CUBO A Mathematical Journal Vol.20, No¯ 01, (01–16). March 2018 http: // dx. doi. org/ 10. 4067/ S0719-06462018000100001 Approximation by Shift Invariant Univariate Sublinear-Shilkret Operators George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A. ganastss@memphis.edu ABSTRACT A very general positive sublinear Shilkret integral type operator is given through a convolution-like iteration of another general positive sublinear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Additionally, two examples of very general specialized operators are presented fulfilling all the above properties, the higher order of approximation of these operators is also considered. RESUMEN Un operador muy general positivo sublineal de tipo integral de Shilkret es dado a través de un iteración de tipo convolución de otro operador general positivo sublineal con una función de tipo escalamiento. Para estos operadores, se entregan condiciones suficientes para invariancia por shifts, conservación de la suavidad global y convergen- cia a la unidad con tasas. Adicionalmente, se presentan dos ejemplos de operadores muy generales especializados que satisfacen todas las propiedades anteriores, también considerando el alto orden de aproximación de estos operadores. Keywords and Phrases: Jackson type inequality, Shilkret integral, modulus of continuity, shift invariant, global smoothness preservation, quantitative approximation. 2010 AMS Mathematics Subject Classification: 41A17, 41A25, 41A35, 41A36. http://dx.doi.org/10.4067/S0719-06462018000100001 Ignacio Castillo Ignacio Castillo Ignacio Castillo 2 George A. Anastassiou CUBO 20, 1 (2018) 1 Introduction Let X,Y be function spaces of functions from R into R+. Let LN : X → Y, N ∈ N, be a sequence of operators with the following properties: (i) (positive homogeneous) LN (αf) = αLN (f) , ∀ α ≥ 0, ∀ f ∈ X. (ii) (Monotonicity) if f,g ∈ X satisfy f ≤ g, then LN (f) ≤ LN (g), ∀ N ∈ N, and (iii) (Subadditivity) LN (f + g) ≤ LN (f) + LN (g) , ∀ f,g ∈ X. We call LN positive sublinear operators. In this article we deal with sequences of Shilkret positive sublinear operators that are con- structed, with the help of Shilkret integral ([5]). Our functions spaces are continuous functions from R into R+. The sequence of operators is generated by a basic operator via dilated trans- lations of convolution type using the Shilkret integral. We prove that our operators possess the following properties: of shift invariance of global smoothness preservation, of convergence to the unit operator with rates. Then we apply our