Fourier series and the Lubkin W-transform

Transcription

1 Fourier series ad the Lubki W-trasform Jaso Boggess, Departmet of Mathematics, Iowa State Uiversity Eric Buch, Departmet of Mathematics, Baylor Uiversity Charles N. Moore, Departmet of Mathematics, Kasas State Uiversity Abstract: We discuss the effect of a particular sequece acceleratio method, the Lubki W- trasform, o the partial sums of Fourier series. We cosider a very geeral class of fuctios with a sigle jump discotiuity, ad prove that this method fails o a large set of poits. Keywords: Fourier series, Lubki W-trasform, covergece acceleratio 2000 Mathematics Subject Classificatio: 65B10, 65T10, 42A20 1 Itroductio Fourier series are a commoly used method of approximatio so it is critical to aalyze their speed of covergece. May Fourier series, i particular those of fuctios with jump discotiuities, coverge rather slowly, so it is atural to attempt to apply extrapolatio methods to accelerate covergece. Various methods of acceleratio have bee applied with some success to the partial sums of Fourier series. I this paper, we ivestigate the applicatio of oe well-kow method of sequece acceleratio the Lubki W-trasform ad its effect o the partial sums of Fourier series. We will show that this trasform behaves poorly whe applied to the Fourier series of a fuctio with a sigle jump discotiuity. Some previous work of oe of the authors, [1] demostrated similar results for aother well-kow trasform- the δ 2 process. For a fuctio f which is itegrable o [ π, π], we defie the Fourier coefficiets by ˆf := 1 2π π π fxe ix dx for each iteger, ad we defie the th partial sum of the Fourier series as S fx := k= ˆfke ikx, where is a positive iteger ad x is i [ π, π]. Whe f is square-itegrable, S fx coverges to fx i the L 2 sese, that is, π π S fx fx 2 dx 0 as. Carleso [4] showed that for square-itegrable f, S fx fx at every poit except for a set of zero Lebesgue measure. Earlier results due to Dii-Lipschitz, Lebesgue ad Dirichlet-Jorda give coditios for poitwise covergece see e.g., Zygmud [12] for these. A typical result is the Dirichlet-Jorda theorem: If f is of bouded variatio over [ π, π], the S fx coverges to fx at each poit of cotiuity of f. For a umerical sequece {s } which has limit s, we say a trasformatio s of s accelerates covergece if there exists a k such that each s depeds oly o s 0,..., s +k ad s coverges to s faster tha s. May sequece trasformatios have bee developed to speed covergece of umerical sequeces which arise i may cotexts see e.g. Breziski ad Redivo-Zaglia [3], Delahaye [5], Sidi [8] or Wimp [11]. Cosider a sequece {a }, = 1, 2, 3..., ad set s = k=0 a k. Let ρ = a +1 /a. The Lubki trasform of s is s := s + a +11 ρ ρ +1 + ρ ρ

