Fractal Approximants on the Circle

Transcription

1 Chaotc Modelng and Smulaton CMSIM 3: , 2018 Fractal Approxmants on the Crcle María A. Navascués 1, Sangta Jha 2, A. K. Bedabrata Chand 2, and María V. Sebastán 1 1 Departmento de Matemátca Aplcada, Escuela de Ingenería y Arqutectura, Unversdad de Zaragoza, Zaragoza, Span. E-mal: manavas@unzar.es, msebast@unzar.es 2 Department of Mathematcs, Indan Insttute of Technology Madras, Chenna, Inda E-mal: chand@tm.ac.n, sangtajha285@gmal.com Abstract. A methodology based on fractal nterpolaton functons s used n ths work to defne new real maps on the crcle generalzng the classcal ones. The power of fractal methodology allows us to generalze any other nterpolant, both smooth and non-smooth, but the mportant fact s that ths technque provdes one of the few methods of non-dfferentable nterpolaton. In ths way, t consttutes a functonal model for chaotc processes. In ths artcle we study a generalzaton of some approxmaton formulae proposed by Dunham Jackson both n classcal and fractal cases. Keywords: Fractal Interpolaton, Trgonometrc Approxmaton, Trgonometrc Interpolaton, Smoothng, Curve Fttng. 1 Introducton Interpolaton and approxmaton are mostly carred out wth smooth functons, but n many practcal stuatons for example we come across sampled sgnals, whch are not smooth. Thus we cannot use classcal nterpolaton n such cases. Fractal nterpolato [2] helps to solve ths problem to a large extent as t nvolves both smooth and non-smooth whch are created dependng on the choce of the scalng parameters, see for nstance [3], [4], [12], [5]. Barnsley [3] and Navascués [10] observed that by a sutable choce of IFS whose elements are selected n terms of a prescrbed contnuous functon f, an entre famly of fractal functons f α, called the α-fractal functons, can be constructed to nterpolate and approxmate f. We gve here a global determnstc method to model perodc sgnals by fractal nterpolaton. Ths paper generalzes a partcular type of approxmants defned by Jackson [8] and extend these approxmants to ts fractals whch are smooth or non-smooth n nature dependng on the choce of scalng factors. The functons proposed have the advantage of ownng an analytcal explct expresson n terms of the samples specfc values of the orgnal functon. Ths fact gves them a partcular mportance n order to Receved: 1 November 2017 / Accepted: 27 March 2018 c 2018 CMSIM ISSN

2 344 Navascués et al. obtan mathematcal representatons of expermental sgnals, from whch only ther values at evenly sampled nodes are known. The lmts proposed prove the convergence of approxmants and ntepolants wth very weak condtons as the samplng frequency s ncreased. The fast evoluton of the programs of advanced calculus enables the use of fractal functons more complcated that mere polynomal and trgonometrc mappngs. In ths way, from the theoretcal pont of vew, new fractal nodal elements are proposed, that provde a generalzaton of the trgonometrc functons. The densty of these mappngs n the space of contnuous and perodc functons s proved. In some cases, the new fractal elements perform better than the classcal Jackson s orgnals. The paper s organzed as follows. In secton 2, we have gven some bref descrpton of α-fractal functons and revew the requred classcal results on unform convergence of polynomal and trgonometrc nterpolaton. A new type of dscrete verson of Jackson approxmants ans ts convergence results proposed n secton 3. These results have been extended to the fractal verson of dscrete Jackson approxmants n secton 4. 2 Prelmnares: In ths secton we shall gather some essental materals that can be found n detals n the references [1], [2], [7], [13], [11]. 2.1 Constructons of fractal functons: In secton 2.1, constructon of contnuous fractal nterpolaton functon based on terated functon system s revewed. Let x 1 < x 2 <... < x N be real numbers, and I = [x 1, x N ], be a closed nterval that contans them. Let a set of data ponts {x, y, N N } be gven, where N k s the frst k natural numbers, and I = [x, x +1 ]. Let L : I I, N N 1 be contractve homeomorphsms such that L x 1 = x, L x N = x Let K = I R and N 1 contnuous mappngs, F : K R be satsfyng F x 1, y 1 = y, F x N, y N = y +1, F x, y F x, y α y y, 2 where x, y, x, y K, α 1, 1, N N 1. Now defne functons w : R 2 R 2 as w x, y = L x, F x, y, N N 1. Theorem 1. Barnsley[2] The IFS I = {K; w, = 1, 2,..., N 1} admts a unque attractor G. G s the graph of a contnuous functon f : I R whch obeys fx = y for = 1, 2,..., N. The prevous functon s called fractal nterpolaton functonfif correspondng to the IFS I. For the mplct representaton of FIF, we proceed as follows: Let G = {g : [x 0, x N ] [c, d] g s contnuous and gx 0 = y 0, gx N = y N }. Then G s a complete metrc space wth respect to unform norm.. Defne a mappng T : G G by T gx = F L 1 x, g L 1 x x

