Ratio Mathematica Volume 37,2019, pp. 85-109 Legendre Wavelet expansion of functions and their Approximations Shyam Lal∗ Indra Bhan † Abstract In this paper, nine new Legendre wavelet estimators of functions having bounded third and fourth derivatives have been obtained.These estimators are new and best approximation in wavelet analysis. Legendre wavelet estimator of a function f of bounded higher order derivatives is better and sharper than the estimator of a function f of bounded less order derivative. Keywords : Legendre Wavelet, Legendre Wavelet Expansion, Orthonormal basis,Legendre Wavelet Approximation . Mathematics Subject Classification:42C40, 65T60, 65L10, 65L60, 65R20. 1 1 Introduction Several researchers have determined the approximation of a functions by trigonometric polynomials in Fourier analysis. In Fourier analysis, a function can be represented generally in one Fourier series. In wavelet analysis, a function can be expanded in many wavelet series corresponding to different wavelets. This is an advantage of wavelet analysis. There is no such advantage in Fourier analysis.Thus a signal can be represented by several wavelet series. Hence Wavelet Analysis is superior to Fourier analysis and has so many applications in Engineering and Technology. The Wavelet approximation of a functions by its Haar wavelet series and related approximations have been studied by Devore[7], Debnath[5], Meyer[9] , Morlet[3], Mhaskar[2], Sablonnière[6] and Lal & Kumar[8]. The purpose of this paper is to discuss the Legendre wavelet series of function having bounded third and fourth derivatives, i.e. 0 ≤ |f ′′′ (x)| < ∞ ∀x ∈ [0, 1] and 0 ≤ |fiv(x)| < ∞ ∀x ∈ [0, 1] and to obtain Legendre wavelet estimators of these functions. This is a significant observation of this research paper that estimate of a function is better and the sharper than the estimate having less order bounded derivative.Therefore comparison of estimated approximations has very importance in Wavelet analysis. ∗Shyam Lal, Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi- 221005, India;shyam lal@rediffmail.com †Indra Bhan, Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi- 221005, India;indrabhanmsc@gmail.com 1Received on September 21st, 2019. Accepted on December 20rd, 2019. Published on December 31st, 2019. doi:10.23755/rm.v37i0.491. ISSN: 1592-7415. eISSN: 2282-8214. c©Shyam Lal and Indra Bhan 85 Shyam Lal and Indra Bhan 2 Definitions and Preliminaries 2.1 Legendre Wavelet Wavelets constitute a family of functions constructed from dilation and translation of a single function ψ ∈ L2(R) , called mother wavelet. We write ψb,a(x) = |a| −1 2 ψ ( x− b a ) , a 6= 0. If we restrict the values of dilation and translation parameter to a = a−n0 , b = mb0a0 −n,a0 > 1,b0 > 0 respectively, the following family of discrete wavelets are constructed: ψn,m(x) = |a0| n 2 ψ(an0x−mb0) The Legendre wavelet over the interval [0,1) is defined as ψn,m(x) = { √ m + 1 2 2 k 2 Pm(2 kx− n̂), n̂− 1 2k ≤ x < n̂ + 1 2k 0 , otherwise, where n = 1, 2, ..., 2k−1 and m = 0, 1, 2, 3, ..., n̂ = 2n − 1 and k is the positive integer. In this definition,the polynomials Pm are Legendre Polynomials of degree m over the interval [-1,1] defined as follows: P0(x) = 1,P1(x) = x (m + 1)Pm+1(x) = (2m + 1)xPm(x) −mPm−1(x) , m = 1, 2, 3, ... The set of {Pm(x) : m = 1, 2, 3, ...} in the Hilbert space L2[−1, 1] is a complete orthogonal set. Orthogonality of Legendre polynomial on the interval [-1,1] implies that 〈Pm,Pn〉 = ∫ 1 −1 Pm(x)Pn(x)dx = { 2 2m + 1 ,m = n 0 , otherwise. for m,n = 0, 1, 2, 3... Furthermore, the set of wavelets ψn,m makes an orthonormal basis in L2[0, 1),i.e.∫ 1 0 ψn,m(x)ψn′m′ (x)dx = δn,n′δm,m′ in which δ denotes Kronecker delta function defined by δn,m = { 1, n=m 0, otherwise. 