2 This is idetical to the colum {θ 2 } of the θ trasform ad the colum {u } of the Levi u- trasform. Whe s is the partial sum of a geometric series, the trasform 1 produces a costat sequece which has each term the sum of the geometric series. This is true for a eve wider class of sequeces; e.g. see Sidi [8] or Wimp [11]. Smith ad Ford [10] used umerical tests to compare differet methods of covergece acceleratio o the partial sums of Fourier series. Usig a set of five poits they tested slowly ad rapidly covergig Fourier series. Amog the methods they tested were the θ ad u traforms, which showed some improvemet of covergece i some cases. Drummod [6] discusses may methods of covergece acceleratio ad icludes discussio of their applicatio to Fourier series. Although with specific series it is possible to get better approximatios eve dramatically better results at some poits the Lubki-W trasformatio does ot i geeral behave well. We will show that for a fairly geeral set of fuctios, ad for a large set of values x [ π, π], this trasform turs the sequece {S fx : = 1, 2,... } ito a sequece without a limit. Theorem 1. Suppose that f C 2 [ π, π] ad that f π fπ. Cosider the sequece Sfx formed by applyig the trasformatio 1 to the sequece S fx. The Sfx diverges at every x of the form x = 2πa, where a 1 4, 1 4 is irratioal. Remarks. Cosider the 2π periodic extesio of f to the whole real lie. The hypotheses of the theorem implies that this fuctio is C 2, except for a sigle jump discotiuity every period. By periodicity, a similar theorem remais valid if this jump occurs aywhere i the iterval [ π, π]. At a jump discotiuity of f, the partial sums of the Fourier series exhibit the Gibbs pheomeo i a eighborhood of the discotiuity, ad may authors report difficulties with acceleratio methods ear such a discotiuity. Our results show that if oe applies the Lubki trasform to the partial sums of a Fourier series with a jump discotiuity, the curiously, difficulties appear away from the discotiuity. The theorem should be compared to those foud i [1] where similar results are show for the δ 2 process. Our proof will follow a few ideas foud i that paper, although here the expressios ivolved i the trasforms are more complicated ad cosequetly the aalysis is more difficult. Although our result cocers divergece, we should metio a few results o covergece. Istead of the partial sums we are cosiderig, cosider separately the series 1 2 ˆf0 + ˆfke ikx ad 1 2 ˆf0 + ˆf ke ikx. Work of Sidi [9] applies ad shows that if, say, f is smooth eough k=1 with a sigle jump discotiuity, the the Lubki trasform ca be used to approximate these accurately, ad cosequetly, a approximatio of f is obtaied upo addig these two approximatios. Breziski [2] has used a similar idea i a study of the effect of the ε algorithm similar to the Lubki trasform o the Gibbs pheomeo. Cosider a Fourier series ad to it add its cojugate series as a imagiary part. Applyig the ε algorithm to the resultig power series is the equivalet to the computatio of Padé approximats ad by takig real parts oe obtais a approximatio of the origial fuctio with the Gibbs pheomeo reduced. 2 The proof of Theorem 1 If S fx deotes the sequece of partial sums of the Fourier series of f, applyig the Lubki trasform 1 results i the sequece of fuctios: k=1 2

3 where ad S fx = S fx + a +11 ρ ρ +1 + ρ ρ +1 2 a = S fx S 1 fx = ˆf e ix + ˆfe ix 3 ρ = a +1 a = ˆf + 1 e i+1x + f + 1e i+1x ˆf e ix + ˆfe ix 4 Sice the fuctios we are cosiderig have bouded variatio, the Dirichlet-Jorda theorem applies so that S fx coverges to fx at every poit of the iterval π, π. Therefore, the theorem will be show if we show that the fractio a +11 ρ ρ +1 +ρ ρ +1 has a subsequece which stays away from 0 at all x of the form x = 2πa, a 1 4, 1 4, a irratioal. Lemma 1. Let f C 2 [ π, π]. The for every iteger, ˆf = 1+1 2i α β f, where α = [fπ f π]/π ad β = [f π f π]/π. This is easily show usig itegratio by parts. Notice also that f the Riema-Lebesgue lemma. See, e.g. Zygmud [12], pg. 45. Usig the result of Lemma 1 i equatio 3 gives = o 1 as by a = ˆf e ix ix + ˆfe 1 +1 = 2i α β f e ix i α β f e ix 5 = α 1+1 six + β 1 cosx f e ix f e ix We wat to substitute 5 ito 4 ad these ito the fractio o the right side of 2. This creates a uwieldly expressio. To deal with this, for coveiece we write a = s + ε where s = α 1+1 six the sie term ad ε = β 1 cosx f e ix f e ix, the error. Notice that s = O 1 ad ε = O 1 as. It will be computatioally simpler if we examie the fractio i with the idices shifted back by oe, that is, we will wat to examie a the behavior of 1 ρ 1 2ρ +ρ 1 ρ. With our otatio, this fractio becomes: s + ε 1 s +1+ε +1 s +ε 1 2 s +1+ε +1 s +ε + s+ε s +1 +ε +1. s 1 +ε 1 s +ε Multiply the umerator ad deomiator by 1s 1 + ε 1 s + ε to obtai 1s 1 + ε 1 s + ε s + ε s +1 ε s 1 + ε 1 s + ε 2s 1 + ε 1 s +1 + ε +1 + s + ε s +1 + ε +1 3