3 Chaotc Modelng and Smulaton CMSIM 3: , [x, x +1 ], N N 1. Now, T s a contracton mappng on the metrc space G,., that s, T g T h α g h, g, h G, where α = α 1,..., α N 1 and α = max{ α : = 1, 2,..., N 1} < 1. Here T possesses a unque fxed pont f G, whch satsfes fx = F L 1 x, f L 1 x x [x, x +1 ]. α-fractal functons: Navascués [10] observed that the theory of FIF can be used to generate a famly of contnuous functons havng fractal characterstcs from a prescrbed contnuous functon. Consder a partton := {x 1, x 2,..., x N } of I satsfyng x 1 < x 2 < < x N, a base functon b satsfyng b CI, b f, bx 1 = fx 1, bx N = fx N and N 1 real numbers α satsfyng α < 1. Defne an IFS through the maps L x = a x + d, F x, y = α y + f L x α bx, N N 1, 3 where L and F satsfes 1 and 2 respectvely and b s defned through the lnear map L : CI CI such that L s bounded wth respect to sup norm and satsfy Lfx 1 = fx 1 and Lfx N = fx N. The correspondng FIF denoted by f,b α = f α s referred as α-fractal functon for f wth respect to a scale vector α base functons b and partton. From 3 the followng unform error bound can be found see for nstance [11], f α f α f α b. 4 1 α 2.2 Some classcal results: Dunham Jackson deduced several nequaltes to compute the unform dstance between a contnuous or dfferentable functon and the space of trgonometrc or algebrac polynomals. For nstance n the perodc case we have the followng results [6],[7],[9]. Theorem 2. Let f C[ π, π] and be perodc. If d nf = df, π n, where df, π n s the mnmum dstance between f and the space then d nf ω { n } π n = a k coskx + b k snkx ; a k, b k R, π n+1 k=0, where ωδ s the modulus of contnuty of f. Theorem 3. If f C 2 [ π, π] and s perodc, then 1 a k = O k 2, b k = O 1 k 2 and f S n = On 1, where S n s the n-th Fourer sum, and a k, b k are ts coeffcents.

4 346 Navascués et al. Theorem 4. If f C[ π, π], s perodc and satsfes a Dn-Lpschtz condton lm δ 0 logδωδ = 0, then the Fourer seres converges unformly to f Ths result s acheved by the Jackson approxmants wth the sngle hypothess of contnuty Secton 3, and let us remember that unform convergence mples convergence n the mean of order p L p -norm on compact ntervals for 1 p. The advantage of the Jackson approxmants s the fact of beng explct n terms of the sampled values of the orgnal functon. 3 Generalzaton of Dscrete Jackson approxmants We consder a type of Jackson approxmants. They are defned n a dscrete way, usng only the samples of the functon to be approached. We consder arbtrary but postve exponents γ > 0. In the Jackson case the value of the exponent s γ = 4 [?], p Consder P m,,γ x = sn mx x 2 γ, where m sn x x 2 2m D m,γ fx = H m,γ fx P m,,γ x, 5 2m Hm,γ 1 = P m,,γ x and x +1 x = π, for = 1, 2,..., 2m 1. 6 m For a postve exponent γ > 0, H m,γ = H m,γ x depends on the varable x although we wll preserve the orgnal notaton omttng t. H m,4 s a constant. An mportant dentty nvolvng the basc functons n 5 s [7], p. 340: For n N and t R, sn nt 2 sn t 2 2 = n + 2 n 1 cost + n 2 cos2t cosn 1t. 7 Lemma 1. For all m = 1, 2,...; γ > 0, and v R : sn mv γ 1. m snv Proof. Usng the dentty 7 for n = m and t = 2v: snmv 2 = m + 2 m 1 cos2v + m 2 cos4v cos2m 1v. snv