results to two specific families of such Shilkret type operators. We continue with the higher order of approximation study of these specific operators, and all results are quantitative. Earlier similar studies have been done by the author, see [3], Chapters 10-17, and [2], Chapters 16, 17. These serve as motivation and inspiration to this work. 2 Background Here we follow [5]. Let F be a σ-field of subsets of an arbitrary set Ω. An extended non-negative real valued function µ on F is called maxitive if µ(∅) = 0 and µ(∪i∈IEi) = sup i∈I µ(Ei) , (1) where the set I is of cardinality at most countable, where {Ei}i∈I is a disjoint collection of sets from F. We notice that µ is monotone and (1) is true even {Ei}i∈I are not disjoint. For more properties of µ see [5]. We also call µ a maxitive measure. Here f stands for a non-negative measurable CUBO 20, 1 (2018) Approximation by Shift Invariant Univariate . . . 3 function on Ω. In [5], Niel Shilkret developed his non-additive integral defined as follows: (N∗) ∫ D fdµ := sup y∈Y {y · µ(D ∩ {f ≥ y})} , (2) where Y = [0,m] or Y = [0,m) with 0 < m ≤ ∞, and D ∈ F. Here we take Y = [0,∞). It is easily proved that (N∗) ∫ D fdµ = sup y>0 {y · µ(D ∩ {f > y})} . (3) The Shilkret integral takes values in [0,∞]. The Shilkret integral ([5]) has the following properties: (N∗) ∫ Ω χEdµ = µ(E) , (4) where χE is the indicator function on E ∈ F, (N∗) ∫ D cfdµ = c(N∗) ∫ D fdµ, c ≥ 0, (5) (N∗) ∫ D sup n∈N fndµ = sup n∈N (N∗) ∫ D fndµ, (6) where fn, n ∈ N, is an increasing sequence of elementary (countably valued) functions converging uniformly to f. Furthermore we have (N∗) ∫ D fdµ ≥ 0, (7) f ≥ g implies (N∗) ∫ D fdµ ≥ (N∗) ∫ D gdµ, (8) where f,g : Ω → [0,∞] are measurable. Let a ≤ f(ω) ≤ b for almost every ω ∈ E, then aµ(E) ≤ (N∗) ∫ E fdµ ≤ bµ(E) ; (9) (N∗) ∫ E 1dµ = µ(E) ; (10) f > 0 almost everywhere and (N∗) ∫ E fdµ = 0 imply µ(E) = 0; (N∗) ∫ Ω fdµ = 0 if and only f = 0 almost everywhere; (N∗) ∫ Ω fdµ < ∞ implies that N(f) := {ω ∈ Ω|f(ω) 6= 0} has σ-finite measure; 4 George A. Anastassiou CUBO 20, 1 (2018) (N∗) ∫ D (f + g)dµ ≤ (N∗) ∫ D fdµ + (N∗) ∫ D gdµ; (11) and ∣ ∣ ∣ ∣ (N∗) ∫ D fdµ − (N∗) ∫ D gdµ ∣ ∣ ∣ ∣ ≤ (N∗) ∫ D |f − g|dµ. (12) From now on in this article we assume that µ : F → [0,+∞). 