86 Legendre Wavelet expansion of functions and their Approximations The function f(x) ∈ L2[0, 1) is expressed in the Legendre wavelet series as : f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψn,m(x) where cn,m = 〈f,ψn,m〉. The (2k−1,M)th partial sums of above series are given by S2k−1,M (f)(x) = 2k−1∑ n=1 M∑ m=0 cn,mψn,m(x) = C Tψ(x) in which C and ψ(x) are 2k−1(M + 1) vectors of the form CT = [c1,0,c1,1, ...c1,M,c2,0,c2,1, ...c2,M, ...,c2k−1,0, ...c2k−1,M ] and ψ(x) = [ψ1,0,ψ1,1, ...ψ1,M,ψ2,0,ψ2,1, ...ψ2,M, ...,ψ2k−1,0, ...ψ2k−1,M ] T 2.2 Legendre Wavelet Approximation Let S2k−1,M (f)(x) denote the (2 k−1,M)th partial sums of the series ∞∑ n=1 ∞∑ m=0 cn,mψn,m(x) i.e. S2k−1,M (f)(x) = 2k−1∑ n=1 M∑ m=0 cn,mψn,m(x) The Legendre wavelet approximation E2k−1,M (f) of a function f ∈ L2[0, 1) by (2k−1,M)th partial sums S2k−1,M (f) of its Legendre Wavelet series is given by E2k−1,M (f) = min‖f −S2k−1,M (f)‖2, (Zygmund[1],pp.115) where ‖f‖2 = (∫ 1 0 |f(x)|2dx )1 2 . If E2k−1,M (f) → 0 as k → ∞, M → ∞. then E2k−1,M (f) is called the best approximation of f of order (2k−1,M) (Zygmund[1],pp.115) 3 Example Express the following function in the Legendre wavelet series : f(t) = t3 ∀t ∈ [0, 1) 87 Shyam Lal and Indra Bhan Proof: f(t) = 2k−1∑ n=1 ∞∑ m=0 cn,mψn,m(t) cn,m = n̂+1 2k∫ n̂−1 2k f(t)ψn,m(t)dt = n̂+1 2k∫ n̂−1 2k t3 ( 2m + 1 2 )1 2 2 k 2 Pm(2 kt− n̂)dt = ( 2m + 1 2 )1 2 2 k 2 1∫ −1 ( v + n̂ 2k )3 Pm(v) dv 2k , v = 2kt− n̂ cn,m = ( 2m + 1 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)Pm(v)dv By above expression cn,0 = ( 1 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)P0(v)dv = ( 1 27k+1 )1 2 (2n̂3 + 2n̂) cn,1 = ( √ 3 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)P1(v)dv = ( √ 3 27k+1 )1 2 ( 2 5 + 2n̂2 ) cn,2 = ( √ 5 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)P2(v)dv = ( √ 5 27k+1 )1 2 ( 4n̂ 5 ) cn,3 = ( √ 7 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)P3(v)dv 88 Legendre Wavelet expansion of functions and their Approximations cn,3 = ( 4 35 )( √ 7 27k+1 )1 2 cn,m = 0, for m ≥ 4 Then, f(t) = 2k−1∑ n=1 cn,0ψn,0(t) + 2k−1∑ n=1 cn,1ψn,1(t) + 2k−1∑ n=1 cn,2ψn,2(t) + 2k−1∑ n=1 cn,3ψn,3(t) Now, ||f||22 = 1 7 = 2k−1∑ n=1 c2n,0||ψn,0|| 2 2 + 2k−1∑ n=1 c2n,1||ψn,1|| 2 2 + 2k−1∑ n=1 c2n,2||ψn,2|| 2 2 + 2k−1∑ n=1 c2n,3||ψn,3|| 2 2 = 2k−1∑ n=1 c2n,0 + 2k−1∑ n=1 c2n,1 + 2k−1∑ n=1 c2n,2 + 2k−1∑ n=1 c2n,3 = 2k−1∑ n=1 [( 1 27k+1 )1 2 (2n̂3 + 2n̂) ]2 + 2k−1∑ n=1  ( √3 27k+1 )1 2 ( 2 5 + 2n̂2 )2 + 2k−1∑ n=1  ( √5 27k+1 )1 2 ( 4n̂ 5 )2 + 2k−1∑ n=1  ( 4 35 )( √ 7 27k+1 )1 2  2 = 1 7 . 4 Theorems In this paper, we prove following new theorems: Theorem (4.1) Let a function f ∈ L2[0, 1) such that its third derivative be bounded ,i.e. 0 ≤ |f ′′′ (x)| < ∞∀ x ∈ [0, 1). Then the Legendre wavelet approximations of f satisfy : (i)E (1) 2k−1,0 (f) = ||f − 2k−1∑ n=1 cn,0ψn,0||2 = O ( 1 2k ) (ii)E (2) 2k−1,1 (f) = ||f − 2k−1∑ n=1 1∑ m=0 cn,mψn,m||2 = O ( 1 22k ) (iii)E (3) 2k−1,2 (f) = ||f − 2k−1∑ n=1 2∑ m=0 cn,mψn,m||2 = O ( 1 23k ) (iv)For f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψn,m, E (4) 2k−1,M (f) = ||f − 2k−1∑ n=1 M∑ m=0 cn,mψn,m||2 89 Shyam Lal and Indra Bhan =  2k−1∑ n=1 ∞∑ m=M+1 c2n,m   1 2 = O ( 1 (2M − 3) 5 2 1 23k ) ,∀M ≥ 2. Theorem (4.2) If a function f ∈ L2[0, 1) having bounded fourth derivative ,i.e. 0 ≤ |fiv(x)| < ∞∀ x ∈ [0, 1). Then its Legendre wavelet approximations are given by (i)E (5) 2k−1,0 (f) = ||f − 2k−1∑ n=1 cn,0ψn,0||2 = O ( 1 2k ) (ii)E (6) 2k−1,1 (f) = ||f − 2k−1∑ n=1 1∑ m=0 cn,mψn,m||2 = O ( 1 22k ) (iii)E (7) 2k−1,2 (f) = ||f − 2k−1∑ n=1 2∑ m=0 cn,mψn,m||2 = O ( 1 23k ) (iv)E (8) 2k−1,3 (f) = ||f − 2k−1∑ n=1 3∑ m=0 cn,mψn,m||2 = O ( 1 24k ) (v)For f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψn,m , E (9) 2k−1,M (f) = ||f − 2k−1∑ n=1 M∑ m=0 cn,mψn,m||2 =  2k−1∑ n=1 ∞∑ m=M+1 c2n,m   1 2 = O ( 1 (2M − 5) 7 2 1 24k ) , ∀ M ≥ 3. 