() CUBO A Mathematical Journal Vol.17, No¯ 01, (65–73). March 2015 Spline left fractional monotone approximation involving left fractional differential operators George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A. ganastss@memphis.edu ABSTRACT Let f ∈ Cs ([−1, 1]), s∈ N and L∗ be a linear left fractional differential operator such that L∗ (f) ≥ 0 on [0, 1]. Then there exists a sequence Qn, n ∈ N of polynomial splines with equally spaced knots of given fixed order such that L∗ (Qn) ≥ 0 on [0, 1]. Furthermore f is approximated with rates fractionally and simultaneously by Qn in the uniform norm. This constrained fractional approximation on [−1, 1] is given via inequalities invoving a higher modulus of smoothness of f(s). RESUMEN Sea f ∈ Cs ([−1, 1]), s∈ N y L∗ un operador diferencial fraccionario lineal izquierdo tal que L∗ (f) ≥ 0 en [0, 1].. Entonces, existe una sucesión Qn, n ∈ N de splines polinomiales con nodos equiespaciados de un orden fijo dado tal que L∗ (Qn) ≥ 0 en [0, 1]. Además, f se aproxima con velocidades fraccionales y simultáneamente por Qn en la norma uniforme. Esta aproximación fraccional restringida a [−1, 1] se encuentra por medio de desigualdades que involucran un módulo alto de suavidad de f(s). Keywords and Phrases: Monotone Approximation, Caputo fractional derivative, fractional linear differential operator, modulus of smoothness, splines. 2010 AMS Mathematics Subject Classification: 26A33, 41A15, 41A17, 41A25, 41A28, 41A29, 41A99. 66 George A. Anastassiou CUBO 17, 1 (2015) 1 Introduction Let [a, b] ⊂ R and for n ≥ 1 consider the partition ∆n with points xin = a + i ( b−a n ) , i = 0, 1, ..., n. Hence ∆n ≡ max1≤i≤n (xin − xi−1,n) = b−a n . Let Sm (∆n) be the space of polynomial splines of order m > 0 with simple knots at the points xin, i = 1, ..., n − 1. Then there exists a linear operator Qn : Qn ≡ Qn (f), mapping B [a, b]: the space of bounded real valued functions f on [a, b], into Sm (∆n) (see [4], p. 224, Theorem 6.18). From the same reference [4], p. 227, Corollary 6.21, we get Corolary 1. Let 1 ≤ σ ≤ m, n ≥ 1. Then for all f ∈ Cσ−1 [a, b] ; r = 0, ..., σ − 1, ∥ ∥ ∥ f(r) − Q(r)n ∥ ∥ ∥ ∞ ≤ C1 ( b − a n )σ−r−1 ωm−σ+1 ( f(σ−1), b − a n ) , (1) where C1 depends only on m, C1 = C1 (m) . By denoting C2 = C1 max0≤r≤σ−1 (b − a) σ−r−1 we obtain Lemma 1. ([1]) Let 1 ≤ σ ≤ m, n ≥ 1. Then for all f ∈ Cσ−1 [a, b]; r = 0, ..., σ − 1, ∥ ∥ ∥ f(r) − Q(r)n ∥ ∥ ∥ ∞ ≤ C2 nσ−r−1 ωm−σ+1 ( f(σ−1), b − a n ) , (2) where C2 depends only on m, σ and b − a. Here ωm−σ+1 is the usual modulus of smoothness of order m − σ + 1. We are motivated by Theorem 1. ([1]) Let h, k, σ, m be integers, 0 ≤ h ≤ k ≤ σ − 1, σ ≤ m and let f ∈ Cσ−1 [a, b]. Let αj (x) ∈ B [a, b], j = h, h + 1, ..., k and suppose that αh (x) ≥ α > 0 or αh (x) ≤ β < 0 for all x ∈ [a, b] . Take the linear differential operator L = k∑ j=h αj (x) [ dj dxj ] (3) and assume, throughout [a, b], L (f) ≥ 0. (4) Then, for every integer n ≥ 1, there is a polynomial spline function Qn (x) of order m with simple knots at { a + i ( b−a n ) , i = 1, ..., n − 1 } such that L (Qn) ≥ 0 throughout [a, b] and ∥ ∥ ∥ f(r) − Q(r)n ∥ ∥ ∥ ∞ ≤ C nσ−k−1 ωm−σ+1 ( f(σ−1), b − a n ) , 0 ≤ r ≤ h. (5) CUBO 17, 1 (2015) Spline left fractional monotone approximation involving left . . . 