A methodology based on fractal interpolation functions is used in this work to define new real maps on the circle generalizing the classical ones. The power of fractal methodology allows us to generalize any other interpolant, both smooth and non-smooth, but the important fact is that this technique provides one of the few methods of non-differentiable interpolation. In this way, it constitutes a func- tional model for chaotic processes. In this article we study a generalization of some approximation formulae proposed by Dunham Jackson both in classical and fractal cases

Chand, A.K. Bedabrata

Jha, Sangita

Navascués Sanagustín, María Antonia

Sebastián Guerrero, María Victoria

English

Repositorio Universidad de Zaragoza

Chaotic Modeling and Simulation (CMSIM) 3: 343–353, 2018Fractal Approximants on the CircleMaŕıa A. Navascués1, Sangita Jha2, A. K. Bedabrata Chand2, and Maŕıa V.Sebastián11 Departmento de Matemática Aplicada, Escuela de Ingenieŕıa y Arquitectura,Universidad de Zaragoza, 500018 Zaragoza, Spain.(E-mail: manavas@unizar.es, msebasti@unizar.es)2 Department of Mathematics, Indian Institute of Technology Madras, 600036Chennai, India(E-mail: chand@iitm.ac.in, sangitajha285@gmail.com)Abstract. A methodology based on fractal interpolation functions is used in thiswork to define new real maps on the circle generalizing the classical ones. The powerof fractal methodology allows us to generalize any other interpolant, both smoothand non-smooth, but the important fact is that this technique provides one of thefew methods of non-differentiable interpolation. In this way, it constitutes a func-tional model for chaotic processes. In this article we study a generalization of someapproximation formulae proposed by Dunham Jackson both in classical and fractalcases.Keywords: Fractal Interpolation, Trigonometric Approximation, Trigonometric In-terpolation, Smoothing, Curve Fitting.1 IntroductionInterpolation and approximation are mostly carried out with smooth functions,but in many practical situations for example we come across sampled signals,which are not smooth. Thus we cannot use classical interpolation in such cases.Fractal interpolatio [2] helps to solve this problem to a large extent as it in-volves both smooth and non-smooth which are created depending on the choiceof the scaling parameters, see for instance ([3], [4], [12], [5]). Barnsley [3] andNavascués [10] observed that by a suitable choice of IFS whose elements are se-lected in terms of a prescribed continuous function f, an entire family of fractalfunctions fα, called the α-fractal functions, can be constructed to interpolateand approximate f. We give here a global deterministic method to model pe-riodic signals by fractal interpolation. This paper generalizes a particular typeof approximants defined by Jackson [8] and extend these approximants to