In this paper, we study wavelet approximation of the Chebyshev polyno-mials of the first, second, third, and fourth kinds. We estimate the wavelet approximation of a function f having bounded first derivatives

H. Mazaheri

S. M. Jesmani

S. shojaeain

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Iranian Journal of Numerical Analysis and OptimizationVol. 14, No. 1, 2024, pp 315–329https://doi.org/10.22067/ijnao.2023.84431.1315https://ijnao.um.ac.ir/Research ArticleWavelet approximation with ChebyshevwaveletsS. M. Jesmani* , H. Mazaheri* and S. Shojaeian,AbstractIn this paper, we study wavelet approximation of the Chebyshev polyno-mials of the first, second, third, and fourth kinds. We estimate the waveletapproximation of a function f having bounded first derivatives.AMS subject classifications (2020): 41A65, 41A52, 46N10.*Corresponding author*Corresponding authorReceived 13 August 2023; revised 7 December 2023; accepted 10 December 2023S. M. JesmaniDepartment of Materials and Metallurgical Engineering, Technical and Vocational Uni-versity(TVU), Tehran, Iran. e-mail: smjesmani@tvu.ac.irH. MazaheriDepartment of Mathematics, Yazd University, 89195-741, Yazd, Iran. e-mail: hmaza-heri@yazd.ac.irSaman ShojaeianDepartment of Mathematics, Yazd University, 89195-741, Yazd, Iran. e-mail: saman-shojaeian1989@yahoo.comHow to cite this articleJesmani, S. M., Mazaheri, H. and Shojaeian, S., Wavelet approximation withChebyshev wavelets. Iran. J. Numer. Anal. Optim., 2024; 14(1): 315-329.https://doi.org/10.22067/ijnao.2023.84431.1315315Jesmani, Mazaheri and Shojaeian 316Keywords: Wavelet approximation; Chebyshev polynomials; Shifted Cheby-shev; Continuous functions.1 IntroductionIn recent years, wavelets have found their way into many different fieldsof science and engineering, particularly, in signal analysis, time-frequencyanalysis, and fast algorithms. Wavelets allow an accurate representation ofseveral functions. The wavelet approximation technique is a new tool forfinding and analyzing unexpected seismic signal processing changes. Thereis scarcely any area of numerical analysis where Chebyshev polynomials donot drop in like surprise visitors, and indeed there are now a number ofsubjects in which these polynomials take a significant position in moderndevelopments including orthogonal polynomials, polynomial approximation,numerical integration, and spectral methods for partial differential equations(see [4, 6]).In Table 1, we have Chebyshev polynomials on the trivial [−1, 1].Table 