67 Moreover, we find ∥ ∥ ∥ f(r) − Q(r)n ∥ ∥ ∥ ∞ ≤ C nσ−r−1 ωm−σ+1 ( f(σ−1), b − a n ) , h + 1 ≤ r ≤ σ − 1, (6) where C is a constant independent of f and n. It depends only on m, σ, L, a, b. Next we specialize on the case of a = −1, b = 1. That is working on [−1, 1] . By Lemma 1 we get Lemma 2. Let 1 ≤ σ ≤ m, n ≥ 1. Then for all f ∈ Cσ−1 ([−1, 1]); j = 0, 1, ..., σ − 1, ∥ ∥ ∥ f(j) − Q(j)n ∥ ∥ ∥ ∞ ≤ C2 nσ−j−1 ωm−σ+1 ( f(σ−1), 2 n ) , (7) where C2 := C2 (m, σ) := C1 (m) 2 σ−1. Since ωm−σ+1 ( f(σ−1), 2 n ) ≤ 2m−σ+1ωm−σ+1 ( f(σ−1), 1 n ) (8) (see [2], p. 45), we get Lemma 3. Let 1 ≤ σ ≤ m, n ≥ 1. Then for all f ∈ Cσ−1 ([−1, 1]); j = 0, 1, ..., σ − 1, ∥ ∥ ∥ f(j) − Q(j)n ∥ ∥ ∥ ∞ ≤ C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (9) where C∗2 := C ∗ 2 (m, σ) := C1 (m) 2 m. We use a lot in this article Lemma 3. In this article we extend Theorem 1 over [−1, 1] to the fractional level. Indeed here L is replaced by L∗, a linear left Caputo fractional differential operator. Now the monotonicity property is only true on the critical interval [0, 1]. Simultaneous fractional convergence remains true on all of [−1, 1] . We make Definition 1. ([3], p. 50) Let α > 0 and ⌈α⌉ = m, (⌈·⌉ ceiling of the number). Consider f ∈ Cm ([−1, 1]). We define the left Caputo fractional derivative of f of order α as follows: ( Dα∗−1f ) (x) = 1 Γ (m − α) ∫x −1 (x − t) m−α−1 f(m) (t) dt, (10) for any x ∈ [−1, 1], where Γ is the gamma function. We set D0∗−1f (x) = f (x) , Dm∗−1f (x) = f (m) (x) , ∀ x ∈ [−1, 1] . (11) 68 George A. Anastassiou CUBO 17, 1 (2015) 2 Main Result Theorem 2. Let h, k, σ, m be integers, 1 ≤ σ ≤ m, n ∈ N, with 0 ≤ h ≤ k ≤ σ − 2 and let f ∈ Cσ−1 ([−1, 1]), with f(σ−1) having modulus of smoothness ωm−σ+1 ( f(σ−1), δ ) there, δ > 0. Let αj (x), j = h, h+1, ..., k be real functions, defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [0, 1]. Let the real numbers α0 = 0 < α1 ≤ 1 < α2 ≤ 2 < ... < ασ−2 ≤ σ−2. Here D αj ∗−1 f stands for the left Caputo fractional derivative of f of order αj anchored at −1. Consider the linear left fractional differential operator L∗ := k∑ j=h αj (x) [ D αj ∗−1 ] (12) and suppose, throughout [0, 1], L∗ (f) ≥ 0. Then, for every integer n ≥ 1, there exists a polynomial spline function Qn (x) of order m > 0 with simple knots at { −1 + i 2 n , i = 1, ..., n − 1 } such that L∗ (Qn) ≥ 0 throughout [0, 1] , and sup −1≤x≤1 ∣ ∣ ( D αj ∗−1f ) (x) − ( D αj ∗−1Qn ) (x) ∣ ∣ ≤ 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (13) j = h + 1, ..., σ − 2. Set lj :≡ sup x∈[−1,1] ∣ ∣α−1h (x) αj (x) ∣ ∣ , h ≤ j ≤ k. (14) When j = 1, ..., h we derive max −1≤x≤1 ∣ ∣ ( D αj ∗−1 f ) (x) − ( D αj ∗−1 Qn ) (x) ∣ ∣ ≤ C∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) · [( k∑ τ=h lτ 2τ−ατ Γ (τ − ατ + 1) )( h−j∑ λ=0 2h−αj−λ λ!