5 Chaotc Modelng and Smulaton CMSIM 3: , Then sn mv γ = 1 m snv m γ m + 2 m 1 cos2v + m 2 cos4v cos2m 1v γ/2 1 + m 1 m + 21 m 1 γ/2 mγ 2 1 m γ m2 γ/ Fg. 1. Graph of the functon fx = sn x cos x 3 n the nterval [0, 2π]. Theorem 5. Let f C[ π, π] be Hölder contnuous such that for x, x [ π, π] fx fx K x x q, 0 < q 1. 8 Then for γ > q + 1, D m,γ f f K 2 π 2 γ π 2m q γ ζγ q + ζγ, where ζs s the Remann zeta functon: ζs = + 1 s. 9 Proof. Accordng to the defntons of H m,γ and D m,γ, and the change we have: x = x + 2u, 2m E m,γ fx = D m,γ fx fx H m,γ fx + 2u fx P m,,γ u. 10

6 348 Navascués et al. Usng the Lpschtz constant and exponent 5 2m E m,γ fx H m,γ K2 q u q P m,,γ x 2u, where due to perodcty, we can assume u [ π/2, π/2]. Consderng ncreasng order n u and denotng them by v 0, v 1,..., v : where, E m,γ fx H m,γ K2 q =0 P m,,γx 2v, 11 π v π + 1 π 2, 12 for = 0, 1,..., 2m 1 [?], p Here P m,,γ x 2v = snmv m snv γ. Now from Lemma 1 snmv 0 γ m snv 0 For 1, usng the nequalty for v [0, π/2], As a consequence, for 1, Usng 11 we obtan E m,γ fx H m,γ K2 q By defnton of v 7: snv 2v π, 14 π m snv m sn m2 2. snmv γ γ. 15 m snv =0 For 0 q 1, + 1 q q + 1, γ 2 γ π snmv γ H m,γ K2 q[ v q 0 m snv + γ ]. 16 v q π q, 0 17 γ 2 γ π q q q γ = 2γ π + 1 q 1 γ. q 1 γ q + 1 γ

7 Chaotc Modelng and Smulaton CMSIM 3: , and thus γ 2 γ π q ζγ q + ζγ, 18 where ζs s the Remann zeta functon 9, convergent for s > 1. If γ > q + 1, collectng all the nequaltes 16, 17, 18: E m,γ fx H m,γ K2 q[ π q + 2 γ π q ] ζγ q + ζγ. 19 Let us bound now H m,γ 6. H 1 m,γ = =0 snmv γ > snmv 0 γ + snmv 1 γ. 20 m snv m snv 0 m snv 1 Snce mv π 2 for = 0, 1, from 7 and 14, snmv m snv 2mv mπ snv 2 π, and thus 20, Fnally 19, γ. Hm,γ 1 > 2 21 π E m,γ fx K 2 π 2 γ π 2m q γ ζγ q + ζγ, f γ > q + 1. The former result s refned n the next Theorem. Theorem 6. Let f C[ π, π] be Hölder contnuous such that for x, x [ π, π], fx fx K x x q, 0 < q 1. Then for γ > q + 1, D m,γ f f K 2 π 2 γ π 2m q q + 2 γ 1 γ q γ 1. Proof. We proceed as n the prevous Theorem untl 11, E m,γ fx H m,γ K2 q =0 Bearng n mnd the nequalty 15 for 2, one obtans E m,γ fx H m,γ K2 q =0 snmv γ. 22 m snv snmv γ H m,γ K2 q v q 0 m snv +vq 1 + γ.

8 350 Navascués et al Fg. 2. Approxmant D m,γf of a sampled sgnal for m = 5, γ = 4Jackson case and γ = 5 n the nterval [ π, π]. By defnton of v 7, v q π q, 0 23 and As before γ 2 γ π q v q π q, m γ 2 γ π q q +1 1 γ = 2γ π + 1 q 1 γ. q 1 γ q + 1 γ. 25 The last sum s part of lower Remann sums of the functons 1/x γ q and 1/x γ, n the nteval [1, + wth unt step, respectvely: f γ > q + 1. Lkewse 1 γ q 1 dx x γ q = 1 γ q + 1, 1 γ dx 1 x γ = 1 γ 1,

