TAYLOR SERIES ARE LIMITS OF LEGENDRE EXPANSIONS. Paul E. Fishback

Transcription

1 TAYLOR SERIES ARE LIMITS OF LEGENDRE EXPANSIONS Paul E. Fishback Abstract. Next to a power series, the classical Legedre series offers the simplest method of represetig a fuctio usig polyomial expasio meas. I 86, Neuma established results for complex Legedre expasios that are aalogous to Taylor s Theorem ad the Cauchy-Hadamard Formula for power series, the primary differece beig that results are stated i terms of ellipses, as opposed to discs, of covergece. After a simple chage of variable, the foci of these ellipses may vary, each leadig to a modified Legedre expasio of the origial fuctio. Our mai result is that as the foci of these ellipses ted to oe aother, the it of the correspodig Legedre expasios is the Taylor series represetatio.. Itroductio. I [], Askey ad Haimo pose the followig questio: What are some basic similarities betwee power series ad Fourier series beyod the fact that they are both ifiite series? A slight geeralizatio of this questio might cosider comparig power series ad other types of series expasios ivolvig orthogoal fuctios. Perhaps the simplest type of such a series expasio is a Legedre series. Such series are costructed usig the classical Legedre polyomials: P 0 (x = P (x = x P (x = (3x P 3 (x = (5x3 3x. P (x = xp (x P (x.

2 Each Legedre polyomial P is a solutio of the secod order Sturm- Liouville differetial equatio [ d ( x dy ] + ( + y = 0. dx dx Take together, the family of fuctios { } = + P forms a complete orthoormal basis for the Hilbert space L ([, ]. A classical result first published i 86 by K. Neuma [], but also cited as a special case of Theorem 9.. i [5], states that for a fuctio f aalytic i a ope disk of radius R >, the Legedre series of f, = =0 ( + t= t= =0 f(tp (t dt P (z, coverges to f(z at all z iterior to the largest ellipse havig foci at (±, 0 i which f is aalytic. I additio, if ρ deotes the semiaxis of this ellipse, the ρ = + t= t= f(tp (t dt This formula is aalogous to the Cauchy-Hadamard Formula for the radius of covergece of a power series. By cosiderig the smallest possible ellipse above, which is iscribed i the circle z = R, oe obtais the iequality. + t= t= f(tp (t dt R + R. If istead, oe oly assumes that f is aalytic i a disk of radius R > 0 about the origi, the for h sufficietly small ad positive, R h >, ad f possesses a Legedre series relative to the iterval [ h, h] give by = =0 + t=h f(tp (t/h dt P (z/h h t= h

3 [4]. A simple chage of variable, together with Neuma s result above, implies that this series coverges to f(z for all z iterior to a ellipse cetered at the origi ad havig foci at (±h, 0. Moreover, + h t=h t= h f(tp (t/h dt R ( h + R. ( h. A Limit Theorem. I oe sese the, a fudametal differece exists betwee the Taylor ad Legedre series expasio, with oe beig valid i a disk cetered at the origi ad the other i a ellipse havig foci at (±h, 0. However, as h approaches zero, the foci of this ellipse move toward the origi. Give this fact, oe might woder whether the correspodig itig value of the Legedre series is simply the Taylor series. The followig theorem establishes that this is precisely the case. Surprisigly, a proof of this somewhat ituitively obvious result, to the best of the author s kowledge, is ot preset i the literature. Theorem. For f aalytic i the ope disk of radius R > 0 at the origi ad for z iterior to this disk, h 0 + = =0 + t=h = f(tp (t/h dt P (z/h = h t= h =0 f ( (0 z. (! I other words, the itig value of the Legedre series expasio at z is merely the Taylor series expasio. Proof. Defie the fuctio g(h = z + h R + R h, which is cotiuous o [0, R] ad satisfies g(0 = z R <. By cotiuity, there exists 0 < h 0 < R so that for all 0 < h h 0, g(h r for some fixed r i [0,. The proof will proceed i two primary steps. I the first step, we will assume h < h 0 ad establish that ( + t=h f(tp (t/h dt P (z/h h t= h r <. 3

4 This iequality will justify passig the it as h 0 + iside the series above. The key igrediet i this process will be a result of Mauro Picoe [3] cocerig the maximum modulus of P at a complex iput w. If ɛ deotes the eccetricity of the ellipse havig ceter at the origi, foci at (±, 0, ad major axis of legth w, the P (w (!!! ( ɛ +. Here the double factorial (!! deotes ( ( 3 3. Applyig Picoe s result ad simplifyig the resultig eccetricity to z /h, we arrive at P (z/h (!!! ( z h +. This iequality, combied with (, yields ( + t=h f(tp (t/h dt P (z/h h t= h R ( h + R h ( (!! =! ( (!! = g(h! ( (!! z + h! R + R h ( z h + ( (!! r!. (3 Rewritig (!!! = (! (!! 4

5 ad usig Stirlig s approximatio,! e π as, we obtai ( (!!! =, so that (3 reduces to r, thereby completig the first step of the proof. Now cosider the it h 0 + t=h f(tp (t/h dt P (z/h, h t= h which may be rewritte as ( + t= f(thp (t dt P (z/h. h 0 t= For each, a applicatio of Taylor s Theorem yields f(th = f(0+f (0th+ f (0 (th f ( (0! (th + f (+ (c (th + ( +! for some c betwee 0 ad t. For each 0 m, t m is orthogoal to P. Also, t= t= t P (t dt = + (! ( +! [6]. Thus, + t= t= f(thp (t dt = h (! (! f ( (0! + O(h +. (4 O the other had, P (z/h = ( (! h (! z + O(h. (5 5

6 Combiig (4 ad (5, we arrive at h t=h f(tp (t/h dt h t= h P (z/h = f ( (0 z,! which completes the proof of the theorem. Ackowledgemet. The author wishes to express his gratitude to Paul Nevai for his suggestio that simplified the proof. Ref ereces. R. Askey ad D. Haimo, Similarities Betwee Fourier ad Power Series, The America Mathematical Mothly, 03 (996, K. Neuma, Über die Etwickelug Eier Fuktio Nach de Kugelfuktioe, Joural für Mathematik, M. Picoe, Maggiorazioe di u Poliomio di Legedre e Delle Derivate i U ellisse a Quello Cofocale, Bollettio dell Uioe Matematica Italiaa, 8 (953, G. Sasoe, Orthogoal Fuctios, Itersciece Publishers, New York, G. Szegö, Orthogoal Polyomials, The America Mathematical Society, Providece, E. T. Whittaker ad G. N. Watso, A Course i Moder Aalysis, Cambridge Uiversity Press, Lodo, 963. Mathematics Subject Classificatio (000: 4C0, 30B0 Paul E. Fishback Departmet of Mathematics Grad Valley State Uiversity Alledale, MI fishbacp@gvsu.edu 6