Γ (h − αj − λ + 1) ) + 2j−αj Γ (j − αj + 1) ] . (15) Finally it holds sup −1≤x≤1 |f (x) − Qn (x)| ≤ C∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ Γ (τ − ατ + 1) + 1 ] . (16) Proof. Set α0 = 0, thus ⌈α0⌉ = 0. We have ⌈αj⌉ = j, j = 1, ..., σ − 2. Let Qn as in Lemma 5. CUBO 17, 1 (2015) Spline left fractional monotone approximation involving left . . . 69 We notice that (x ∈ [−1, 1]) ∣ ∣ ( D αj ∗−1f ) (x) − ( D αj ∗−1Qn ) (x) ∣ ∣ = 1 Γ (j − αj) ∣ ∣ ∣ ∣ ∫x −1 (x − t) j−αj−1 f(j) (t) dt − ∫x −1 (x − t) j−αj−1 Q(j)n (t) dt ∣ ∣ ∣ ∣ = (17) 1 Γ (j − αj) ∣ ∣ ∣ ∣ ∫x −1 (x − t) j−αj−1 ( f(j) (t) − Q(j)n (t) ) dt ∣ ∣ ∣ ∣ ≤ 1 Γ (j − αj) ∫x −1 (x − t) j−αj−1 ∣ ∣ ∣ f(j) (t) − Q(j)n (t) ∣ ∣ ∣ dt (9) ≤ (18) 1 Γ (j − αj) (∫x −1 (x − t) j−αj−1 dt ) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = 1 Γ (j − αj) (x + 1) j−αj (j − αj) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = (19) (x + 1) j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) ≤ 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) . (20) Hence ∥ ∥D αj ∗−1 f − D αj ∗−1 Qn ∥ ∥ ∞,[−1,1] ≤ 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (21) j = 0, 1, ..., σ − 2. We set ρn := C ∗ 2ωm−σ+1 ( f(σ−1), 1 n )   k∑ j=h lj 2j−αj Γ (j − αj + 1) n σ−j−1  . (22) I. Suppose, throughout [0, 1], αh (x) ≥ α > 0. Let Qn (x), x ∈ [−1, 1], the polynomial spline of order m > 0 with simple knots at the points xin, i = 1, ..., n − 1, on [−1, 1] (xin = −1 + i 2 n , i = 0, 1, ..., n, here ∆n = 2 n ), so that max −1≤x≤1 ∣ ∣ ∣ ∣ D αj ∗−1 ( f (x) + ρn xh h! ) − ( D αj ∗−1 Qn ) (x) ∣ ∣ ∣ ∣ ≤ 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (23) j = 0, 1, ..., σ − 2. When j = h + 1, ..., σ − 2, then max −1≤x≤1 ∣ ∣ ( D αj ∗−1 f ) (x) − ( D αj ∗−1 Qn ) (x) ∣ ∣ ≤ 70 George A. Anastassiou CUBO 17, 1 (2015) 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (24) proving (13). For j = 1, ..., h we find that D αj ∗−1 ( xh h! ) = h−j∑ λ=0 (−1) λ (x + 1) h−αj−λ λ!Γ (h − αj − λ + 1) . (25) Therefore we get from (23) max −1≤x≤1 ∣ ∣ ∣ ∣ ∣ ( D αj ∗−1 f ) (x) + ρn ( h−j∑ λ=0 (−1) λ (x + 1) h−αj−λ λ!Γ (h − αj − λ + 1) ) − ( D αj ∗−1 Qn ) (x) ∣ ∣ ∣ ∣ ∣ ≤ (26) 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , j = 1, ..., h. Therefore we get for j = 1, ..., h, that max −1≤x≤1 ∣ ∣ ( D αj ∗−1f ) (x) − ( D αj ∗−1Qn ) (x) ∣ ∣ ≤ (27) ρn ( h−j∑ λ=0 2h−αj−λ λ!Γ (h − αj − λ + 1) ) + 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = C∗2ωm−σ+1 ( f(σ−1), 1 n )   k∑ j=h l j 2 j−α j Γ ( j − α j + 1 ) nσ−j−1   · ( h−j∑ λ=0 2h−αj−λ λ!Γ (h − αj − λ + 1) ) + 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = C∗2ωm−σ+1 ( f(σ−1), 1 n )     k∑ j=h l j 2 j−α j Γ ( j − α j + 1 ) 1 nσ−j−1   · (28) ( h−j∑ λ=0 2h−αj−λ λ!Γ (h − αj − λ + 1) ) + 2j−αj Γ (j − αj + 1) 1 nσ−j−1 ] ≤ C∗2ωm−σ+1 ( f(σ−1), 1 n ) 1 nσ−k−1     k∑ j=h l j 2j−αj Γ ( j − α j + 1 )   · (29) ( h−j∑ λ=0 2h−αj−λ λ!Γ (h − αj − λ + 1) ) + 2j−αj Γ (j − αj + 1) ] . CUBO 17, 1 (2015) Spline left fractional monotone approximation involving left . . . 71 Hence for j = 1, ..., h we derived (15): max −1≤x≤1 ∣ ∣ ( D αj ∗−1 f ) (x) − ( D αj ∗−1 Qn ) (x) ∣ ∣ ≤ C∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) · [( k∑ τ=h lτ 2τ−ατ Γ (τ − ατ + 1) )( h−j∑ λ=0 2h−αj−λ λ!