1: Chebyshev polynomialsChebyshev Polynomials Shifted Chebyshev w (weight) w∗ (shifted weight)Tn(x) = cos n θ T ∗n(x) = Tn(2x− 1) 1√1−x21√x−x2Un(x) =sin (n+1) θsin θ U∗n(x) = Un(2x− 1)√1− x2√x− x2Vn(x) =cos(n+ 12 )θcos( θ2 )V ∗n (x) = Vn(2x− 1)√1+x1−x√x1−xWn(x) =sin(n+ 12 )θsin( θ2 )W ∗n(x) =Wn(2x− 1)√1−x1+x√1−xxWith above definition, we obtain the fundamental recurrence relation:Wn+1(x) = 2xWn(x)−Wn−1(x), n ≥ 2, W0(x) = 1, W1(x) = 2x+ 1.In Table 2, we have Chebyshev wavelets, for k ∈ N, m ≥ 0, and n =0, 1, 2, . . . , 2k.Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329317 Wavelet approximation with Chebyshev waveletsTable 2: Chebyshev waveletsSequences Chebyshev Polynomials Wavelet ChebyshevTn,m 2k2 Tm(2kt− n), t ∈ [ n2k, n+12k],0 otherwise.Un,m 2k2Um(2kt− n), t ∈ [ n2k, n+12k],0 otherwise.Vn,m 2k2 Vm(2kt− n), t ∈ [ n2k, n+12k],0 otherwise.Wn,m 2k2Wm(2kt− n), t ∈ [ n2k, n+12k],0 otherwise.In the following, we suppose ψn,m = Tn,m, Un,m, Vn,m,Wn,m. A functionf ∈ L2[0, 1) is expanded by wavelet series asf(t) =∞∑n=0∞∑m=0cn,mψn,m(t),wherecn,m =∫ 10f(t)ψn,m(t)ω∗n,m(t)dt,and ωn,m is the weight function of pseudo Chebyshevs. Also,∫ 10ψn(x)ψm(x)ω∗n,m(t)dx = L,whereL =π2 , m = n, ψ = T, U,0, m ̸= n, ψ = T,W,U, V,π, m = n, ψ = V,W,ωn,m(t) =1√1−t2 if ψ = T,√1− t2 if ψ = U,√1+t1−t if ψ = V,√1−t1+t if ψ =W.Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329Jesmani, Mazaheri and Shojaeian 318Definition 1 (Multiresolution analysis). A sequence of closed subspaces {Vj}of L2(R) (inner product space), j ∈ Z, is called a multiresolution in L2(R) ifit satisfies the following conditions:(i) Vj ⊂ Vj+1;(ii) f(x) ∈ Vj ⇐⇒ f(2x) ∈ Vj+1;(iii) f(x) ∈ V0 ⇐⇒ (x+ 1) ∈ V0;(iv) ∪∞−∞Vj is dense in L2((R), and ∩∞−∞Vj = {0};(v) There exists a function ϕ ∈ V0 such that the collection {ϕ(x–k) : k ∈ Z}is a Riesz basis of V0.Definition 2. Let Pn(f) be the orthogonal projection of L2([0, 1]) onto Vn.ThenPnf =∞∑−∞an,kϕn,k, n = 1, 2, 3, . . . ,wherean,k = ⟨f, ϕn,k⟩.ThusPn(f) =∞∑−∞⟨f, ϕn,k⟩ϕn,k, n = 1, 2, 3, . . . ,Definition 3. The wavelet approximation of Chebyshev polynomial is de-fined byEn(f) = ∥f–Pn(f)∥2 =∫ 1a(f(t)–Pn(f)(t))2dt = O(ϕ(n)),wheref(t) =∞∑n=0∞∑m=0cn,mVn,m(t)orf(t) =∞∑n=0∞∑m=0cn,mWn,m(t).If En(f) → 0 as n → ∞, then En(f) is called the best approximation of f(see [1, 2, 3]).We conclude this section by a list of known lemmas.Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329319 Wavelet approximation with Chebyshev waveletsLemma 1. [5] Let f ∈ L2([0, 1]) be a continuous function such that forconstant P > 0, |f”(x)| ≤ P . If f(t) =∑∞n=0∑∞m=0 cn,mVn,m(t) is expandedin terms of Chebyshev polynomial of the third kind. Then the Chebyshevwavelet approximation E2k,l(t) of f by (2k, l)th partial sums2k−2,M (t) =2k−2∑n=1M−1∑m=0cn,mψn,m(t), (1)of its Chebyshev wavelet series in L2w∗[0, 1] is given by∥f − s2k−2,M∥22 = O(122k(M − 1) 32). M ≥ 1 (M is resolution)Lemma 2. [5] Let f ∈ L2([0, 1]) be a continuous function such that for con-stant P > 0, |f”(x)| ≤ P . If f(t) =∑∞n=0∑∞m=0 cn,mWn,m(t) is expanded interms of the Chebyshev polynomial of the fourth kind. Then the Chebyshevwavelet approximation E2k,l(t) of f by (2k, l)th partial sums2k−2,M (t) =2k−2∑n=1M−1∑m=0cn,mψn,m(t), (2)of its Chebyshev wavelet series in L2w∗[0, 1] is given byE22k−1,l = O(1M !2M(k+1)).2 Main resultsIn this section, we consider wavelet approximation in Chebyshev polynomialsof first, second, third, and fourth kinds and Haar wavelet. We estimate thewavelet approximation of a function f having bounded first derivatives.Theorem 1. Let f ∈ L2([0, 1]) be a continuous function and let for constantP > 0, |f ′(x)| ≤ P .(i) If f(t) =∑∞n=0∑∞m=0 cn,mTn,m(t) is expanded in terms of Chebyshevpolynomial of the first kind. Then the Chebyshev wavelet approxima-tion E2k,l(t) of f by (2k, l)th partial sum isIran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329Jesmani, Mazaheri and Shojaeian 320s2k−1,l =∞∑n=2k−1+1∞∑m=lcn,mTn,m,andE2k,l(f) = ∥f − s2k,l∥22 = O(P(l + 1)32).(ii) If f(t) =∑∞n=0∑∞m=0 cn,mUn,m(t) is expanded in terms of Chebyshevpolynomial of the second kind. Then the Chebyshev wavelet approxi-mation E2k–1,l(t) of f by (2k–1, l)th partial sum iss2k−1,l =∞∑n=2k−1+1∞∑m=lcn,mUn,m,andE2k−1,l(f) = ∥f − s2k−1,l∥2 = O(P122k4 − 1(2l + 4(l + 1)(l + 3))12 ).(iii) If f(t) =∑∞n=0∑∞m=0 cn,mVn,m(t) is expanded in terms of Chebyshevpolynomial of the third kind. Then the Chebyshev wavelet approxima-tion E2k,l(t) of f by (2k, l) th partial sum iss2k−1,l =∞∑n=2k−1+1∑m=0cn,mVn,m,andE22k,l(f) = ∥f−s2k,l∥22 = O(2− k2−1(ln l(l + 1)(l + 3)− 1l + 1−2781l + 2−1l)).(iv) If f(t) =∑∞n=0∑∞m=0 cn,mWn,m(t) is expanded in terms of Chebyshevpolynomial of the fourth kind. Then the Chebyshev wavelet approxi-mation E2k,l(t) of f by (2k, l)th partial sum iss2k−1,l =∞∑n=2k−1+1∑m=0cn,mWn,m,andE22k−1,l = O(2− 3k2 −3(14(l + 1)+1l+1l + 3+53ln ll + 3)).Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329321 Wavelet approximation with Chebyshev waveletsProof. (i) We havecn,m = < f(t), Tn,m >w1= 2k2∫ 10f(t)Tm(2kt− n)w1(ktk − n)dt= 2k2∫ π0f(cos θ + n2k)Tm(cos θ)1√1− cos2 θsin θdθ= 2k2∫ π0f(cos θ + n2k) cos(mθ)dθ= 2k2 [(f(cos θ + n2k)(− 1msinmθ))|π0+12km∫ π0f ′(cos θ + n2k) sin θ sin(mθ)dθ]= 2k2 [12km∫ π0f ′(cos θ + n2k) sin θ sin(mθ)dθ]≤ 2 k2 [ P2km∫ π0sin θ sin(mθ)dθ]≤ 2 k2 (− P2km)[− 1msin θ cos(mθ)|π0 −1m∫ π0cos(mθ) cos(θ)dθ]≤ P2k2m2(∫ π0cos(mθ) cos(θ)dθ).Therefore|cn,m| =P2k2m2|∫ π0cos(mθ) cos(θ)dθ|≤ P2k2m2∫ π0| cos(mθ)|| cos(θ)|dθ≤ Pπ2k2m2|cn,m|2 ≤P 2π22km4.For Tn,m∥Tn,m∥22 = 2k∫ 10|Tm(2kt− n)|2w1(2kt− n)dt≤ 2k∫ 1−1Tm(2kt− n)2w1(2kt− n)dt=π22k= π2k−1,Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329Jesmani, Mazaheri and Shojaeian 322andE22k−1,l = ∥f − s2k−1,l∥22= ∥∞∑n=0∞∑m=0cn,mTn,m −∞∑n=2k−1+1l∑m=0cn,mTn,m∥22= ∥2k−1∑n=0∞∑m=l+1cn,mTn,m∥22=2k−1∑n=0∞∑m=l+1|cn,m|2∥Tn,m∥22≤ π22k−1∑n=0∞∑m=l+1P 2π2m4=P 2π322k−1∑n=0∞∑m=l+11m4.It is follows thatE2k−1,l = O(P(l + 1)32).(ii) We have∥Un,m∥22 = 2k∫ 1a|Um(2kt− n)|2w2(2kt− n)dt= 2k∫ 1aUm(2kt− n)2w2(2kt− n)dt=π22k= π2k−1.Also,cn,m = < f(t), Un,m >w= 2k2∫ 10f(t)Um(2kt− n)w1(2kt− n)dt= 2k2∫ π0f(cos θ + n2k)Um(cos θ)sin θ sin θdθ= 2k2∫ π0f(cos θ + n2k)sin(m+ 1)θsin θ sin2θdθIran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329323 Wavelet approximation with Chebyshev wavelets= 2k2∫ π0f(cos θ + n2k) sin(m+ 1)θ sin θdθ= 2k2∫ π0f(cos θ + n2k)12(cos(mθ)− cos(m+ 2)θ)dθ= 2k2−1∫ π0f(cos θ + n2k)(cos(mθ)− cos(m+ 2)θ)dθ= 2k2−1[f(cos θ + n2k)(1msin(mθ)− 1m+ 2sin(m+ 2)]π0+12k2+1∫ π0f ′(cos θ + n2k) sin θ( 1msin(mθ)− 1m+ 2sin(m+ 2))dθ≤ P2k2+1∫ π0sin θ( 1msin(mθ)− 1m+ 2sin(m+ 2))dθ=P2k2+1[sin θ( 1m2cosmθ + 1(m+ 2)2cos(m+ 2)θ]π0+P2k2+1∫ π0cos θ( 1m2cosmθ + 1(m+ 2)2cos(m+ 2)θ)dθ.Therefore|cn,m| ≤P2k2+1|∫ π0cos θ( 1m2cosmθ + 1(m+ 2)2cos(m+ 2)θ)dθ|≤ P2k2+1∫ π0| cos θ|( 1m2| cosmθ|+ 1(m+ 2)2| cos(m+ 2)|(θ)dθ≤ P2k2+1∫ π0(1m2+1(m+ 2)2)dθandE22k−1,l = ∥f − s2k−1,l∥22= ∥∞∑n=0∞∑m=0cn,mUn,m −∞∑n=2k−1+1l∑m=0cn,mTn,m∥22= ∥2k−1∑n=0∞∑m=l+1cn,mUn,m∥22=2k−1∑n=0∞∑m=l+1|cn,m|2∥Un,m∥22≤ P2k2+12k−1∑n=0∞∑m=l+1(1m2+1(m+ 2)2)Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329Jesmani, Mazaheri and Shojaeian 324≤ P2k2+12k−1∑n=0(2l + 4(l + 1)(l + 3))≤ P2k2+12k−1∑n=0(2l + 4(l + 1)(l + 3)).ThereforeE2k−1,l = ∥f − s2k−1,l∥2 = O(P122k4 − 1(2l + 4(l + 1)(l + 3))12 ).