3 Univariate Theory This section is motivated and inspired by [3] and [4]. Let L be the Lebesgue σ− algebra on R, and the set function µ : L → [0,+∞], which is assumed to be maxitive. Let CU (R,R+) be the space of uniformly continuous functions from R into R+, and C(R,R+) the space of continuous functions from R into R+. For any f ∈ CU (R,R+) we have ω1 (f,δ) < +∞, δ > 0, where ω1 (f,δ) := sup x,y∈R: |x−y|≤δ |f(x) − f(y)| , δ > 0, is the first modulus of continuity. Let {tk}k∈Z be a sequence of positive sublinear operators that map CU (R,R+) into C(R,R+) with the property (tk (f)) (x) := l0 ( f ( 2−k· )) (x) , ∀ x ∈ R, ∀ f ∈ CU (R,R+) . (13) For a fixed a > 0 we assume that sup u,y∈R: |u−y|≤a |t0 (f,u) − f(y)| ≤ ω1 ( f, ma + n 2r ) , ∀ f ∈ CU (R,R+) , (14) where m ∈ N, n ∈ Z+, r ∈ Z. Let ψ : R → R+ which is Lebesgue measurable, such that (N∗) ∫a −a ψ(u)dµ(u) = 1. (15) We define the positive sublinear-Shilkret operators (T0 (f)) (x) := (N ∗) ∫a −a (t0f) (x − u)ψ(u)dµ(u) , (16) and (Tk (f)) (x) := ( T0 ( f ( 2−k· )))( 2kx ) , ∀ k ∈ Z, ∀ x ∈ R. (17) CUBO 20, 1 (2018) Approximation by Shift Invariant Univariate . . . 5 Therefore it holds (Tk (f)) (x) = (N ∗ ) ∫a −a ( t0 ( f ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) = (18) (N∗) ∫a −a (tk (f)) ( 2kx − u ) ψ(u)dµ(u) , ∀ x ∈ R, ∀ k ∈ Z. Indeed here we have (Tk (f)) (x) (8) ≤ (N∗) ∫a −a ∥ ∥tk (f) ( 2kx − · )∥ ∥ ∞,[−a,a] ψ(u)dµ(u) (5) = ∥ ∥tk (f) ( 2kx − · )∥ ∥ ∞,[−a,a] ( (N∗) ∫a −a ψ(u)dµ(u) ) = (19) ∥ ∥tk (f) ( 2kx − · ) ∥ ∥ ∞,[−a,a] < +∞. Hence (Tk (f)) (x) ∈ R+ is well-defined. Let f,g ∈ M (R,R+) (Lebesgue measurable functions) where X ∈ A, A ⊂ R is a Lebesgue measurable set. We derive that ∣ ∣ ∣ ∣ (N∗) ∫ A f(x)dµ(x) − N∗ ∫ A g(x)dµ(x) ∣ ∣ ∣ ∣ (12) ≤ (N∗) ∫ A |f(x) − g(x)|dµ(x) . (20) We need Definition 3.1. Let fα (·) := f(· + α), α ∈ R, and Φ be an operator. If Φ(fα) = (Φf)α, then Φ is called a shift invariant operator. We give Theorem 3.2. Assume that ( t0 ( f ( 2−k · +α )))( 2ku ) = ( t0 ( f ( 2−k· )))( 2k (u + α) ) , (21) for all k ∈ Z, α ∈ R fixed, all u ∈ R and any f ∈ CU (R,R+). Then Tk is a shift invariant operator for all k ∈ Z. Proof. We have that (Tk (f(· + α))) (x) = (Tk (fα)) (x) (18) = (N∗) ∫a −a ( t0 ( fα ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) = (N∗) ∫a −a ( t0 ( f ( 2−k · +α )))( 2kx − u ) ψ(u)dµ(u) = 6 George A. Anastassiou CUBO 20, 1 (2018) (N∗) ∫a −a ( t0 ( f ( 2−k · +α )))( 2k ( x − 2−ku )) ψ(u)dµ(u) (21) = (22) (N∗) ∫a −a ( t0 ( f ( 2−k· )))( 2k ( x − 2−ku + α )) ψ(u)dµ(u) = (N∗) ∫a −a ( t0 ( f ( 2−k· )))( 2k (x + α) − u ) ψ(u)dµ(u) (18) = (Tk (f)) (x + α) , that is Tk (fα) = (Tk (f))α , (23) proving the claim. It follows the global smoothness of the operators Tk. Theorem 3.3. For any f ∈ CU (R,R+) assume that, for all u ∈ R, |(t0 (f)) (x − u) − (t0 (f)) (y − u)| ≤ ω1 (f, |x − y|) , (24) for any x,y ∈ R. Then ω1 (Tkf,δ) ≤ ω1 (f,δ) , ∀ δ > 