5 Proofs 5.1 Proof of the Theorem (4.1) (i) The error e(0)n (x) between f(x) and its expression over any subinterval is defined as e (0) n (x) = cn,0ψn,0(x) −f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) ,n = 1, 2, 3, ...2k−1 ||e(0)n || 2 2 = n̂+1 2k∫ n̂−1 2k (e(0)n (x)) 2dx = n̂+1 2k∫ n̂−1 2k (c2n,0ψ 2 n,m(x) + (f(x)) 2 − 2cn,0ψn,0(x)f(x))dx 90 Legendre Wavelet expansion of functions and their Approximations = c2n,0 n̂+1 2k∫ n̂−1 2k ψ2n,0(x)dx + n̂+1 2k∫ n̂−1 2k (f(x))2dx− 2cn,0 n̂+1 2k∫ n̂−1 2k f(x)ψn,0(x)dx = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0. (5.1) Now, n̂+1 2k∫ n̂−1 2k (f(x))2dx = 1 2k−1∫ 0 ( f ( n̂− 1 2k + h ))2 dh,x = n̂− 1 2k + h = 1 2k−1∫ 0 [ f ( n̂− 1 2k ) + hf ′ ( n̂− 1 2k ) + h2 2 f ′′ ( n̂− 1 2k ) + h3 6 f ′′′ ( n̂− 1 2k + θh )]2 , 0 < θ < 1 by Taylor′s expansion = 1 2k−1∫ 0 ( f ( n̂− 1 2k ))2 dh + 1 2k−1∫ 0 h2 ( f ′ ( n̂− 1 2k ))2 dh + 1 2k−1∫ 0 h4 4 ( f ′′ ( n̂− 1 2k ))2 dh + 1 2k−1∫ 0 h6 36 ( f ′′′ ( n̂− 1 2k + θh ))2 dh + 1 2k−1∫ 0 2hf ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) dh + 1 2k−1∫ 0 h2f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) dh + 1 2k−1∫ 0 h3 3 f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 1 2k−1∫ 0 h3f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) dh + 1 2k−1∫ 0 h4 3 f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 1 2k−1∫ 0 h5 6 f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh = 2 2k ( f ( n̂− 1 2k ))2 + 8 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 8 5 1 25k ( f ′′ ( n̂− 1 2k ))2 + 1 36 1 2k−1∫ 0 h6 ( f ′′′ ( n̂− 1 2k + θh ))2 dh + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) 91 Shyam Lal and Indra Bhan + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 1 3 1 2k−1∫ 0 h3f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 4 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) dh + 1 3 1 2k−1∫ 0 h4f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 1 6 1 2k−1∫ 0 h5f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh. (5.2) Now, cn,0 = < f(x),ψn,0(x) > = n̂+1 2k∫ n̂−1 2k f(x)ψn,0(x)dx = 2 k−1 2 n̂+1 2k∫ n̂−1 2k f(x)dx = 2 k−1 2 1 2k−1∫ 0 f ( n̂− 1 2k + h ) dh,x = n̂− 1 2k + h = 2 k−1 2 1 2k−1∫ 0 [ f ( n̂− 1 2k ) + hf ′ ( n̂− 1 2k ) + h2 2 f ′′ ( n̂− 1 2k ) + h3 6 f ′′′ ( n̂− 1 2k + θh )] dh = 2 k−1 2   2 2k f ( n̂− 1 2k ) + 2 22k f ′ ( n̂− 1 2k ) + 4 3 1 23k f ′′ ( n̂− 1 2k ) + 1 6 1 2k−1∫ 0 h3f ′′′ ( n̂− 1 2k + θh ) dh   . Next, c2n,0 = 2 2k ( f ( n̂− 1 2k ))2 + 2 23k ( f ′ ( n̂− 1 2k ))2 + 8 9 1 25k ( f ′′ ( n̂− 1 2k ))2 + 2k 2   1 2k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh   2 + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 1 3 1 2k−1∫ 0 h3f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh 92 Legendre Wavelet expansion of functions and their Approximations + 8 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 2 2k f ′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh + 4 3 1 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh. (5.3) Now, by using equations (5.1), (5.2) and (5.3) we have ||e(0)n || 2 2 = 2 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 32 45 1 25k ( f ′′ ( n̂− 1 2k ))2 + 1 36 1 2k−1∫ 0 h6 ( f ′′′ ( n̂− 1 2k + θh ))2 dh − 2k 2   1 2k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh   2 + 4 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 1 3 f ′ ( n̂− 1 2k ) 12k−1∫ 0 h4f ′′′ ( n̂− 1 2k + θh ) dh− 2 2k f ′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh + 1 6 f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h5f ′′′ ( n̂− 1 2k + θh ) dh− 4 3 1 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh = I1 + I2 + I3 − I4 + I5 + I6 − I7 + I8 − I9, say. Since |f ′ (x)| ≤ M1, |f ′′ (x)| ≤ M2, |f ′′′ (x)| ≤ M3,∀x ∈ [0, 1), therefore |I1| ≤ 2 3 1 23k M21 |I2| ≤ 32 45 1 25k M22 |I3| ≤ 32 63 1 27k M23 |I4| ≤ 2 9 1 27k M23 |I5| ≤ 4 3 1 24k M1M2 |I6| ≤ 32 15 1 25k M1M3 |I7| ≤ 4 3 1 25k M1M3 |I8| ≤ 16 9 1 26k M2M3 |I9| ≤ 8 9 1 26k M2M3. 