(iii) We havecn,m = < f(t), Vn,m >w= 2k2∫ 10f(t)Vm(2kt− n)w1(2kt− n)dt= −2 k2∫ π0f(cos θ + n2k)cos(m+ 12 )θcos( θ2 )√1 + cos θ1− cos θ sin θdθ= −2 k2∫ π0f(cos θ + n2k)cos(m+ 12 )θcos( θ2 )√2 cos2( θ2 )2 sin2( θ2 )2 sin θ2cos θ2dθ= −2 k2∫ π0f(cos θ + n2k)2 cos(m+ 12)θ cos θ2dθ= −2 k2∫ π0f(cos θ + n2k)(cos(m+ 1)θ + cosmθ)dθ= −2 k2 [f(cos θ + n2k)(1m+ 1sin(m+ 1)θ + 1msinmθ)]π0+2k2∫ π0f ′(cos θ + n2k)sin θ2k(1m+ 1sin(m+ 1)θ + 1msinmθ)dθ= 2k2∫ π0f ′(cos θ + n2k)sin θ2k(1m+ 1sin(m+ 1)θ + 1msinmθ)dθ≤ 2− k2 P∫ π0sin θ( 1m+ 1sin(m+ 1)θ + 1msinmθ))dθ= 2−k2 P [(1m+ 1sin θ sin(m+ 1)θ + 1msin θ sinmθ)]π0−2− k2−1P∫ π0cos θ( 1m+ 1sin(m+ 1)θ + 1m(sinmθ)dθ= 2−k2−1P∫ π0(1m+ 1cos θ sin(m+ 1)θ − 1mcos θ sinmθ)= 2−k2−1P∫ π0(12(m+ 1)(sinmθ + sin(m+ 2)θ)Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329325 Wavelet approximation with Chebyshev wavelets− 12m(sin(m− 1)θ + sin(m+ 1)θ))dθ= 2−k2−1P [− 12m(m+ 1)cosmθ − 1(m+ 1)(m+ 2)cos(m+ 2)θ+12m(m− 1)cos(m− 1)θ + 1m(m+ 1)cos(m+ 1)θ]π0= 2−k2−1P [− 12m(m+ 1)(cosmπ − 1)− 1(m+ 1)(m+ 2)(cos(m+ 2)π − 1)+12m(m− 1)(cos(m− 1)π − 1) + 1m(m+ 1)(cos(m+ 1)π − 1)],and|cn,m| = 2−k2−1P | − 12m(m+ 1)cos(mπ − 1)− 1(m+ 1)(m+ 2)(cos(m+ 2)π − 1)+12m(m− 1)(cos(m− 1)π − 1) + 1m(m+ 1)(cos(m+ 1)π − 1)|≤ 2− k2−1P ( 1m(m+ 1)+2(m+ 1)(m+ 2)+1m(m− 1)+2m(m+ 1))|cn,m|2 = 2−k−2P 2(1(m2(m+ 1)2+4(m+ 1)2(m+ 2)2+1m2(m− 1)2+4m2(m+ 1)2+2m(m+ 2)(m+ 1)2+1m2(m+ 1)(m− 1)+2m2(m+ 1)2+2m(m+ 1)(m− 1)(m+ 2)+4m(m+ 2)(m+ 1)2+2m2(m− 1)(m+ 1))= 2−k−2P 2(4(m+ 1)2(m+ 2)2+1m2(m− 1)2+6m2(m+ 1)2+6m(m+ 2)(m+ 1)2+3m2(m+ 1)(m− 1)+2m(m+ 1)(m− 1)(m+ 2))= −10 1m+ 41m2+ 101m+ 1+ 81(m+ 1)2− 161m− 1+1(m− 1)2− 2321m+ 2.For Vn,m, we haveIran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329Jesmani, Mazaheri and Shojaeian 326∥Vn,m∥22 = 2−k2∫ 10|Vm(2kt− n)|2w(t)dt≤ 2− k2∫ 1−1Vm(2kt− n)2√1 + t1− tdt= 2−k2π2= 2−k2−1π.ThereforeE22k−1,l = ∥f − s2k−1,l∥22= ∥∞∑n=0∞∑m=0cn,mVn,m −∞∑n=2k−1+1l∑m=0cn,mVn,m∥22= ∥2k−1∑n=0∞∑m=l+1cn,mVn,m∥22=2k−1∑n=0∞∑m=l+1|cn,m|2∥Vn,m∥22≤ 2− k2−1π∞∑m=l+1(−10 1m+ 41m2+ 101m+ 1+81(m+ 1)2− 1m− 1+1(m− 1)2− 10 1m+ 2.= 2−k2−1π(−10 ln(l + 1)− 12(l + 1)+ 10 ln(l + 2)−2781l + 2+ ln l − 1l− ln(l + 3)).= 2−k2−1π(10 ln l(l + 1)(l + 3)− 1l + 1− 2781l + 2− 1l).ThereforeE22k−1,l = O(2− k2−1(ln l(l + 1)(l + 3)− 1l + 1− 2781l + 2− 1l).(iv) We havecn,m = < f(t),Wn,m >w= 2k2∫ 10f(t)Wm(2kt− n)√1− t1 + tdtIran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329327 Wavelet approximation with Chebyshev wavelets= −2 k2∫ π0f(cos θ + n2k)sin(m+ 12 )θsin( θ2 )√1− cos θ1 + cos θ sin θdθ= −2 k2∫ π0f(cos θ + n2k)sin(m+ 12 )θsin( θ2 )√2 sin2( θ2 )2 cos2( θ2 )2 sin θ2cos θ2dθ= −2 k2∫ π0f(cos θ + n2k)2 sin(m+ 12)θ sin θ2dθ= −2 k2∫ π0f(cos θ + n2k)(− cos(m+ 1)θ + cosmθ)dθ= −2 k2 [f(cos θ + n2k)(− 1m+ 1sin(m+ 1)θ + 1msinmθ)]π0+2k2∫ π0f ′(cos θ + n2k)sin θ2k(− 1m+ 1sin(m+ 1)θ + 1msinmθ)dθ= 2k2∫ π0f ′(cos θ + n2k)sin θ2k(− 1m+ 1sin(m+ 1)θ + 1msinmθ)dθ≤ 2− k2 