0. (25) Proof. We observe that |(T0 (f)) (x) − (T0 (f)) (y)| = ∣ ∣ ∣ ∣ (N∗) ∫a −a (t0f) (x − u)ψ(u)dµ(u) − (N ∗ ) ∫a −a (t0f) (y − u)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (26) (N∗) ∫a −a |(t0f) (x − u) − (t0f) (y − u)|ψ(u)dµ(u) (by (24), (5)) ≤ ω1 (f, |x − y|) ( (N∗) ∫a −a ψ(u)dµ(u) ) (15) = ω1 (f, |x − y|) . So that |(T0 (f)) (x) − (T0 (f)) (y)| ≤ ω1 (f, |x − y|) . (27) From (17), (27) we get |(Tk (f)) (x) − (Tk (f)) (y)| (17) = ∣ ∣ ( T0 ( f ( 2−k· )))( 2kx ) − ( T0 ( f ( 2−k· )))( 2ky )∣ ∣ ≤ (28) ω1 ( f ( 2−k· ) ,2k |x − y| ) = ω1 (f, |x − y|) , i.e. global smoothness for Tk has been proved. The convergence of Tk to the unit operator, as k → +∞, k with rates follows: CUBO 20, 1 (2018) Approximation by Shift Invariant Univariate . . . 7 Theorem 3.4. For f ∈ CU (R,R+), under the assumption (14), we have |(Tk (f)) (x) − f(x)| ≤ ω1 ( f, ma + n 2k+r ) , (29) where m ∈ N, n ∈ Z+, k,r ∈ Z. Proof. We notice that |(Tk (f)) (x) − f(x)| (17) = ∣ ∣ ( T0 ( f ( 2−k· )))( 2kx ) − f(x) ∣ ∣ (18) = ∣ ∣ ∣ ∣ (N∗) ∫a −a ( t0 ( f ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) − f(x) ∣ ∣ ∣ ∣ (15) = ∣ ∣ ∣ ∣ (N∗) ∫a −a ( t0 ( f ( 2−k· )))( 2kx − u ) ψ(u)dµ(u) − (N∗) ∫a −a f(x)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (N∗) ∫a −a ∣ ∣ ( t0 ( f ( 2−k· )))( 2kx − u ) − f(x) ∣ ∣ψ(u)dµ(u) = (30) (N∗) ∫a −a ∣ ∣ ( t0 ( f ( 2−k· )))( 2kx − u ) − f ( 2−k· )( 2kx )∣ ∣ψ(u)dµ(u) (14) ≤ (here ∣ ∣ ( 2kx − u ) − 2kx ∣ ∣ = |u| ≤ a) ω1 ( f ( 2−k· ) , ma + n 2r )( (N∗) ∫a −a ψ(u)dµ(u) ) (15) = ω1 ( f ( 2−k· ) , ma + n 2r ) · 1 = ω1 ( f, ma + n 2k+r ) , (31) proving the claim. We give some applications. For each k ∈ Z, we define (i) (Bkf) (x) := (N ∗) ∫a −a f ( x − u 2k ) ψ(u)dµ(u) , (32) i.e., here (tk (f)) (u) = f ( u 2k ) , and (t0 (f)) (u) = f(u) , (33) are continuous in u ∈ R. Also for k ∈ Z, we define (ii) (Γk (f)) (x) := (N ∗ ) ∫a −a γfk ( 2kx − u ) ψ(u)dµ(u) , (34) 8 George A. Anastassiou CUBO 20, 1 (2018) where (tk (f)) (u) = γ f k (u) := n∑ j=0 wjf ( u 2k + j 2kn ) , (35) n ∈ N, wj ≥ 0, n∑ j=0 wj = 1, is continuous in u ∈ R. Notice here that (t0 (f)) (u) = γ f 0 (u) = n∑ j=0 wjf ( u + j n ) (36) is also continuous in u ∈ R. Indeed we have (Γk (f)) (x) = (N ∗) ∫a −a   n∑ j=0 wjf ( ( x − u 2k ) + j 2kn )  ψ(u)dµ(u) . (37) Clealry here we have (Bk (f)) (x) = ( B0 ( f ( 2−k· )))( 2kx ) , and (Γk (f)) (x) = ( Γ0 ( f ( 2−k· )))( 2kx ) , (38) ∀ k ∈ Z, ∀ x ∈ R. We give Proposition 3.5. Bk,Γk are shift invariant operators. Proof. (i) For Bk operators: Here t0f = f. Hence ( t0 ( f ( 2−k · +α )))( 2ku ) = f ( 2−k2ku + α ) = f(u + α) = (39) ( t0 ( f ( 2−k· )))( 2k (u + α) ) . (ii) For Γk operators: (t0 (f)) (u) = n∑ j=0 wjf ( u + j n ) . Hence ( t0 ( f ( 2−k · +α )))( 2ku ) = n∑ j=0 wjf ( 2−k ( 2ku + j n ) + α ) = n∑ j=0 wjf ( 2−k ( 2k (u + α) + j n )) = ( t0 ( f ( 2−k· )))( 2k (u + α) ) , (40) proving the claim. CUBO 20, 1 (2018) Approximation by Shift Invariant Univariate . . . 