odd. It is of course true that () lim = Φ(y)ϕ (k) (y)dy = o matter what may be the fuctio Φ(y) itegrable i the sese of Lebesgue ad with a itegrable square. Hece we have lim = µ / F (y)q (k) (a y)dy =. O the iterval µ y µ + µ the fuctio Q (k) (a y) takes oly the values ± ± ad except for oe of these umbers as costat factor has the value ϕ (k) (y). It is thus true that lim = (µ / )+(µ / ) µ / F (y)q (k) (a y)dy =. From the correspodig result for each of the itegrals i (9) ad a similar treatmet of the last itegral i the right-had member of (8) we have () lim = ρ/ ν F (y)q (k) (a y)dy = lim F (y)q (k) = (a y)dy =. (ρ+)/ ν This well-kow fact follows from the covergece of the series proved from the iequality Z where a (k) = R Φ(y)ϕ(k) (y)dy. Σ(a (k) ) (Φ(x) a ϕ a ϕ a () ϕ() a (k) ϕ (k) ) dx

11 5 J. L. Walsh: Normal Orthogoal Fuctios. We shall obtai a upper limit for the secod itegral i (8) by the secod law of the mea. We otice that (ρ+)/ ν Q (k) (a y)dy ξ whatever may be the value of ξ. I fact this relatio is immediate if is small ad it follows for the larger values of by virtue of the method of costructio of the Q (k). Moreover if ν ad if ξ = ρ this itegral has ν the value zero. We therefore have from the secod law of the mea ν (ρ+)/ ν F (y)q (k) ρ/ ν ( ) ρ + (ρ+)/ ν +F ν ξ (a y)dy = F ( ρ ν ) ξ Q (k) (a y)dy ( ρ )] = [F (ρ+)/ ν (a) F ν ξ Q (k) ρ/ ν Q (k) (a y)dy. (a y)dy ρ By a proper choice of the poit we ca make the factor of this last ν itegral as small as desired; the etire expressio will be as small as desired for sufficietly large. The relatios () are idepedet of the choice of ρ so (7) is completely proved for the fuctio F ν. A similar proof applies to F so (7) ca be cosidered as completely proved for the origial fuctio F (x). The uiform covergece of (5) as stated i Theorem IV follows from the uiform cotiuity of F (x) ad will be readily established by the reader. 4. Further Expasio of the set ϕ. The least square iterpretatio already give for the partial sums ad the expressio of the ϕ s i terms of the f s show that if the terms of (5) are grouped as i Theorems II ad III the questio of covergece or divergece of the series at a poit depeds merely o that poit ad the ature of the fuctio F (x) i the eighborhood of that poit. This same fact for series (5) whe the terms are ot grouped follows from (8) ad () if F (x) is itegrable ad with a itegrable square. We shall further exted this result ad prove: Theorem V. If F (x) is ay itegrable fuctio the the covergece or divergece of the series (5) at a poit depeds merely o that poit ad the behaviour of the fuctio i the eighborhood of that poit. If i particular F (x) is of limited variatio i the eighborhood of a poit x = a ad if a is dyadically ratioal or if F (a ) = F (a + ) the series (5) coverges for x = a to the value [F (a ) + F (a + )]. If F (x) is ot oly of limited variatio but is also cotiuous i two eighborhoods oe o each side of a

12 J. L. Walsh: Normal Orthogoal Fuctios. 6 ad if a is dyadically ratioal or if F (a ) = F (a + ) the covergece of (5) is uiform i the eighborhood of the poit a. Theorem V follows immediately from the reasoig already give ad from () proved without restrictio o Φ; we state the theorem for ay bouded ormal orthogoal set of fuctios ψ : Theorem VI. If ψ (x)} is a uiformly bouded set of ormal orthogoal fuctios o the iterval ( ) ad if Φ(x) is ay itegrable fuctio the () lim = Φ(x)ψ (x)dx =. Deote by E the poit set which cotais all poits of the iterval for which Φ(x) > N; we choose N so large that Φ(x) dx < ɛ E where ɛ is arbitrary. Deote by E the poit set complemetary to E; the we have Φ(x)ψ (x)dx = Φ(x)ψ (x)dx + Φ(x)ψ (x)dx. E E It follows from the proof of () already idicated that the secod itegral o the right approaches zero as becomes ifiite. The first itegral is i absolute value less tha Mɛ whatever may be the value of where M is the uiform boud of the ψ. It therefore follows that these two itegrals ca be made as small as desired first by choosig ɛ sufficietly small ad the by choosig sufficietly large. It is iterestig to ote that Theorem VI breaks dow if we omit the hypothesis that the set ψ is uiformly bouded. I fact Theorem VI does ot hold for Haar s set χ. Thus cosider the fuctio Φ(x) = (x ) ν ν <. We have Φ(x)χ ( +) (x)dx = /+/ (x ) ν dx /+/ (x /+/ ) ν dx = ( ) ν (/) [ν ]. ν Theorem VI is proved by essetially this method for the set ψ (x) = si πx by Lebesgue Aales scietifiques de l école ormale supérieure ser. 3 Vol. XX 93. See also Hobso Fuctios of a Real Variable (97) p. 675 ad Lebesgue Aales de la Faculté des Sciece de Toulouse ser 3 Vol. I (99) pp. 5 7 especially p. 5. /