itsfractals which are smooth or non-smooth in nature depending on the choiceof scaling factors. The functions proposed have the advantage of owning ananalytical explicit expression in terms of the samples (specific values) of theoriginal function. This fact gives them a particular importance in order toReceived: 1 November 2017 / Accepted: 27 March 2018c© 2018 CMSIM ISSN 2241-0503344 Navascués et al.obtain mathematical representations of experimental signals, from which onlytheir values at evenly sampled nodes are known. The limits proposed provethe convergence of approximants and intepolants with very weak conditionsas the sampling frequency is increased. The fast evolution of the programs ofadvanced calculus enables the use of fractal functions more complicated thatmere polynomial and trigonometric mappings. In this way, from the theoreticalpoint of view, new fractal nodal elements are proposed, that provide a gener-alization of the trigonometric functions. The density of these mappings in thespace of continuous and periodic functions is proved. In some cases, the newfractal elements perform better than the classical Jackson’s originals.The paper is organized as follows. In section 2, we have given some briefdescription of α-fractal functions and review the required classical results onuniform convergence of polynomial and trigonometric interpolation. A newtype of discrete version of Jackson approximants ans its convergence resultsproposed in section 3. These results have been extended to the fractal versionof discrete Jackson approximants in section 4.2 Preliminaries:In this section we shall gather some essential materials that can be found indetails in the references ([1], [2], [7], [13], [11]).2.1 Constructions of fractal functions:In section 2.1, construction of continuous fractal interpolation function based oniterated function system is reviewed. Let x1 < x2 < . . . < xN be real numbers,and I = [x1, xN ], be a closed interval that contains them. Let a set of datapoints {(xi, yi), i ∈ NN} be given, where Nk is the first k natural numbers, andIi = [xi, xi+1]. Let Li : I → Ii, i ∈ NN−1 be contractive homeomorphisms suchthatLi(x1) = xi, Li(xN ) = xi+1. (1)Let K = I × R and N − 1 continuous mappings, Fi : K → R be satisfyingFi(x1, y1) = yi, Fi(xN , yN ) = yi+1, |Fi(x, y)− Fi(x, y′)| ≤ |αi||y − y′|, (2)where (x, y), (x, y′) ∈ K, αi ∈ (−1, 1), i ∈ NN−1. Now define functionswi : R2 → R2 as wi(x, y) = (Li(x), Fi(x, y)),∀i ∈ NN−1.Theorem 1. (Barnsley[2]) The IFS I = {K;wi, i = 1, 2, . . . , N − 1} admitsa unique attractor G. G is the graph of a continuous function f : I → R whichobeys f(xi) = yi for i = 1, 2, . . . , N.The previous function is called fractal interpolation function(FIF) correspond-ing to the IFS I. For the implicit representation of FIF, we proceed as follows:Let G = {g : [x0, xN ] → [c, d] | g is continuous and g(x0) = y0, g(xN ) = yN}.Then G is a complete metric space with respect to uniform norm ‖.