93 Shyam Lal and Indra Bhan Therefore, ||e(0)n || 2 2 ≤ |I1| + |I2| + |I3| + |I4| + |I5| + |I6| + |I7| + |I8| + |I9| ≤ 2 3 1 23k M21 + 32 45 1 25k M22 + 32 63 1 27k M23 + 2 9 1 27k M23 + 4 3 1 24k M1M2 + 32 15 1 25k M1M3 + 4 3 1 25k M1M3 + 16 9 1 26k M2M3 + 8 9 1 26k M2M3 = 2 3 1 23k M21 + 32 45 1 25k M22 + 56 63 1 27k M23 + 4 3 1 24k M1M2 + 52 15 1 25k M1M3 + 24 9 1 26k M2M3 < 2 23k [ M21 + ( M2 2k )2 + ( M3 22k )2 + 2M1M2 2k + 2M1M3 22k + 2M2M3 23k ] = 2 23k ( M1 + M2 2k + M3 22k )2 = 2M2 23k ( 1 + 1 2k + 1 22k )2 ,M = max[M1,M2,M3]. Next, (E (1) 2k−1,0 (f))2 = 1∫ 0  2k−1∑ n=1 e(0)n (x)  2 dx = 1∫ 0 2k−1∑ n=1 (e(0)n (x)) 2dx + 2 2k−1∑ n=1 2k−1∑ n6=n′ 1∫ 0 e(0)n (x)e (0′) n (x)dx = 2k−1∑ n=1 1∫ 0 (en(x)) 2dx, due to disjoint supports of en and e ′ n = 2k−1∑ n=1 ||e(0)n || 2 2 ≤ (2k−1) 2M2 23k ( 1 + 1 2k + 1 22k )2 = M2 22k ( 1 + 1 2k + 1 22k )2 . Then, E (1) 2k−1,0 (f) ≤ M 2k ( 1 + 1 2k + 1 22k ) ≤ M ( 1 2k + 1 2k + 1 2k ) = 3M ( 1 2k ) = O ( 1 2k ) . 94 Legendre Wavelet expansion of functions and their Approximations (ii)e(1)n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) −f(x) , x ∈ [ n̂− 1 2k , n̂ + 1 2k ) ||e(1)n || 2 2 = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0 − c 2 n,1. (5.4) Now, consider cn,1 = < f(x),ψn,1(x) > = n̂+1 2k∫ n̂−1 2k f(x)ψn,1(x)dx = √ 3 2 2 k 2 n̂+1 2k∫ n̂−1 2k f(x)P1(2 kx− n̂)dx = √ 3 2 2 k 2 1 2k−1∫ 0 f ( n̂− 1 2k + h ) P1(2 kh− 1)dh, x = n̂− 1 2k + h = √ 3 2 2 k 2 1 2k−1∫ 0 f ( n̂− 1 2k + h ) (2kh− 1)dh = √ 3 2 2 k 2 1 2k−1∫ 0 f ( n̂− 1 2k ) (2kh− 1)dh + √ 3 2 2 k 2 1 2k−1∫ 0 f ′ ( n̂− 1 2k ) h(2kh− 1)dh + √ 3 2 2 k 2 1 2k−1∫ 0 f ′′ ( n̂− 1 2k ) h2 2 (2kh− 1)dh + √ 3 2 2 k 2 1 2k−1∫ 0 f ′′′ ( n̂− 1 2k + θh ) h3 6 (2kh− 1)dh cn,1 = √ 3 2 2 k 2 [ 2 3 1 22k f ′ ( n̂− 1 2k ) + 2 3 1 23k f ′′ ( n̂− 1 2k )] + √ 3 2 2 k 2 1 2k−1∫ 0 f ′′′ ( n̂− 1 2k + θh ) h3 6 (2kh− 1)dh. Now, c2n,1 = 2 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 2 3 1 25k ( f ′′ ( n̂− 1 2k ))2 + 3 2 2k   1 2k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh   2 + 4 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) 95 Shyam Lal and Indra Bhan + 2 2k f ′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh + 2 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh. (5.5) By using equations (5.2), (5.3), (5.4) and (5.5), we have ||e(1)n || 2 2 = 2 45 1 25k ( f ′′ ( n̂− 1 2k ))2 + 1 36 1 2k−1∫ 0 h6 ( f ′′′ ( n̂− 1 2k + θh ))2 dh − 4 3 1 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k ) dh− 2k 2   1 2k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh   2 + 1 6 f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h5f ′′′ ( n̂− 1 2k + θh ) dh− 3 2 2k   1 2k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh   2 − 2 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh = I1 + I2 + I3 + I4 + I5 + I6 + I7,say. Therefore |I1| ≤ 2 45 1 25k M22 |I2| ≤ 32 63 1 27k M23 |I3| ≤ 8 9 1 26k M2M3 |I4| ≤ 2 9 1 27k M23 |I5| ≤ 16 9 1 26k M2M3 |I6| ≤ 18 75 1 27k M23 |I7| ≤ 12 15 1 26k M2M3 ||e(1)n || 2 2 ≤ |I1| + |I2| + |I3| + |I4| + |I5| + |I6| + |I7| ≤ 2 45 1 25k M22 + 32 63 1 27k M23 + 8 9 1 26k M2M3 + 2 9 1 27k M23 + 16 9 1 26k M2M3 + 18 75 1 27k M23 + 12 15 1 26k M2M3 96 Legendre Wavelet expansion of functions and their Approximations = 2 45 1 25k M22 + 1528 1575 1 27k M23 + 144 45 1 26k M2M3 < 2 25k ( M22 + ( M3 2k )2 + 2M2M3 2k ) = 2 25k M2 ( 1 + 1 2k )2 ,M = max[M2,M3]. Next, (E (2) 2k−1,1 (f))2 = 2k−1∑ n=1 ||e(1)n || 2 2 ≤ (2k−1) 2 25k M2 ( 1 + 1 2k )2 = M2 24k ( 1 + 1 2k )2 . Then, E (2) 2k−1,1 (f) ≤ M 22k ( 1 + 1 2k ) = O ( 1 22k ) . (iii) e(2)n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) + cn,2ψn,2(x) −f(x) , x ∈ [ n̂−1 2k , n̂+1 2k ) Similarly, it can be proved that E (3) 2k−1,2 (f) = O ( 1 23k ) . (iv) 0 ≤ |f ′′′ (x)| < M1 ,∀x ∈ [0, 1) cn,m = 1∫ 0 f(x)ψn,m(x)dx = n̂+1 2k∫ n̂−1 2k f(x) √ 2m + 1 2 2 k 2 Pm(2 kx− n̂)dx = √ 2m + 1 2k+1 1∫ −1 f ( n̂ + t 2k ) Pm(t)dt 97 Shyam Lal and Indra Bhan = √ 2m + 1 2k+1 1∫ −1 f ( n̂ + t 2k ) d(Pm+1(t) −Pm−1(t)) 2m + 1 = ( 1 2k+1(2m + 1) )1 2 ×  {f (n̂ + t 2k ) (Pm+1(t) −Pm−1(t)) }1 −1 − 1∫ −1 