Γ (h − αj − λ + 1) ) + 2j−αj Γ (j − αj + 1) ] . (30) When j = 0 from (23) we obtain max −1≤x≤1 ∣ ∣ ∣ ∣ f (x) + ρn xh h! − Qn (x) ∣ ∣ ∣ ∣ ≤ C∗2 nσ−1 ωm−σ+1 ( f(σ−1), 1 n ) . (31) And max −1≤x≤1 |f (x) − Qn (x)| ≤ ρn h! + C∗2 nσ−1 ωm−σ+1 ( f(σ−1), 1 n ) = (32) C∗2 h! ωm−σ+1 ( f(σ−1), 1 n ) ( k∑ τ=h lτ 2τ−ατ Γ (τ − ατ + 1) n σ−τ−1 ) + C∗2 nσ−1 ωm−σ+1 ( f(σ−1), 1 n ) = C∗2ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ Γ (τ − ατ + 1) n σ−τ−1 + 1 nσ−1 ] ≤ (33) C∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ Γ (τ − ατ + 1) + 1 ] . Proving max −1≤x≤1 |f (x) − Qn (x)| ≤ C∗2 nσ−k−1 ωm−σ+1 ( f(σ−1), 1 n ) [ 1 h! k∑ τ=h lτ 2τ−ατ Γ (τ − ατ + 1) + 1 ] , (34) So that (16) is established. Also if 0 ≤ x ≤ 1, then α−1h (x) L ∗ (Qn (x)) = α −1 h (x) L ∗ (f (x)) + ρn (x + 1) h−αh Γ (h − αh + 1) + (35) k∑ j=h α−1 h (x) αj (x) [ D αj ∗−1 Qn (x) − D αj ∗−1 f (x) − ρn h! D αj ∗−1 xh ] (23) ≥ ρn (x + 1) h−αh Γ (h − αh + 1) −   k∑ j=h lj 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n )   = 72 George A. Anastassiou CUBO 17, 1 (2015) ρn (x + 1) h−αh Γ (h − αh + 1) − ρn = ρn [ (x + 1) h−αh Γ (h − αh + 1) − 1 ] = ρn [ (x + 1) h−αh − Γ (h − αh + 1) Γ (h − αh + 1) ] ≥ ρn [ 1 − Γ (h − αh + 1) Γ (h − αh + 1) ] ≥ 0. (36) Explanation: We know that Γ (1) = 1, Γ (2) = 1, and Γ is convex and positive on (0, ∞) . Here 0 ≤ h − αh < 1 and 1 ≤ h − αh + 1 < 2. Thus Γ (h − αh + 1) ≤ 1 and 1 − Γ (h − αh + 1) ≥ 0. Hence L∗ (Qn (x)) ≥ 0, x ∈ [0, 1] . II. Suppose on [0, 1] that αh (x) ≤ β < 0. Let Qn (x), x ∈ [−1, 1], be the polynomial spline of order m > 0, (as before), so that max −1≤x≤1 ∣ ∣ ∣ ∣ D αj ∗−1 ( f (x) − ρn xh h! ) − ( D αj ∗−1 Qn ) (x) ∣ ∣ ∣ ∣ ≤ 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) , (37) j = 0, 1, ..., σ − 2. Similarly as before we obtain again inequalities of convergence (13), (15) and (16). Also if 0 ≤ x ≤ 1, then α−1h (x) L ∗ (Qn (x)) = α −1 h (x) L ∗ (f (x)) − ρn (x + 1) h−αh Γ (h − αh + 1) + (38) k∑ j=h α−1h (x) αj (x) [ D αj ∗−1 Qn (x) − D αj ∗−1 f (x) + ρn h! ( D αj ∗−1 xh ) ] (37) ≤ − ρn (x + 1) h−αh Γ (h − αh + 1) + k∑ j=h lj 2j−αj Γ (j − αj + 1) C∗2 nσ−j−1 ωm−σ+1 ( f(σ−1), 1 n ) = (39) ρn ( 1 − (x + 1) h−αh Γ (h − αh + 1) ) = ρn ( Γ (h − αh + 1) − (x + 1) h−αh Γ (h − αh + 1) ) ≤ (40) ρn ( 1 − (x + 1) h−αh Γ (h − αh + 1) ) ≤ 0, and hence again L∗ (Qn (x)) ≥ 0, x ∈ [0, 1] . Received: May 2014. Accepted: October 2014. CUBO 17, 1 (2015) Spline left fractional monotone approximation involving left . . . 73 References [1] G.A. Anastassiou, Spline monotone approximation with linear differential operators, Approxi- mation Theory and its Applications, Vol. 5 (4) (1989), 61-67. [2] R. De Vore, G. Lorentz, Constructive Approximation, Springer-Verlag, Heidelberg, New York, 1993. [3] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Vol. 2004, 1st edition, Springer, New York, Heidelberg, 2010. [4] L.L. Schumaker, Spline functions: Basic Theory, John Wiley and Sons, Inc., New York, 1981. Introduction Main Result