13 7 J. L. Walsh: Normal Orthogoal Fuctios. Wheever ν it is clear that () caot hold ad if ν > there is a sub-sequece of the sequece i () which actually becomes ifiite. We tur ow from the study of the covergece of such a series expasio as (5) to the study of the summability of such expasios ad are to prove Theorem VII. If F (x) is cotiuous i the closed iterval ( ) the series (5) is summable uiformly i the etire iterval to the sum F (x). If F (x) is itegrable i the iterval ( ) ad if F (a ) ad F (a + ) exist ad if either F (a ) = F (a + ) or a is dyadically ratioal the the series (5) is summable for x = a to the value [F (a )+F (a+)]. If F (x) is cotiuous i the eighborhood of the poit x = a or if a is dyadically ratioal ad F (x) cotiuous i the eighborhood of a except for a fiite jump at a the summability is uiform throughout a eighborhood of that poit. I this theorem ad below the term summability idicates summability by the first Cesàro mea. We shall fid it coveiet to have for referece the followig (3) Lemma. Suppose the series (b + b + + b ) + (b + + b b + +(b k + + b k b k+ ) + coverges to the sum B ad that the sequece (4) b b + b 3b + b + b 3 3 ( )b + ( )b + + b ( )b + + b ( )b + ( )b + + b + b + + ( )b + + b + b + + b + + ( )b + + b + ( )b + +( )b b coverges to zero. The the series (5) b + b + b 3 + is summable to the sum B. This lemma ivolves merely a trasformatio of the formulas ivolvig the limit otios. Isert zeroes i series (3) so that the paretheses are

14 J. L. Walsh: Normal Orthogoal Fuctios. 8 respectively the -th -th 3 -th terms of the ew series; this ew series coverges to the sum B ad hece is summable to the sum B. The termby-term differece of the ew series ad (5) is the series (6) b + b + + b (b + b b ) + b + + b b (b + + b b ) + which is to be show to be summable to the sum zero. The sequece correspodig to the summatio of (6) is precisely (4). A sufficiet coditio for the covergece to zero of (4) is that we have idepedetly of m (7) lim k= mb k + + (m )b k b k +m m = m k+ k for from a geometric poit of view each term of the sequece (4) is the ceter of gravity of a umber of terms such as occur i (7) each term weighted accordig to the umber of b i that appears i it. A (ɛ δ)-proof ca be supplied with o difficulty. For the case of Theorem VII let us assume F (x) itegrable ad that F (a ) ad F (a + ) exist. The series (5) is to be idetified with the series (5) ad (3) with (5) after the terms are grouped as i Theorem III. The sum that appears i (7) is the for x = a (8) m [mϕ () (a)ϕ () (y) + (m )ϕ () (a)ϕ () (y) + +ϕ (m) (a)ϕ (m) (y)]f (y)dy m. We shall prove that (8) formed for the fuctio F (y) defied i (6) ad for a dyadically irratioal has the limit zero as becomes ifiite. Let us otice that (9) m mϕ () (a)ϕ () (y) + (m )ϕ () (a)ϕ () (y) + +ϕ (m) (a)ϕ (m) (y) dy =. This follows directly from (3) ad (4). The value of the itegral i (9) is uchaged it we replace a by ay dyadic irratioal b. Choose < b < so that all the fuctios ϕ ϕ ϕ ϕ m are positive for x = b. The the itegrad i (9) ca be reduced merely to mϕ (y) so (9) is proved. Let us cosider the itegral (8) formed for the fuctio F (y) to be divided as i (8) where as before ρ ν < a < ρ + ν ad let us deote by () () () (3) respectively the etire itegral ad its three parts. The () ca be made as small as desired simply by