1 2k f ′ ( n̂ + t 2k ) (Pm+1(t) −Pm−1(t))dt   = ( 1 23k+1(2m + 1) )1 2   1∫ −1 f ′ ( n̂ + t 2k ) (Pm−1(t) −Pm+1(t))dt   = ( 1 23k+1(2m + 1) )1 2   1∫ −1 f ′ ( n̂ + t 2k ) (Pm−1(t))dt− 1∫ −1 f ′ ( n̂ + t 2k ) (Pm+1(t))dt   = ( 1 23k+1(2m + 1) )1 2 1∫ −1 [ f ′ ( n̂ + t 2k ) d(Pm(t) −Pm−2(t)) (2m− 1) −f ′ ( n̂ + t 2k ) d(Pm+2(t) −Pm(t)) (2m + 3) ] = ( 1 25k+1(2m + 1) )1 2 1∫ −1 [ f ′′ ( n̂ + t 2k ) (Pm+2(t) −Pm(t)) (2m + 3) −f ′′ ( n̂ + t 2k ) d(Pm(t) −Pm−2(t)) (2m− 1) ] = ( 1 25k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(Pm+3(t) −Pm+1(t)) (2m + 5) ] − ( 1 25k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(Pm+1(t) −Pm−1(t)) (2m + 1) ] + ( 1 25k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(Pm−1(t) −Pm−3(t)) (2m− 3) ] − ( 1 25k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(Pm+1(t) −Pm−1(t)) (2m + 1) ] = ( 1 27k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (Pm+1(t) −Pm−1(t)) (2m + 1) − (Pm+3(t) −Pm+1(t)) (2m + 5) ] dt − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (Pm−1(t) −Pm−3(t)) (2m− 3) − (Pm+1(t) −Pm−1(t)) (2m + 1) ] dt 98 Legendre Wavelet expansion of functions and their Approximations = ( 1 27k+1(2m + 1) )1 2 × 1∫ −1 f ′′′ ( n̂ + t 2k )[ 2(2m + 3)Pm+1(t) − (2m + 5)Pm−1(t) − (2m + 1)Pm+3(t) (2m + 1)(2m + 5)(2m + 3) ] dt − ( 1 27k+1(2m + 1) )1 2 × 1∫ −1 f ′′′ ( n̂ + t 2k )[ 2(2m− 1)Pm−1(t) − (2m + 1)Pm−3(t) − (2m− 3)Pm+1(t) (2m + 1)(2m− 1)(2m− 3) ] dt. Let τ1(t) = 2(2m + 3)Pm+1(t) − (2m + 5)Pm−1(t) − (2m + 1)Pm+3(t) τ2(t) = 2(2m− 1)Pm−1(t) − (2m + 1)Pm−3(t) − (2m− 3)Pm+1(t) Then, cn,m = ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3)(2m + 5)   1∫ −1 f ′′′ ( n̂ + t 2k ) τ1(t)dt   − ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m− 1)(2m− 3)   1∫ −1 f ′′′ ( n̂ + t 2k ) τ2(t)dt   |cn,m| ≤ ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3)(2m + 5)   1∫ −1 ∣∣∣∣f′′′ ( n̂ + t 2k )∣∣∣∣ |τ1(t)|dt   + ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m− 1)(2m− 3)   1∫ −1 ∣∣∣∣f′′′ ( n̂ + t 2k )∣∣∣∣ |τ2(t)dt|   ≤ M1 ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3)(2m + 5) 1∫ −1 |τ1(t)|dt + M1 ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m− 1)(2m− 3) 1∫ −1 |τ2(t)|dt. (5.6) Consider, 1∫ −1 |τ1(t)|dt = 1∫ −1 1.|τ1(t)|dt ≤   1∫ −1 12.dt   1 2   1∫ −1 |τ1(t)|2dt   1 2 99 Shyam Lal and Indra Bhan = √ 2   1∫ −1 (2(2m + 3)Pm+1(t) − (2m + 5)Pm−1(t) − (2m + 1)Pm+3(t)) 2 dt   1 2 = √ 2   1∫ −1 [ 4(2m + 3)2P2m+1(t) + (2m + 5) 2P2m−1(t) + (2m + 1) 2P2m+3(t) ] dt   1 2 = √ 2 [ 4(2m + 3)2 2 2m + 3 + (2m + 5)2 2 2m− 1 + (2m + 1)2 2 2m + 7 ]1 2 by orthogonality condition on Pm = 2 [ 4(2m + 3) + (2m + 5)2 2m− 1 + (2m + 1)2 2m + 7 ]1 2 ≤ 2 [ 4(2m + 3)(2m− 1) + (2m + 5)2 + (2m + 1)2 (2m− 1) ]1 2 = 2 [ 24m2 + 40m + 14 2m− 1 ]1 2 = 2 √ 2 [ (2m + 1)(6m + 7) 2m− 1 ]1 2 ≤ 2 √ 6 [ (2m + 1)(2m + 3) (2m− 1) ]1 2 . (5.7) Now , 1∫ −1 |τ2(t)|dt = 1∫ −1 1.|τ2(t)|dt = √ 2   1∫ −1 [2(2m− 1)Pm−1(t) − (2m + 1)Pm−3(t) − (2m− 3)Pm+1(t)]2dt   1 2 = √ 2   1∫ −1 [ (2m− 3)2Pm+1(t) + (2m + 1)2P2m−3(t) + 4(2m− 1) 2P2m−1(t) ] dt   1 2 = √ 2 [ (2m− 3)2 2 (2m + 3) + (2m + 1)2 2 2m− 5 + 4(2m− 1)2 2 2m− 1 ]1 2 by orthogonality condition on Pm = 2 [ (2m− 3)2 (2m + 3) + (2m + 1) (2m− 5) + 4(2m− 1) ]1 2 ≤ 2 [ (2m− 3)2 + (2m + 1)2 + 4(2m− 1)(2m− 5) 2m− 5 ]1 2 100 Legendre Wavelet expansion of functions and their Approximations = 2 [ 24m2 − 56m + 30 2m− 5 ]1 2 = 2 √ 2 [ (2m− 3)(6m− 5) 2m− 5 ]1 2 ≤ 2 √ 6 [ (2m− 3)(2m− 1) (2m− 5) ]1 2 . (5.8) Now , by using equations (5.6), (5.7) and (5.8) we have |Cn,m| ≤ M1 ( 1 27k+1(2m + 1) )1 2 [ 2 √ 6 (2m− 3) 5 2 + 2 √ 6 (2m− 5) 5 2 ] ≤ M1 ( 1 27k+1(2m + 1) )1 2 [ 4 √ 6 (2m− 5) 5 2 ] ≤ 4 √ 6M1 2 7k+1 2 1 (2m− 5)3 . Therefore, |Cn,m| ≤ 4 √ 6M1 2 7k+1 2 1 (2m− 5)3 ,∀m ≥ 3. (5.9) S2k−1,M (f)(x) = 2k−1∑ n=1 M∑ m=0 cn,mψn,m(x) f(x) −S2k−1,M (f)(x) = 2k−1∑ n=1 ∞∑ m=0 cn,mψn,m(x) − 2k−1∑ n=1 M∑ m=0 cn,mψn,m(x) = 2k−1∑ n=1 M∑ m=0 cn,mψn,m(x) + 2k−1∑ n=1 ∞∑ m=M+1 cn,mψn,m(x) − 2k−1∑ n=1 M∑ m=0 cn,mψn,m(x) = 2k−1∑ n=1 ∞∑ m=M+1 cn,mψn,m(x). Then, ||f −S2k−1,M (f)|| 2 2 = 1∫ 0  2k−1∑ n=1 ∞∑ m=M+1 cn,mψn,m(x)  2 dx = 2k−1∑ n=1 ∞∑ m=M+1 c2n,m, by orthogonality property of ψn,m 101 Shyam Lal and Indra Bhan ≤ 2k−1∑ n=1 ∞∑ m=M+1 ( 4 √ 6M1 2 7k+1 2 1 (2m− 5)3 )2 , by (5.9) = 96M21 2k−1∑ n=1 1 27k+1 ∞∑ m=M+1 1 (2m− 5)6 = 96M21 4 1 26k ∞∫ M+1 1 (2m− 5)6 dm = 12M21 5 1 26k 1 (2M − 3)5 ∴ E (4) 2k−1,M (f) ≤ 2 √ 3M1√ 5 1 23k(2M − 3) 5 2 = O ( 1 (2M − 3) 5 2 23k ) , M ≥ 2. 