4 We complete the proof of the theorem usig two lemmas: the first shows that if x satisfies the hypotheses of the theorem, the there exists a subsequece { } such that the deomiator of 6 is bouded by C. The secod demostrates that for this subsequece, the umerator of 6 is bouded below by c for some costat c. Thus, alog this subsequece, 6 caot coverge to 0, thus causig the sequece Sfx i equatio to ot coverge to fx. Let us first examie the deomiator. Notice that upo multiplyig out the terms, we ed up with various terms ivolvig 1 multiplied by ε s ad s s. Recallig that s = O 1 ad ε = O 1 the ay of these terms which ivolve a product of a ε ad a s or two ε s will be O 1 as. Thus, to obtai our desired cotrol of the deomiator, we must examie the term 1 s 1 s 2s 1 s +1 + s s +1. Usig the defiitio of s ad simplifyig, this becomes: α 2 si 1x six si 1x si + 1x six si + 1x. The, the fact that the deomiator of 6 is bouded by c theorem follows from: for x as i the hypotheses of the Lemma 2. Give x = 2πa, a 1 4, 1 4, a irratioal, there exists a ifiite umber of such that si 1x si x si 1x si + 1x si x si + 1x < π 7 Proof. Notice that sice si 1x six si + 1x si 1x six si + 1x si 1x six + 2 si + 1x si 1x x si + 1x 4 < 8 it suffices to show that there exists a sequece { } such that si 1x si x + 2 si 1x si + 1x x si + 1x 48π < 9 Employig trigoometric idetities o the terms iside the absolute value, we obtai si x si x cosx cos x six + 2 si x cosx cos x six si x cosx + cos x six x si x cosx + cos x six = 2 si x cosx + 2 si x cos 2 x 2 cos 2 x si x Now cosider the equatio si β cosx + si β cos 2 x cos 2 β si x = 0. Rearragig we have ta si x β = cosx 1 + cosx

5 The square of the taget fuctio has rage [0, +. Give ay x π 2, π 2 there exists a β 0, π 2 such that equatio 11 is satisfied. Fix x such that x has the form x = 2πa, a 1 4, 1 4, a is irratioal, ad let β be give by 11. Cosider the fuctio gy = siy x siy + 2 siy x siy + x y siy + x. The gβ = 0 ad g y 8 for all y. We claim that there exists a sequece depedig o x such that x β mod 2π 6π. Assume this claim mometarily. The for these, g x = g x gβ 8 x β mod 2π 48π. This shows 9 ad combied with 8 gives 7. We eed to substatiate the claim. This follows from a result due to Chebyshev see A. Ya. Kichi [7], p. 39: For a arbitrary irratioal umber a ad arbitrary real umber γ, there exists a ifiite sequece of positive itegers { } such that a m γ 3 for some iteger m which depeds o. Apply this with a as above ad γ = β 2π ad multiply the resultig equatio by 2π to obtai x 2πm β 6π. This is the claim, which fiishes the proof of the Lemma. We must ow examie the umerator of 6. Multiplyig out the umerator i 6 yields various terms. Each will cotai the product 1 multiplied by three terms each of which could be of the type s or a ε. Ay such product which cotais oe or more of the ε will be at most O 1. Cosequetly, to show that the umerator of 6 is bouded below by c for all as i the previous lemma, it suffices to estimate the term 1s k 1s k s k s k +1 which, after usig the defiitio of s ad simplifyig becomes: α 3 1 si 1x si x si x k + 1x + 1 The followig lemma will the give the desired estimate of the umerator. Lemma 3. For x ad { } as i Lemma 2, the expressio α 3 1 si 1x si x si x k + 1x + 1 is bouded below, for large, by c for some costat c which depeds o x but ot. Proof. Sice α3 1 si x si 1x si x α 3 1 si x si 1x si x k + 1x + 1 k + 1x it suffices to show that α3 1 si x si 1x si x k + 1x > c < α3 k that is, α 3 1 si x si 1x si x + 1x > c. 12 5