9 Next we show that the operators Bk,Γk possess the property of global smoothness preservation. Theorem 3.6. For all f ∈ CU (R,R+) and all δ > 0 we have ω1 (Bkf,δ) ≤ ω1 (f,δ) , and ω1 (Γkf,δ) ≤ ω1 (f,δ) . (41) Proof. (i) For Bk operators: Here t0f = f, therefore |(t0 (f)) (x − u) − (t0 (f)) (y − u)| = |f(x − u) − f(y − u)| ≤ ω1 (f, |x − y|) . (42) (ii) For Γk operators: We observe that |(t0 (f)) (x − u) − (t0 (f)) (y − u)| = ∣ ∣γf0 (x − u) − γ f 0 (y − u) ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wj ( f ( x − u + j n ) − f ( y − u + j n )) ∣ ∣ ∣ ∣ ∣ ∣ ≤ n∑ j=0 wj ∣ ∣ ∣ ∣ f ( x − u + j n ) − f ( y − u + j n )∣ ∣ ∣ ∣ ≤ ω1 (f, |x − y|)   n∑ j=0 wj   = ω1 (f, |x − y|) , (43) proving the claim. The operators Bk,Γk, k ∈ Z, converge to the unit operator with rates presented next. Theorem 3.7. For k ∈ Z, |(Bk (f)) (x) − f(x)| ≤ ω1 ( f, a 2k ) , and |(Γk (f)) (x) − f(x)| ≤ ω1 ( f, a+1 2k ) . (44) Proof. (i) For Bk operators: Here (t0 (f)) (u) = f(u) and sup u,y∈R |u−y|≤a |(t0 (f)) (u) − f(y)| = sup u,y∈R |u−y|≤a |f(u) − f(y)| = ω1 (f,a) , (45) and we use Theorem 3.4. (ii) For Γk operators: Here we see that sup u,y∈R |u−y|≤a |(t0 (f)) (u) − f(y)| = sup u,y∈R |u−y|≤a ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wjf ( u + j n ) − f(y) ∣ ∣ ∣ ∣ ∣ ∣ ≤ 10 George A. Anastassiou CUBO 20, 1 (2018) sup u,y∈R |u−y|≤a n∑ j=0 wj ∣ ∣ ∣ ∣ f ( u + j n ) − f(y) ∣ ∣ ∣ ∣ ≤ sup u,y∈R |u−y|≤a n∑ j=0 wjω1 ( f, ∣ ∣ ∣ ∣ u + j n − y ∣ ∣ ∣ ∣ ) ≤ (46) sup u,y∈R |u−y|≤a n∑ j=0 wjω1 ( f, j n + |u − y| ) ≤   n∑ j=0 wj  ω1 (f,1 + α) = ω1 (f,α + 1) . By (29) we are done. 4 Higher order of Approximation Here all are as in Section 3. See also earlier our work [1], and [2], Chapter 16. We give Theorem 4.1. Let f ∈ CN (R,R+), N ≥ 1. Consider the Shilkret-sublinear operators (Bkf) (x) = (N ∗ ) ∫a −a f ( x − u 2k ) ψ(u)dµ(u) , ∀ k ∈ Z, ∀ x ∈ R. Then |(Bkf) (x) − f(x)| ≤ N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + aN 2kNN! ω1 ( f(N), a 2k ) . (47) If f(N) is uniformly continuous or bounded and continuous, then as k → +∞ we obtain that (Bkf) (x) → f(x) pointwise with rates. Proof. Since f ∈ CN (R,R+), N ≥ 1, by Taylor’s formula we have f ( x − u 2k ) − f(x) = N∑ i=1 f(i) (x) i! ( − u 2k )i + (48) ∫x− u 2k x ( f(N) (t) − f(N) (x) ) ( x − u 2k − t )N−1 (N − 1) ! dt. Call Γu (x) := ∣ ∣ ∣ ∣ ∣ ∫x− u 2k x ( f(N) (t) − f(N) (x) ) ( x − u 2k − t )N−1 (N − 1) ! dt ∣ ∣ ∣ ∣ ∣ . (49) Next we estimate Γu (x), where u ∈ [−a,a] . i) Case of −a ≤ u ≤ 0, then x ≤ x − u 2k . Then Γu (x) ≤ ∫x− u 2k x ∣ ∣ ∣ f(N) (t) − f(N) (x) ∣ ∣ ∣ ( x − u 2k − t )N−1 (N − 1) ! dt ≤ CUBO 20, 1 (2018) Approximation by Shift Invariant Univariate . . . 11 ∫x− u 2k x ω1 ( f(N), |t − x| ) ( x − u 2k − t )N−1 (N − 1) ! dt ≤ ω1 ( f(N), |u| 2k )∫x− u 2k x ( x − u 2k − t )N−1 (N − 1) ! dt ≤ (50) ω1 ( f(N), a 2k ) ( − u 2k )N N! ≤ ω1 ( f(N), a 2k ) aN 2kNN! . That is, when −a ≤ u ≤ 0, then Γu (x) ≤ ω1 ( f(N), a 2k ) aN 2kNN! . (51) ii) Case of 0 ≤ u ≤ a, then x ≥ x − u 2k . Then Γu (x) = ∣ ∣ ∣ ∣ ∣ ∫x x− u 2k ( f(N) (t) − f(N) (x) ) ( t − x + u 2k )N−1 (N − 1) ! dt ∣ ∣ ∣ ∣ ∣ ≤ ∫x x− u 2k ∣ ∣ ∣ f(N) (t) − f(N) (x) ∣ ∣ ∣ ( t − x + u 2k )N−1 (N − 1) ! dt ≤ (52) ∫x x− u 2k ω1 ( f(N), |t − x| ) ( t − x + u 2k )N−1 (N − 1) ! dt ≤ ω1 ( f(N), |u| 2k )∫x x− u 2k ( t − x + u 2k )N−1 (N − 1) ! dt ≤ ω1 ( f(N), a 2k ) ( u 2k )N N! ≤ ω1 ( f(N), a 2k ) aN 2kNN! . (53) That is, when 0 ≤ u ≤ a, then Γu (x) ≤ ω1 ( f(N), a 2k ) aN 2kNN! . (54) We proved that Γu (x) ≤ ω1 ( f(N), a 2k ) aN 2kNN! := ρ ≥ 0, (55) ∀ k ∈ Z, ∀ x ∈ R, |u| ≤ a. By (48) we get that (|u| ≤ a) ∣ ∣ ∣ f ( x − u 2k ) − f(x) ∣ ∣ ∣ ≤ N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + ρ. (56) We observe that |(Bkf) (x) − f(x)| = 12 George A. Anastassiou CUBO 20, 1 (2018) ∣ ∣ ∣ ∣ (N∗) ∫a −a f ( x − u 2k ) ψ(u)dµ(u) − (N∗) ∫a −a f(x)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (57) (N∗) ∫a −a ∣ ∣ ∣ f ( x − u 2k ) − f(x) ∣ ∣ ∣ ψ(u)dµ(u) ≤ ( N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + ρ ) ( (N∗) ∫a −a ψ(u)dµ(u) ) (15) = ( N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + ρ ) · 1 = (58) N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! ai 2ki + aN 2kNN! ω1 ( f(N), a 2k ) , proving the claim. Corollary 4.2. Let f ∈ C1 (R,R+). Then |(Bkf) (x) − f(x)| ≤ a 2k ( |f′ (x)| + ω1 ( f′, a 2k )) , (59) ∀ k ∈ Z, ∀ x ∈ R. Proof. By (47) for N = 1. We also present Theorem 4.3. Let f ∈ CN (R,R+), N ≥ 1. Consider the Shilkret-sublinear operators (Γk (f)) (x) = (N ∗) ∫a −a   n∑ j=0 wjf ( ( x − u 2k ) + j 2kn )  ψ(u)dµ(u) , (60) ∀ k ∈ Z, ∀ x ∈ R. Then |(Γkf) (x) − f(x)| ≤ N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + (a + 1) N N!2kN ω1 ( f(N), a + 1 2k ) . (61) If f(N) is uniformly continuous or bounded and continuous, then as k → +∞ we obtain that (Γkf) (x) → f(x) , pointwise with rates. Corollary 4.4. Let f ∈ C1 (R,R+). Then |(Γkf) (x) − f(x)| ≤ (a + 1) 2k [ |f′ (x)| + ω1 ( f′, a + 1 2k )] , (62) ∀ k ∈ Z, ∀ x ∈ R. Proof. By (61) for N = 1. CUBO 20, 1 (2018) Approximation by Shift Invariant Univariate . . . 