5.2 Proof of the Theorem(4.2) (i) The error e∗(0)n (x) between f(x) and its expression over any subinterval is defined as e∗(0)n (x) = cn,0ψn,0(x) − f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) ,n = 1, 2, 3, ...2k−1 Now consider, n̂+1 2k∫ n̂−1 2k (f(x))2dx = 1 2k−1∫ 0 ( f ( n̂− 1 2k + h ))2 dh,x = n̂− 1 2k + h = 1 2k−1∫ 0 [ f ( n̂− 1 2k ) + hf ′ ( n̂− 1 2k ) + h2 2 f ′′ ( n̂− 1 2k ) + h3 6 f ′′′ ( n̂− 1 2k ) + h4 24 fiv ( n̂− 1 2k + θh )]2 dh = 2 2k ( f ( n̂− 1 2k ))2 + 8 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 8 5 1 25k ( f ′′ ( n̂− 1 2k ))2 + 32 63 1 27k ( f ′′′ ( n̂− 1 2k ))2 + 1 2k−1∫ 0 h8 576 ( fiv ( n̂− 1 2k + θh ))2 dh + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) 102 Legendre Wavelet expansion of functions and their Approximations + 1 2k−1∫ 0 h4 12 f ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 4 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 32 15 1 25k f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k−1∫ 0 h5 12 f ′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 16 9 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k−1∫ 0 h6 24 f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 1 2k−1∫ 0 h7 72 f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 4 3 1 24k f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) . Now, cn,0 = 2 k−1 2 [ 2 2k f ( n̂− 1 2k ) + 2 22k f ′ ( n̂− 1 2k ) + 4 3 1 23k f ′′ ( n̂− 1 2k ) + 2 3 1 24k f ′′′ ( n̂− 1 2k )] + 2 k−1 2   1 2k−1∫ 0 h4 24 fiv ( n̂− 1 2k + θh ) dh   . Next, c2n,0 = 2 2k ( f ( n̂− 1 2k ))2 + 2 23k ( f ′ ( n̂− 1 2k ))2 + 8 9 1 25k ( f ′′ ( n̂− 1 2k ))2 + 2 9 1 27k ( f ′′′ ( n̂− 1 2k ))2 + 2k 2   1 2k−1∫ 0 h4 24 fiv ( n̂− 1 2k + θh ) dh   2 + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 4 3 1 24k f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k−1∫ 0 h4 12 f ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 8 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 4 3 1 25k f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k 1 2k−1∫ 0 h4 12 f ′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 8 9 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 4 3 1 22k 1 2k−1∫ 0 h4 24 f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 2 3 1 23k 1 2k−1∫ 0 h4 24 f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh. 103 Shyam Lal and Indra Bhan Since |f ′ (x)| ≤ M1, |f ′′ (x)| ≤ M2, |f ′′′ (x)| ≤ M3 and |fiv(x)| ≤ M4, ∀x ∈ [0, 1) therefore, ||e∗(0)n || 2 2 ≤ 2 23k ( M1 + M2 2k + M3 22k + M4 23k )2 . ∴ E (5) 2k−1,0 (f) = O ( 1 2k ) . (ii) The error e∗(1)n (x) between f(x) and its expression over any subinterval is defined as e∗ (1) n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) −f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) ,n = 1, 2, 3, ...2k−1 ||e∗(1)n ||22 = n̂+1 2k∫ n̂−1 2k (e∗ (1) n (x)) 2dx = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0 − c 2 n,1. Now, cn,1 = √ 3 2 2 k 2 [ 2 3 1 22k f ′ ( n̂− 1 2k ) + 2 3 1 23k f ′′ ( n̂− 1 2k ) + 2 5 1 24k f ′′′ ( n̂− 1 2k )] + √ 3 2 2 k 2 1 2k−1∫ 0 h4 24 (2kh− 1)fiv ( n̂− 1 2k + θh ) dh. Next, c2n,1 = 2 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 2 3 1 25k ( f ′′ ( n̂− 1 2k ))2 + 6 25 1 27k ( f ′′′ ( n̂− 1 2k ))2 + 3 2 2k   1 2k−1∫ 0 h4 24 (2kh− 1)fiv ( n̂− 1 2k + θh ) dh   2 + 2 2k 1 2k−1∫ 0 h4 24 (2kh− 1)f ′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 4 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 4 5 1 25k