15 9 J. L. Walsh: Normal Orthogoal Fuctios. proper choice of the poit ρ ( ρ ν for the iterval ν ρ + ) we ca make F (y) F (a) uiformly small we have established (9) ad we have also (ρ+)/ ν ρ/ ν [mϕ () (a)ϕ () (y) + (m )ϕ () (a)ϕ () (y) ν + + ϕ (m) (a)ϕ (m) (y)]f (a)dy = if merely > ν. The itegral () is the average of m itegrals of the type that appear i (8): ρ/ ν F (y)q (k) (a y)dy k = m. Thus the etire itegral () approaches zero as becomes ifiite. Treatmet i a similar way of the itegral (3) proves that () approaches zero. It is likewise true that (8) formed for the fuctio F (y) also approaches zero as becomes ifiite. This completes the proof of the secod setece i Theorem VII for a dyadic irratioal; we omit the proof for a dyadic ratioal. The uiformity of the cotiuity of F (x) gives us readily the remaiig parts of Theorem VII. 5. Not Every Cotiuous Fuctio Ca Be Expaded i Terms of the ϕ. The summability of the expasios of cotiuous fuctios i terms of the fuctios ϕ is aother poit of resemblace of those fuctios to the Fourier sie ad cosie fuctios. Still aother poit of resemblace which we shall ow establish is that there exists a cotiuous fuctio whose expasio i terms of the ϕ s does ot coverge at every poit of the iterval. Our proof rests o a beautiful theorem due to Haar by virtue of which the existece of such a cotiuous fuctio will be show if we prove merely that (4) K (k) (a y) dy is ot bouded uiformly for all ad k. The poit a is a poit of divergece of the expasio of the cotiuous fuctio ad for our particular case may be chose ay poit of the iterval ( ). We shall study (4) i detail merely for a dyadically irratioal; the itegral (4) is idepedet of the poit a chose if a is dyadically irratioal. L.c. p.335. This coditio holds for ay set of ormal orthogoal fuctios ad is ecessary as well as sufficiet if a slight restrictio is added.

16 J. L. Walsh: Normal Orthogoal Fuctios. The itegral (4) is bouded uiformly for all the values if k = so it will be sufficiet to cosider the itegral c (k) = The followig table shows the value of c (k) value of k: Q (k) (a y) dy. for small values of ad for each = = 3 = = We have the geeral formulas c () = c (+) = c (k) = c (k) + c (k+) + = [c(k) + c (k+) ] + so the c (k) are ot uiformly bouded. Theorem VIII. If a poit a is arbitrarily chose there will exist a cotiuous fuctio whose ϕ-developmet does ot coverge at a. 6. The Approximatio to a Fuctio at a Discotiuity. We have cosidered i 3 ad 4 with a fair degree of completeess the ature of the approach to F (x) of the formal developmet of a arbitrary fuctio F (x) i the eighborhood of a poit of cotiuity of F (x). We shall ow cosider the approach to F (x) of this formal developmet i the eighborhood of a poit of discotiuity of F (x). We study this problem merely for a fuctio which is costat except for a sigle discotiuity a fiite jump but this leads directly to similar results for ay fuctio F (x) at a isolated discotiuity which is a fiite jump if F (x) is of such a ature that the expasio of F (x) would coverge uiformly i the eighborhood of the poit of discotiuity were that discotiuity removed by the additio of a fuctio costat except for a fiite jump. Let us cosider the fuctio x < a f(x) = a < x.

17 J. L. Walsh: Normal Orthogoal Fuctios. If a is dyadically ratioal f(x) ca be expressed as a fiite sum of fuctios ϕ ad thus is represeted uiformly if we make the defiitio f(a) = [f(a ) + f(a + )]; this follows from the evidet possibility of expadig f(x) i terms of the fuctios f f f (). If the poit a is dyadically irratioal f(x) caot be expaded i terms of the ϕ. The formal developmet of f(x) coverges i fact for every value of x other tha a ad diverges for x = a. The covergece for x a follows ideed from Theorem IV. We proceed to demostrate the divergece. Use the dyadic otatio The partial sum S (k) (x) = ϕ (x) a = a + a + a a = or. f(y)ϕ (y)dy + ϕ (x) f(y)ϕ (y)dy + + ϕ (k) (x) f(y)ϕ (k) (y)dy is i the sese of least squares the best approximatio to f(x) that ca be formed from the fuctios ϕ ϕ ( ϕ (k). It is therefore true that whe r k = o every subiterval r + ) o which f(x) is costat ( S (k) m (x) is also costat ad equal to f(x). O that subiterval m + ) which cotais the poit a S (k) (x) has the value (5) a m = a + + a + + a which lies betwee zero ad uity. Thus S (k) (x)[ > ] is a fuctio with two poits of discotiuity ad which takes o three distict values at its totality of poits of cotiuity. The ifiite series correspodig to the sequece (5) is (6) ( a + a 3 + a ) + ( a3 + a a ( a4 + + a a a 3 ( a5 + + a a a 4 ) ) ) +. A discotiuity at x = or x = is slightly differet [compare the first footote of ]. Uder the preset defiitio of the ϕ s it acts like a artificial discotiuity i the iterior of the iterval ad has o effect o the sequece represetig the fuctio. This was poited out for the set χ by Faber Jahresbericht der deutsche Mathematiker- Vereiigug Vol. 9 (9) pp. 4.