‖∞.Define a mapping T : G 7→ G by (Tg)(x) = Fi(L−1i (x), g ◦ L−1i (x)) ∀x ∈Chaotic Modeling and Simulation (CMSIM) 3: 343–353, 2018 345[xi, xi+1], i ∈ NN−1.Now, T is a contraction mapping on the metric space (G, ‖.‖∞), that is,‖Tg − Th‖∞ ≤ |α|∞‖g − h‖, g, h ∈ G,where α = (α1, . . . , αN−1) and |α|∞ = max{|αi| : i = 1, 2, . . . , N − 1} < 1.Here T possesses a unique fixed point f ∈ G, which satisfiesf(x) = Fi(L−1i (x), f ◦ L−1i (x)) ∀x ∈ [xi, xi+1].α-fractal functions: Navascués [10] observed that the theory of FIF canbe used to generate a family of continuous functions having fractal charac-teristics from a prescribed continuous function. Consider a partition ∆ :={x1, x2, . . . , xN} of I satisfying x1 < x2 < · · · < xN , a base function b satisfy-ing b ∈ C(I), b 6= f, b(x1) = f(x1), b(xN ) = f(xN ) and N − 1 real numbers αisatisfying |αi| < 1. Define an IFS through the mapsLi(x) = aix+ di, Fi(x, y) = αiy + f ◦ Li(x)− αib(x), i ∈ NN−1, (3)where Li and Fi satisfies (1) and (2) respectively and b is defined through thelinear map L : C(I)→ C(I) such that L is bounded with respect to sup normand satisfy Lf(x1) = f(x1) and Lf(xN ) = f(xN ).The corresponding FIF denoted by fα∆,b = fα is referred as α-fractal functionfor f with respect to a scale vector α base functions b and partition ∆.From (3) the following uniform error bound can be found (see for instance[11]),‖fα − f‖∞ ≤|α|∞1− |α|∞‖fα − b‖∞. (4)2.2 Some classical results:Dunham Jackson deduced several inequalities to compute the uniform distancebetween a continuous (or differentiable) function and the space of trigonometricor algebraic polynomials. For instance in the periodic case we have the followingresults ([6],[7],[9]).Theorem 2. Let f ∈ C[−π, π] and be periodic. If d∗n(f) = d(f, πn), whered(f, πn) is the minimum distance between f and the spaceπn ={ n∑k=0(ak cos(kx) + bk sin(kx)); ak, bk ∈ R},then d∗n(f) ≤ ω(πn+1), where ω(δ) is the modulus of continuity of f .Theorem 3. If f ∈ C2[−π, π] and is periodic, then|ak| = O( 1k2), |bk| = O( 1k2)and‖f − Sn‖∞ = O(n−1),where Sn is the n-th Fourier sum, and ak, bk are its coefficients.346 Navascués et al.Theorem 4. If f ∈ C[−π, π], is periodic and satisfies a Dini-Lipschitz condi-tion limδ→0log(δ)ω(δ) = 0, then the Fourier series converges uniformly to fThis result is achieved by the Jackson approximants with the single hypothesisof continuity (Section 3), and let us remember that uniform convergence impliesconvergence in the mean of order p (Lp-norm) on compact intervals for 1 ≤p ≤ ∞.The advantage of the Jackson approximants is the fact of being explicit interms of the sampled values of the original function.3 Generalization of Discrete Jackson approximantsWe consider a type of Jackson approximants. They are defined