f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 4 5 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) 104 Legendre Wavelet expansion of functions and their Approximations + 2 22k 1 2k−1∫ 0 h4 24 (2kh− 1)f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 6 5 1 23k 1 2k−1∫ 0 h4 24 (2kh− 1)f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh Therefore, ||e∗(1)n || 2 2 ≤ 2 25k ( M2 + M3 2k + M4 22k )2 . Then, E (6) 2k−1,1 (f) = O ( 1 22k ) . (iii) The error e∗(2)n (x) between f(x) and its expression over any subinterval is defined as e∗ (2) n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) + cn,2ψn,2(x) −f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) , n = 1, 2, 3, ...2k−1 ||e∗(2)n || 2 2 = n̂+1 2k∫ n̂−1 2k (e(2)n (x)) 2dx = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0 − c 2 n,1 − c 2 n,2. Now, cn,2 = < f(x),ψn,2(x) > = √ 5 2 2 k 2 2 15 [ 1 23k f ′′ ( n̂− 1 2k ) + 1 24k f ′′′ ( n̂− 1 2k )] + √ 5 2 2 k 2 1 2   1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)fiv ( n̂− 1 2k + θh ) dh   . Next, c2n,2 = 2 45 1 25k ( f ′′ ( n̂− 1 2k ))2 + 2 45 1 27k ( f ′′′ ( n̂− 1 2k ))2 +   1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)fiv ( n̂− 1 2k + θh ) dh   2 + 4 45 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) 105 Shyam Lal and Indra Bhan + 1 3 1 22k 1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 1 3 1 23k 1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh. Therefore, ||e∗(2)n || 2 2 ≤ 2 27k ( M3 + M4 2k )2 . Then, E (7) 2k−1,2 (f) = O ( 1 23k ) . (iv) The error e∗(3)n (x) between f(x) and its expression over any subinterval is defined as e∗ (3) n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) + cn,2ψn,2(x) + cn,3ψn,3(x) − f(x) , x ∈ [ n̂−1 2k , n̂+1 2k ) , n = 1, 2, 3, ...2k−1 Similarly , it can be proved that E (8) 2k−1,3 (f) = O ( 1 24k ) . (v) Following the proof of Theorem (4.1)(iv) we have cn,m = ( 1 27k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (Pm+1(t) −Pm−1(t)) (2m + 1) − (Pm+3(t) −Pm+1(t)) (2m + 5) ] dt − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (Pm−1(t) −Pm−3(t)) (2m− 3) − (Pm+1(t) −Pm−1(t)) (2m + 1) ] dt = ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm+2(t) −Pm(t)) (2m + 3) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm(t) −Pm−2(t)) (2m− 1) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 5) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm+4(t) −Pm+2(t)) (2m + 7) ] 106 Legendre Wavelet expansion of functions and their Approximations + ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 5) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm+2(t) −Pm(t)) (2m + 3) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m− 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm(t) −Pm−2(t)) (2m− 1) ] + ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m− 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm−2(t) −Pm−4(t)) (2m− 5) ] + ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm+2(t) −Pm(t)) (2m + 3) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(Pm(t) −Pm−2(t)) (2m− 1) ] = ( 1 29k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3) × 1∫ −1 fiv ( n̂ + t 2k )[ (Pm(t) −Pm−2(t)) (2m− 1) − (Pm+2(t) −Pm(t)) (2m + 3) ] dt + ( 1 29k+1(2m + 1) )1 2 1 (2m + 3)(2m + 5) × 1∫ −1 fiv ( n̂ + t 2k )[ (Pm+4(t) −Pm+2(t)) (2m + 7) − (Pm+2(t) −Pm(t)) (2m + 3) ] dt + ( 1 29k+1(2m + 1) )1 2 1 (2m− 1)(2m + 1) × 1∫ −1 fiv ( n̂ + t 2k )[ (Pm(t) −Pm−2(t)) (2m− 1) − (Pm+2(t) −Pm(t)) (2m + 3) ] dt + ( 1 29k+1(2m + 1) )1 2 1 (2m− 1)(2m− 3) × 1∫ −1 fiv ( n̂ + t 2k )[ (Pm(t) −Pm−2(t)) (2m− 1) − (Pm−2(t) −Pm−4(t)) (2m− 5) ] dt. |cn,m| ≤ ( 1 29k )1 2 8 √ 6M2 (2m− 7)4 , (∵ |fiv(x)| ≤ M2 ∀ x ∈ [0, 1)). 107 Shyam Lal and Indra Bhan Next, ||f −S2k−1,M (f)|| 2 2 = 2k−1∑ n=1 ∞∑ m=M+1 C2n,m ≤ 2k−1∑ n=1 ∞∑ m=M+1 (( 1 29k )1 2 8 √ 6M2 (2m− 7)4 )2 = 48M22 7 1 28k 1 (2M − 5)7 ∴ E (9) 2k−1,M (f) = √ 48 7 M2 24k 1 (2M − 5) 7 2 = O ( 1 (2M − 5) 7 2 1 24k ) , ∀ M ≥ 3. 