9 Chaotc Modelng and Smulaton CMSIM 3: , f γ > 1. Collectng all the nequaltes 23, 24, 25: E m,γ fx H m,γ K2 q[ π q+ π q+2 γ π q 2m Fnally, 1 γ q γ 1 E m,γ fx K 2 π 2 γ π 2m q q + 2 γ 1 γ q , γ 1 ]. f γ > q Fractal verson of dscrete Jackson approxmants Some easy and elegant observatons on fractal functons n conjuncton wth the theorems from secton 3, provde ther correspondng fractal versons. Such knd of fractal approxmants possess many desrable propertes as ther tradtonal counterparts and also reassures the ubquty of fractal functons. As descrbed n secton 2, to get the fractal verson we have to perturb the bass functon P m,,γ x usng sutable base functons b,m,γ, sutable scalng vector α m and partton of [ π, π] and defne the fractal dscrete Jackson approxmants as 2m Dm,γfx α = H m,γ fx Pm,,γx. α The followng generalzes the Theorem 6. Theorem 7. Let f C[ π, π] be Hölder contnuous such that for x, x [ π, π], fx fx K x x q, 0 < q 1. Then for γ > q + 1, D α m,,γ f f K 2 π 2 γ π 2m q γ ζγ q + ζγ +m π 2 γ f α 1 α max 1 2m P α m,,γ b m,,γ, where α, b m,,γ are the sutable scalng vector and bass functon used to construct the fractal perturbaton of P m,,γ. Proof. From the trangle nequalty we have: D α m,γf f D α m,γf D m,γ f + D m,γ f f. Under the stated hypothess from Theorem 5 we have: D m,γ f f K 2 π 2 γ π 2m q γ ζγ q + ζγ. 26

10 352 Navascués et al Fg. 3. Approxmant D α m,γf of a sampled sgnal for m = 12, γ = 5 and n the nterval [ π, π] for dfferent values of α. Now 2m Dm,γfx α D m,γ fx = H m,γ fx Pm,,γx α P m,,γ x 2m H m,γ fx Pm,,γx α P m,,γ x 2m H m,γ 2m f Pm,,γx α P m,,γ x π 2 γ m f max P m,,γ α P m,,γ. 1 2m 27 Fnally, usng 4, 26 and 27the proof follows. The next result s the refned verson of the above theorem. Theorem 8. Let f C[ π, π] be Hölder contnuous such that for x, x [ π, π], fx fx K x x q, 0 < q 1. Then for γ > q + 1, D m,γ f f K 2 π 2 γ π 2m q q + 2 γ 1 γ q m π 2 γ f α 1 α γ 1 max P m,,γ α b m,,γ, 1 2m

11 Chaotc Modelng and Smulaton CMSIM 3: , where α, b m,,γ are the sutable scalng vector and bass functon used to construct the fractal perturbaton of P m,,γ. Proof. The result follows from Theorem 6 and the trangle nequalty D α m,γf f D α m,γf D m,γ f + D m,γ f f. Corollary 1. If f C[ π, π] s Hölder contnuous wth exponent q 0 < q 1 and γ > q + 1 and f we select scalng factors α as α 1 m, for = q+1 1, 2,... N 1 where N s the number of partton of [ π, π], then Dm,γf α converges unformly to f as m tends to nfnty. The order of convergence Om q does not depend on γ. References 1. N. I. Acheser. Theory of Approxmaton, Dover Publ., New York, M.F. Barnsley. Fractal functons and nterpolaton, Constr. Approx., 2, , M. F. Barnsley. Fractals Everywhere, Academc Press Inc., Orlando, Florda, M. F. Barnsley and A. N. Harrngton. The calculus of fractal nterpolaton functons, J. Approx. Theory, 571, 14-34, A. K. B. Chand, and G. P. Kapoor. Generalzed cubc splne fractal nterpolaton functons, SIAM J. Numer. Anal., 442, , E. W. Cheney. Approxmaton Theory, AMS Chelsea Publ., Provdence P. J. Davs. Interpolaton and Approxmaton, Dover Publ., New York 2nd. ed D. Jackson. On approxmaton by trgonometrc sums and polynomals. Trans. Am. Math. Soc., 13, , G. G. Lorentz. Approxmaton of Functons, AMS Chelsea Publ., Provdence Rhode Island, M. A. Navascués. Fractal polynomal nterpolaton, Z. Anal. Anwend., 252, , M. A. Navascués and A. K. B. Chand, Fundamental sets of fractal functons, Acta Appl. Math., 1003, , M. A. Navascués and M. V. Sebastán, Smooth fractal nterpolaton, J. Ineq. Appl., 78734, 1-20, J. Szabados, and P. Vértes. Interpolaton of Functons, World Sc. Publ., Sngapore, 1990.