in a discreteway, using only the samples of the function to be approached. We considerarbitrary but positive exponents γ > 0. In the Jackson case the value of theexponent is γ = 4 ([?], p. 456). Consider Pm,i,γ(x) =∣∣∣ sin(m(xi−x)2 )m sin(xi−x2) ∣∣∣γ ,Dm,γ(f)(x) = Hm,γ2m∑i=1f(xi)Pm,i,γ(x), (5)whereH−1m,γ =2m∑i=1Pm,i,γ(x) and xi+1 − xi =πm, for i = 1, 2, . . . , 2m− 1. (6)For a positive exponent γ > 0, Hm,γ = Hm,γ(x) depends on the variable xalthough we will preserve the original notation omitting it. Hm,4 is a constant.An important identity involving the basic functions in (5) is ([7], p. 340):For n ∈ N and t ∈ R,( sin(nt2 )sin( t2 ))2= n+ 2((n− 1) cos(t) + (n− 2) cos(2t) + . . .+ cos((n− 1)t)). (7)Lemma 1. For all m = 1, 2, . . .; γ > 0, and v ∈ R :∣∣∣ sin (mv)m sin(v)∣∣∣γ ≤ 1.Proof. Using the identity (7) for n = m and t = 2v:( sin(mv)sin(v))2= m+ 2((m− 1) cos(2v) + (m− 2) cos(4v) + . . .+ cos(2(m− 1)v)).Chaotic Modeling and Simulation (CMSIM) 3: 343–353, 2018 347Then∣∣∣ sin (mv)m sin(v)∣∣∣γ = 1mγ|m+ 2((m− 1) cos(2v) + (m− 2) cos(4v) + . . .+ cos(2(m− 1)v))|γ/2≤ 1mγ(m+ 2(1 + (m− 1)2)(m− 1))γ/2≤ 1mγ(m2)γ/2.0 1 2 3 4 5 6−0.2−0.15−0.1−0.0500.050.10.15Fig. 1. Graph of the function f(x) = (sinx cosx)3 in the interval [0, 2π].Theorem 5. Let f ∈ C[−π, π] be Hölder continuous such that forx, x′ ∈ [−π, π]|f(x)− f(x′)| ≤ K|x− x′|q, 0 < q ≤ 1. (8)Then for γ > q + 1,‖Dm,γ(f)− f‖∞ ≤ K2 (π2 )γ( π2m )q(1 + 2γ(ζ(γ − q) + ζ(γ))),where ζ(s) is the Riemann zeta function:ζ(s) =+∞∑i=11is. (9)Proof. According to the definitions of Hm,γ and Dm,γ , and the changexi = x+ 2ui,we have:|Em,γ(f)(x)| = |Dm,γ(f)(x)− f(x)| ≤ Hm,γ2m∑i=1|f(x+ 2ui)− f(x)|Pm,i,γ(u).(10)348 Navascués et al.Using the Lipschitz constant and exponent (5)|Em,γ(f)(x)| ≤ Hm,γK2q2m∑i=1|ui|q|Pm,i,γ(xi − 2ui),where due to periodicity, we can assume ui ∈ [−π/2, π/2]. Considering increas-ing order in |ui| and denoting them by v0, v1, . . . , v2m−1 :|Em,γ(f)(x)| ≤ Hm,γK2q2m−1∑i=0vqi Pm,i,γ(xi − 2vi), (11)where,πi4m≤ vi ≤π(i+ 1)4m≤ π2, (12)for i = 0, 1, . . . , 2m− 1 ([?], p. 458). Here Pm,i,γ(xi − 2vi) = | sin(mvi)m sin(vi) |γ . Nowfrom Lemma 1 ∣∣∣ sin(mv0)m sin(v0)∣∣∣γ ≤ 1. (13)For i ≥ 1, using the inequalitysin(v) ≥ 2vπ, (14)for v ∈ [0, π/2],m sin(vi) ≥ m sin( πi4m)≥ m2 i4m≥ i2.As a consequence, for i ≥ 1, ∣∣∣ sin(mvi)m sin(vi)∣∣∣γ ≤ (2i)γ. (15)Using (11) we obtain|Em,γ(f)(x)| ≤ Hm,γK2q2m−1∑i=0vqi∣∣∣ sin(mvi)m sin(vi)∣∣∣γ ≤ Hm,γK2q[vq0 + 2m−1∑i=1vqi(2i)γ].