6 Conclusions (1) After discussing the Legendre wavelet approximation of a function f with bounded third and fourth derivatives ,it is trivial to find out the wavelet estimators of a function f of bounded first and second derivatives . (2)The estimates of the Theorems (4.1) an(4.2) are obtained as following: (i)E (1) 2k−1,0 (f) = O ( 1 2k ) → 0 as k →∞ (ii)E (2) 2k−1,1 (f) = O ( 1 22k ) → 0 as k →∞ (iii)E (3) 2k−1,2 (f) = O ( 1 23k ) → 0 as k →∞ (iv)E (4) 2k−1,M (f) = O ( 1 (2M−3) 5 2 1 23k ) → 0 as k →∞,M →∞ (v)E (5) 2k−1,0 (f) = O ( 1 2k ) → 0 as k →∞ (vi)E (6) 2k−1,1 (f) = O ( 1 22k ) → 0 as k →∞ (vii)E (7) 2k−1,2 (f) = O ( 1 23k ) → 0 as k →∞ (viii)E (8) 2k−1,3 (f) = O ( 1 24k ) → 0 as k →∞ (ix)E (9) 2k−1,M (f) = O ( 1 (2M−5) 7 2 1 24k ) → 0 as k →∞,M →∞ Then E (1) 2k−1,0 (f),E (2) 2k−1,1 (f),E (3) 2k−1,2 (f),E (4) 2k−1,M (f),E (5) 2k−1,1 (f),E (6) 2k−1,1 (f),E (7) 2k−1,2 (f), E (8) 2k−1,3 (f),E (9) 2k−1,M (f) are best possible Legendre wavelet approximation in Wavelet Analysis. (3)Legendre wavelet estimators of a function f with bounded fourth order derivative is better and sharper than the estimator of a function f of bounded third order derivative. (4) Legendre wavelet estimator of a function f of bounded higher order derivatives is better and sharper than the estimator of a function f of bounded less order derivatives. 108 Legendre Wavelet expansion of functions and their Approximations 7 Acknowledgments Shyam Lal, one of the authors, is thankful to DST - CIMS for encouragement to this work. Indra Bhan, one of the authors, is grateful to C.S.I.R. (India) for providing financial assistance in the form of Junior Research Fellowship vide Ref. No. 18/12/2016 (ii) EU-V Dated:01-07-2017 for this research work. References [1] A. Zygmund , Trigonometric Series Volume I, Cambridge University Press, 1959. [2] H. N. Mhaskar, “ Polynomial operators and local smoothness classes on the unit interval, II,” Jaen J. Approx., Vol. 1, No. 1(2009), pp. 1-25. [3] J. Morlet, G. Arens, E. Fourgeau and D. Giard, Wave propagation and sampling theory, part I; Complex signal and scattering in multilayer media, Geophysics 47(1982) No. 2, 203-221. [4] J. Morlet, G. Arens, E. Fourgeau and D. Giard, Wave propagation and sampling theory, part II; sampling theory complex waves, Geophysics 47(1982) no. 2, 222-236. [5] L. Debnath, Wavelet Transform and their applications, Birkhauser Bostoon, Massachusetts-2002. [6] P. Sablonnière, “ Rational Bernstein and spline approximation. A new approach, ” Jaen J. Approx., Vol. 1, No. 1(2009), pp. 37-53. [7] R. A. Devore, Nonlinear Approximation, Acta Numerica, Vol. 7, Cambridge University Press,Cambridge(1998), pp. 51-150. [8] Shyam Lal and Susheel Kumar “Best Wavelet Approximation of function belonging to Generalized Lipschitz Class using Haar Scaling function,” Thai Journal of Mathematics, Vol. 15(2017), No. 2, pp. 409-419. [9] Y. Meyer (1993)(Toulouse(1992))(Y. Meyer and S. Roques , eds) Frontieres, Gif-sur-Yvette, Wavelets their post and their future, Progress in Wavelet analysis and applications, pp. 9-18. [10] Lal, Shyam, and Indra Bhan. ”Approximation of Functions Belonging to Generalized Hölder’s Class H(ω)α [0, 1) by First Kind Chebyshev Wavelets and Its Applications in the Solution of Linear and Nonlinear Differential Equations.” International Journal of Applied and Computational Mathematics 5.6 (2019): 155. [11] Lal, Shyam, and Rakesh. ”The approximations of a function belonging Hölder classHα[0, 1)by second kind Chebyshev wavelet method and applications in solutions of differential equation.” International Journal of Wavelets, Multiresolution and Information Processing 17.01 (2019): 1850062. 109 Introduction Definitions and Preliminaries Legendre Wavelet Legendre Wavelet Approximation Example Theorems Proofs Proof of the Theorem (4.1) Proof of the Theorem(4.2) Conclusions Acknowledgments