6 Agai, usig trigoometric idetities, we rewrite the trigoometric terms i 12 as si x si x cosx cos x six si x x cosx + cos x six = si x si x cosx + si x cos 2 x cos 2 x si x si x cos x six Notice that the terms i the large paretheses match equatio 10. As i the proof of Lemma 1, these terms are bouded above by 48π. Cosider the term si x cos x six. By cotiuity, sice x β, mod 2π, the si x cos x six si β cosβ six. Sice x 0, six 0, ad sice β 0, π 2, si2 β 0 ad cosβ 0. Thus, si x cos x six si β cosβ six 0 ad this allows us to coclude Remarks ad further results The theorem states that for x = 2πa, a 1 4, 1 4 S fx does ot coverge to fx. It is quite likely that this covergece fails at other x also but we have ot bee able to show this. The methods used i the proof apply i other situatios. For example, cosider the slowly covergig series si jx j=0 j. With oly slight alteratios, left to the reader, the proof of Theorem 1 ca be adapted to show: Theorem 2. Set S x = si jx j=0 j ad let Sx be the sequece of fuctios obtaied by applyig 1 to S x. The Sx fails to coverge at ay poit x = 2πa where a 1 4, 3 4 is irratioal. Cosider the fuctio fx = x. Figure 1 shows the partial sums of its Fourier series, S fx, for = 25 ad 100 o the iterval [0, π]. f as well as all S f are odd fuctios. The trasformed partial sums Sfx are show for the same values of i Figure 2. The graphs show the difficulties which occur at a dese set of values i π 2, π 2. Figure 3 ivestigates whether the trasformed sequece provides a better approximatio by graphig fx S fx fx Sfx. At poits where this is positive, Sfx is the better approximatio; where this is egative, S fx provides the better approximatio. It would be iterestig to be able to a priori determie at which x the trasform speeds covergece, but this seems difficult. It is likely that the methods i this paper could be applied to show that other sequece acceleratio trasforms may give ureliable results whe applied to Fourier series. We suspect that similar difficulties ca be prove to exist for other trasforms. Curretly, each trasform cosidered seems to require a differet aalysis so it would also be of iterest to develop more geeral methods for these type of theorems. These are topics for future study. Refereces [1] Abebe, E., Graber, J., ad Moore, C. N.: Fourier series ad the δ 2 process, preprit [2] Breziski, C.: Extrapolatio algorithms for filterig series of fuctios, ad treatig the Gibbs pheomeo. Numerical Algorithms 36, [3] Breziski, C., Redivo-Zaglia, M.: Extrapolatio Methods. Theory ad Practice. North- Hollad, Amsterdam 1991 [4] Carleso, L.: O covergece ad growth of partial sums of Fourier series. Acta. Math. 116,

7 [5] Delahaye, J.-P.: Sequece Trasformatios, Spriger Series i Computatioal Mathematics 11. Spriger Verlag, Berli, 1988 [6] Drummod, J.E.: Covergece speedig, covergece ad summability. Joural of Computatioal ad Applied Mathematics 11, [7] Kichi, A. Ya.: Cotiued Fractios. The Uiversity of Chicago Press, Chicago 1964 [8] Sidi, A.: Practical Extrapolatio Methods, Cambridge Moographs o Applied ad Computatioal Mathematics 10. Cambridge Uiversity Press, Cambridge 2003 [9] Sidi, A.: A covergece ad stability study of the iterated Lubki trasform ad the θ algorithm. Math. Comp. 72, [10] Smith, D. A. ad Ford, W. F.: Numerical Comparisos of Noliear Covergece Accelerators. Mathematics of Computatio 38, [11] Wimp, J.: Sequece Trasformatios ad Their Applicatios. Academic Press, New York 1981 [12] Zygmud, A.: Trigoometric Series, Secod Editio. Cambridge Uiversity Press, Cambridge

8 Figure 1: Partial sums for = 25 ad = Figure 2: Trasformed partial sums for = 25 ad = x x Figure 3: fx S fx fx S fx for = 25 ad = 100 8