(16)By definition of vi (7):vq0 ≤( π4m)q, (17)2m−1∑i=1vqi(2i)γ≤ 2γ( π4m)q 2m−1∑i=1(i+ 1)q1iγ.For 0 ≤ q ≤ 1, (i+ 1)q ≤ (iq + 1),2m−1∑i=1vqi(2i)γ≤ 2γ( π4m)q 2m−1∑i=1(iq + 1)1iγ= 2γ( π4m)q 2m−1∑i=1( 1iγ−q+1iγ)Chaotic Modeling and Simulation (CMSIM) 3: 343–353, 2018 349and thus2m−1∑i=1vqi(2i)γ≤ 2γ( π4m)q(ζ(γ − q) + ζ(γ)), (18)where ζ(s) is the Riemann zeta function (9), convergent for s > 1.If γ > q + 1, collecting all the inequalities ((16), (17), (18)):|Em,γ(f)(x)| ≤ Hm,γK2q[( π4m)q+ 2γ( π4m)q(ζ(γ − q) + ζ(γ))]. (19)Let us bound now Hm,γ (6).H−1m,γ =2m−1∑i=0∣∣∣ sin(mvi)m sin(vi)∣∣∣γ > ∣∣∣ sin(mv0)m sin(v0)∣∣∣γ + ∣∣∣ sin(mv1)m sin(v1)∣∣∣γ . (20)Since mvi ≤ π2 for i = 0, 1, from (7) and (14),sin(mvi)m sin(vi)≥ 2mvimπ sin(vi)≥ 2π,and thus (20),H−1m,γ > 2( 2π)γ. (21)Finally (19),|Em,γ(f)(x)| ≤K2(π2)γ(π2m)q(1 + 2γ(ζ(γ − q) + ζ(γ)),if γ > q + 1. The former result is refined in the next Theorem.Theorem 6. Let f ∈ C[−π, π] be Hölder continuous such that forx, x′ ∈ [−π, π],|f(x)− f(x′)| ≤ K|x− x′|q, 0 < q ≤ 1.Then for γ > q + 1,‖Dm,γ(f)− f‖∞ ≤ K2 (π2 )γ( π2m )q(1 + 2q + 2γ(1γ−(q+1) +1γ−1)).Proof. We proceed as in the previous Theorem until (11),|Em,γ(f)(x)| ≤ Hm,γK2q2m−1∑i=0vqi∣∣∣ sin(mvi)m sin(vi)∣∣∣γ . (22)Bearing in mind the inequality (15) for i ≥ 2, one obtains|Em,γ(f)(x)| ≤ Hm,γK2q2m−1∑i=0vqi∣∣∣ sin(mvi)m sin(vi)∣∣∣γ ≤ Hm,γK2q(vq0+vq1+2m−1∑i=2vqi(2i)γ).350 Navascués et al.0 1 2 3 4 5 6−0.1−0.0500.050.10 1 2 3 4 5 6−0.1−0.0500.050.1Fig. 2. Approximant Dm,γ(f) of a sampled signal for m = 5, γ = 4(Jackson case)and γ = 5 in the interval [−π, π].By definition of vi (7),vq0 ≤( π4m)q, (23)vq1 ≤( π2m)q, (24)and2m−1∑i=2vqi(2i)γ≤ 2γ( π4m)q 2m−1∑i=2(i+ 1)q1iγ.As before2m−1∑i=2vqi(2i)γ≤ 2γ( π4m)q 2m−1∑i=2(iq+1)1iγ= 2γ( π4m)q 2m−1∑i=2( 1iγ−q+1iγ). (25)The last sum is part of lower Riemann sums of the functions 1/xγ−q and 1/xγ ,in the inteval [1,+∞) with unit step, respectively:2m−1∑i=21iγ−q≤∫ ∞1dxxγ−q=1γ − (q + 1),if γ > q + 1. Likewise2m−1∑i=21iγ≤∫ ∞1dxxγ=1γ − 1,Chaotic Modeling and Simulation (CMSIM) 3: 343–353, 2018 351if γ > 1. Collecting all the inequalities ((23), (24), (25)):|Em,γ(f)(x)| ≤ Hm,γK2q[( π4m)q+( π2m)q+2γ( π4m)q( 1γ − (q + 1)+1γ − 1)].Finally,|Em,γ(f)(x)| ≤K2(π2)γ(π2m)q(1 + 2q + 2γ( 1γ − (q + 1)+1γ − 1)),if γ > q + 1. 4 Fractal version of discrete Jackson approximantsSome easy and elegant observations on fractal functions in conjunction withthe theorems from section 3, provide their corresponding fractal versions. Suchkind of fractal approximants possess many desirable properties as their tradi-tional counterparts and also reassures the ubiquity of fractal functions.As described in section 2, to get the fractal version we have to perturb thebasis function Pm,i,γ(x) using suitable base functions bi,m,γ , suitable scalingvector αm and partition of [−π, π] and define the fractal discrete Jackson ap-proximants asDαm,γ(f)(x) = Hm,γ2m∑i=1f(xi)Pαm,i,γ(x).The following generalizes the Theorem 6.Theorem 7. Let f ∈ C[−π, π] be Hölder continuous such that forx, x′ ∈ [−π, π],|f(x)− f(x′)| ≤ K|x− x′|q, 0 < q ≤ 1.Then for γ > q + 1,‖Dαm,i,γ(f)− f‖∞ ≤ K2 (π2 )γ( π2m )q(1 + 2γ(ζ(γ − q) + ζ(γ)))+m(π2 )γ‖f‖∞ |α|∞1−|α|∞ max1≤i≤2m ‖Pαm,i,γ − bm,i,γ‖∞,where α, bm,i,γ are the suitable scaling vector and basis function used to con-struct the fractal perturbation of Pm,i,γ .Proof. From the triangle inequality we have:‖Dαm,γ(f)− f‖∞ ≤ ‖Dαm,γ(f)−Dm,γ(f)‖∞ + ‖Dm,γ(f)− f‖∞.Under the stated hypothesis from Theorem 5 we have:‖Dm,γ(f)− f‖∞ ≤K2(π2)γ(π2m)q(1 + 2γ(ζ(γ − q) + ζ(γ))). (26)352 Navascués et al.