P∫ π0sin θ( 1m+ 1sin(m+ 1)θ + 1msinmθ))dθ= 2−k2 P [(1m+ 1sin θ sin(m+ 1)θ + 1msin θ sinmθ)]π0−2− k2−1P∫ π0cos θ( 1m+ 1sin(m+ 1)θ + 1m(sinmθ)dθ= 2−k2−1P∫ π0(1m+ 1cos θ sin(m+ 1)θ − 1mcos θ sinmθ)= 2−k2−1P∫ π0(12(m+ 1)(sinmθ + sin(m+ 2)θ)− 12m(sin(m− 1)θ + sin(m+ 1)θ))dθ= 2−k2−1P [− 12m(m+ 1)cosmθ − 1(m+ 1)(m+ 2)cos(m+ 2)θ+12m(m− 1)cos(m− 1)θ + 1m(m+ 1)cos(m+ 1)θ]π0= 2−k2−1P [− 12m(m+ 1)cosmπ − 1(m+ 1)(m+ 2)cos(m+ 2)π+12m(m− 1)cos(m− 1)π + 1m(m+ 1)cos(m+ 1)π].Therefore|cn,m| = 2−k2−1P | − 12m(m+ 1)cosmπ − 1(m+ 1)(m+ 2)cos(m+ 2)π+12m(m− 1)cos(m− 1)π + 1m(m+ 1)cos(m+ 1)π|Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329Jesmani, Mazaheri and Shojaeian 328≤ 2− k2−1P | 32m(m+ 1)+1(m+ 1)(m+ 2)+12m(m− 1)]≤ 2− k2−1P ( 12m+12(m+ 1)+1m− 1− 1m+ 2)|cn,m|2 = 2−k−2P 2(14m2+14(m+ 1)2+1(m− 1)2+1(m+ 2)2+12m(m+ 1)+1m(m− 1)+1m(m+ 2)+1(m+ 1)(m− 1)+1(m+ 1)(m+ 2)+2(m− 1)(m+ 2))= 2−k−2P 2(14m2+14(m+ 1)2+1(m− 1)2+1(m+ 2)2+531m− 1− 531m+ 2),E22k−1,l = ∥f − s2k−1,l∥22= ∥∞∑n=0∞∑m=0cn,mVn,m −∞∑n=2k−1+1l∑m=0cn,mWn,m∥22= ∥2k−1∑n=0∞∑m=l+1cn,mWn,m∥22=2k−1∑n=0∞∑m=l+1|cn,m|2∥Wn,m∥22≤ 2− 3k2 −3πP 2∞∑m=l+1(14m2+14(m+ 1)2+1(m− 1)2+1(m+ 2)2+531m− 1− 531m+ 2)= 2−3k2 −3πP 2(14(l + 1)+1l+1l + 3+53ln ll + 3).ThereforeE22k−1,l = O(2− 3k2 −3(14(l + 1)+1l+1l + 3+53ln ll + 3)).Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329329 Wavelet approximation with Chebyshev waveletsReferences[1] Çakmak, M. and Uslu, K.A. generalization of Chebyshev polynomialswith well-known kinds and transition relations, Acta Univ. ApulensisMath. Inform. 57 (2019), 19–30.[2] Dehghan, M. and Eslahchi, M.R. Best uniform polynomial approxima-tion of some rational functions, Comput. Math. Appl. 59(1) (2010),382–390.[3] Eslahchi, M.R. and Dehghan, M. The best uniform polynomial approxi-mation to the class of the form 1(a2±x2) , Nonlinear Anal. 71(3-4) (2009),740–750.[4] Mason, J.C. and Handscomb, D.C. Chebyshev polynomials, Chapmanand Hall/CRC, Boca Raton, FL, 2003.[5] Nigam, H.K., Mohapatra, R.N. and Murari, K. Wavelet approximationof a function using Chebyshev wavelets, Thai J. Math. (2020), 197–208.[6] Rivlin, T.J. An introduction to the approximation of functions, Correctedreprint of the original. Dover Books on Advanced Mathematics. DoverPublications, Inc., New York, 1981.Iran. J. Numer. Anal. Optim., Vol. 14, No. 1, 2024, pp 315–329

Wavelet approximation with Chebyshev wavelets

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Wavelet approximation with Chebyshev wavelets

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