18 J. L. Walsh: Normal Orthogoal Fuctios. Not all umbers a after a certai poit ca be zero ad ot all of them ca be uity so the geeral term of the series (6) caot approach zero ad the sequece (5) caot coverge. It is likewise true that the sequece (5) is ot always summable ad if summable may ot be summable to the value. Thus if we choose a = the sequece (5) is summable to the sum 3. Likewise the sequece S(k) (x) for x = a ad where we cosider all values of ad k is summable to the value 3. The geeral behaviour of S (k) (x) for f(x) where we do ot make the restrictio k = is quite easily foud from the behaviour for k = ad the relatio ϕ (i) (a) f(y)ϕ (i) (y)dy = ϕ (k) (a) f(y)ϕ (k) (y)dy which holds for all values of i k ad. I fact there occurs a pheomeo quite aalogous to Gibbs s pheomeo for Forier s series. For the set ϕ the approximatig fuctios are uiformly bouded. The peaks of the approximatig fuctio S (k) disappear etirely for k = but reappear (usually altered i height) for larger values of. It is clear that the facts cocerig the approximatig curves for f(x) hold without essetial modificatio for a fuctio of limited variatio at a simple fiite discotiuity ad that the facts for the summatio of the approximatig sequece hold without essetial modificatio for a fuctio cotiuous except at a simple fiite discotiuity. 7. The Uiqueess of Expasios. We ow study the possibility of a series of the form (7) a ϕ (x) + a ϕ (x) + + a ϕ (x) + which coverges o x to the sum zero with the possible exceptio of a certai umber of poits x. Faber has poited out that there exists a series of the fuctios χ (k) (x) which coverges to zero except at oe sigle poit ad the covergece is uiform except i the eighborhood of that poit. We state for referece the easily proved Lemma. If the series (7) coverges for eve oe dyadically irratioal value of x the lim = a =. L. c. p..

19 3 J. L. Walsh: Normal Orthogoal Fuctios. This lemma results immediately from the fact that ϕ (k) (x) = ± if x is dyadically ratioal. We shall ow use this lemma to establish Theorem IX. If the series (7) coverges to the sum zero uiformly except i the eighborhood of a sigle value of x the a = for every. We phrase the argumet to apply whe this exceptioal value x is dyadically irratioal. If x > we have for x a ϕ (x) + a ϕ (x) + + a ϕ (x) + = (a + a )ϕ (x) + (a + a 3 )ϕ (x) + (a 4 + a 5 )ϕ (x) + = for every value of y = x. The we have from the uiformity of the covergece (8) a + a = a + a 3 = a 4 + a 5 =. If x < 3 4 we have for 3 4 x or for y y = 4x 3 a ϕ (x) + a ϕ (x) + + a ϕ (x) + = (a a + a a 3 )ϕ (y) + (a 4 a 5 + a 6 a 7 )ϕ (y) +(a 4 a 4+ + a 4+ a 4+3 )ϕ (y) + =. From the uiformity of the covergece we have or from (8) If x > 5 8 we have for 5 8 x 3 4 or for y y = 8x 5 a a + a a 3 = a 4 a 5 + a 6 a 7 = a = a = a = a 3 a 4 = a 5 = a 6 = a a ϕ (x) + a ϕ (x) + = (a a a + a 3 a 4 + a 5 + a 6 a 7 )ϕ (y) (a 8 a 9 a + a a + a 3 + a 4 a 5 )ϕ (y) + =. The each of these coefficiets must vaish ad hece a = a = a = a 3 = a 4 = a 5 = a 6 = a7. This lemma is closely coected with a geeral theorem due to Osgood Trasactios of the America Mathematical Society Vol. (99) pp See also Placherel Mathematische Aale Vol. 68 (99 9) pp

20 J. L. Walsh: Normal Orthogoal Fuctios. 4 Cotiuatio i this way together with the Lemma shows that every a must vaish. This reasoig is typical ad does ot essetially deped o our umerical assumptios about x. The Theorem IX is proved. The reasoig is precisely similar if istead of the hypothesis of Theorem IX we admit the possibility of a fiite umber of poits i the eighborhood of each of which the covergece is ot assumed uiform: Theorem X. If the series a ϕ (x) + a ϕ (x) + + a ϕ (x) + coverges to the sum zero uiformly x except i the eighborhood of a fiite umber of poits the = a = a = = a =. Harvard Uiversity May 9.

21 5 J. L. Walsh: Normal Orthogoal Fuctios. This paper has bee copied from the origial article published i the America Joural of Mathematics 93 volume 45 pages 5 4. It has bee prepared by Neil Johso usig TexShop for OSX i plai L A TEX with additioal maths symbols from the amssymb package. All errors are mostly due to me although I did spot (ad correct i a couple of places) mior errors i the origial. Neil Johso Cambridge December 3.