−3 −2 −1 0 1 2 3−0.2−0.15−0.1−0.0500.050.10.15−3 −2 −1 0 1 2 3−0.2−0.15−0.1−0.0500.050.10.15Fig. 3. Approximant Dαm,γ(f) of a sampled signal for m = 12, γ = 5 and in theinterval [−π, π] for different values of α.Now|Dαm,γ(f)(x)−Dm,γ(f)(x)| = |Hm,γ2m∑i=1f(xi)(Pαm,i,γ)(x)− Pm,i,γ(x)|≤ Hm,γ2m∑i=1|f(xi)||(Pαm,i,γ)(x)− Pm,i,γ(x)|≤ Hm,γ2m‖f‖∞2m∑i=1|(Pαm,i,γ)(x)− Pm,i,γ(x)|≤ (π2)γm‖f‖∞ max1≤i≤2m‖Pαm,i,γ − Pm,i,γ‖∞.(27)Finally, using (4), (26) and (27)the proof follows. The next result is the refined version of the above theorem.Theorem 8. Let f ∈ C[−π, π] be Hölder continuous such that forx, x′ ∈ [−π, π],|f(x)− f(x′)| ≤ K|x− x′|q, 0 < q ≤ 1.Then for γ > q + 1,‖Dm,γ(f)− f‖∞ ≤ K2 (π2 )γ( π2m )q(1 + 2q + 2γ(1γ−(q+1) +1γ−1))+m(π2 )γ‖f‖∞ |α|∞1−|α|∞ max1≤i≤2m ‖Pαm,i,γ − bm,i,γ‖∞,Chaotic Modeling and Simulation (CMSIM) 3: 343–353, 2018 353where α, bm,i,γ are the suitable scaling vector and basis function used to con-struct the fractal perturbation of Pm,i,γ .Proof. The result follows from Theorem 6 and the triangle inequality‖Dαm,γ(f)− f‖∞ ≤ ‖Dαm,γ(f)−Dm,γ(f)‖∞ + ‖Dm,γ(f)− f‖∞.Corollary 1. If f ∈ C[−π, π] is Hölder continuous with exponent q (0 < q ≤ 1)and γ > q + 1 and if we select scaling factors α as |αi| ≤ 1mq+1 , for i =1, 2, . . . N − 1 where N is the number of partition of [−π, π], then Dαm,γ(f)converges uniformly to f as m tends to infinity. The order of convergenceO(m−q) does not depend on γ.References1. N. I. Achieser. Theory of Approximation, Dover Publ., New York, 1992.2. M.F. Barnsley. Fractal functions and interpolation, Constr. Approx., 2, 303-329,1986.3. M. F. Barnsley. Fractals Everywhere, Academic Press Inc., Orlando, Florida, 1988.4. M. F. Barnsley and A. N. Harrington. The calculus of fractal interpolation func-tions, J. Approx. Theory, 57(1), 14-34, 1989.5. A. K. B. Chand, and G. P. Kapoor. Generalized cubic spline fractal interpolationfunctions, SIAM J. Numer. Anal., 44(2), 655-676, 2006.6. E. W. Cheney. Approximation Theory, AMS Chelsea Publ., Providence 2000.7. P. J. Davis. Interpolation and Approximation, Dover Publ., New York (2nd. ed.)1976.8. D. Jackson. On approximation by trigonometric sums and polynomials. Trans. Am.Math. Soc., 13, 491-515, 1912.9. G. G. Lorentz. Approximation of Functions, AMS Chelsea Publ., Providence(Rhode Island), 1986.10. M. A. Navascués. Fractal polynomial interpolation, Z. Anal. Anwend., 25(2),401-418, 2005.11. M. A. Navascués and A. K. B. Chand, Fundamental sets of fractal functions, ActaAppl. 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Fractal approximants on the circle

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