13 Proof. of Theorem 4.3. Since f ∈ CN (R), N ≥ 1, by Taylor’s formula we get n∑ j=0 wjf ( ( x − u 2k ) + j 2kn ) − f(x) = N∑ i=1 f(i) (x) i! n∑ j=0 wj ( − u 2k + j 2kn )i + (63) n∑ j=0 wj ∫(x− u 2k )+ j 2kn x ( f(N) (t) − f(N) (x) ) ( ( x − u 2k ) + j 2kn − t )N−1 (N − 1) ! dt. Call ε(x,u,j) := ∫(x− u 2k )+ j 2kn x ( f(N) (t) − f(N) (x) ) ( ( x − u 2k ) + j 2kn − t )N−1 (N − 1) ! dt. (64) We estimate ε(x,u,j). Here |u| ≤ a. i) case of u ≤ j n , iff u 2k ≤ j 2kn , iff x ≤ x − u 2k + j 2kn . Hence |ε(x,u,j)| ≤ ∫(x− u 2k )+ j 2kn x ∣ ∣ ∣ f(N) (t) − f(N) (x) ∣ ∣ ∣ ( ( x − u 2k ) + j 2kn − t )N−1 (N − 1) ! dt ≤ (65) ∫(x− u 2k )+ j 2kn x ω1 ( f(N), |t − x| ) ( ( x − u 2k ) + j 2kn − t )N−1 (N − 1) ! dt ≤ ω1 ( f(N), [ j 2kn − u 2k ])∫(x− u 2k )+ j 2kn x ( ( x − u 2k ) + j 2kn − t )N−1 (N − 1) ! dt ≤ ω1 ( f(N), a + 1 2k ) ( j 2kn − u 2k )N N! ≤ ω1 ( f(N), a + 1 2k ) (a + 1) N 2kNN! . (66) For u ≤ j n , we hve proved that |ε(x,u,j)| ≤ ω1 ( f(N), a + 1 2k ) (a + 1) N 2kNN! . (67) ii) case of u ≥ j n , iff u 2k ≥ j 2kn , iff x ≥ x − u 2k + j 2kn . We observe that |ε(x,u,j)| = 14 George A. Anastassiou CUBO 20, 1 (2018) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∫x (x− u 2k )+ j 2kn ( f(N) (t) − f(N) (x) ) ( t − [ ( x − u 2k ) + j 2kn ])N−1 (N − 1) ! dt ∣ ∣ ∣ ∣ ∣ ∣ ∣ ≤ (68) ∫x (x− u 2k )+ j 2kn ∣ ∣ ∣ f(N) (t) − f(N) (x) ∣ ∣ ∣ ( t − [ ( x − u 2k ) + j 2kn ])N−1 (N − 1) ! dt ≤ ∫x (x− u 2k )+ j 2kn ω1 ( f(N), |t − x| ) ( t − [ ( x − u 2k ) + j 2kn ])N−1 (N − 1) ! dt ≤ ω1 ( f(N), u 2k − j 2kn )∫x (x− u 2k )+ j 2kn ( t − [ ( x − u 2k ) + j 2kn ])N−1 (N − 1) ! dt ≤ ω1 ( f(N), a + 1 2k ) ( u 2k − j 2kn )N N! ≤ ω1 ( f(N), a + 1 2k ) (a + 1) N 2kNN! . (69) So when u ≥ j n , we proved that |ε(x,u,j)| ≤ ω1 ( f(N), a + 1 2k ) (a + 1) N 2kNN! . (70) Therefore it always holds |ε(x,u,j)| ≤ ω1 ( f(N), a + 1 2k ) (a + 1) N 2kNN! . (71) Consequently we derive n∑ j=0 wj |ε(x,u,j)| ≤ ω1 ( f(N), a + 1 2k ) (a + 1) N 2kNN! := ψ. (72) By (63) we find ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wjf ( ( x − u 2k ) + j 2kn ) − f(x) ∣ ∣ ∣ ∣ ∣ ∣ ≤ N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + ψ. (73) Therefore we get |(Γk (f)) (x) − f(x)| = ∣ ∣ ∣ ∣ ∣ ∣ (N∗) ∫a −a   n∑ j=0 wjf ( ( x − u 2k ) + j 2kn )  ψ(u)dµ(u) − (N∗) ∫a −a f(x)ψ(u)dµ(u) ∣ ∣ ∣ ∣ ∣ ∣ (20) ≤ (74) (N∗) ∫a −a ∣ ∣ ∣ ∣ ∣ ∣ n∑ j=0 wjf ( ( x − u 2k ) + j 2kn ) − f(x) ∣ ∣ ∣ ∣ ∣ ∣ ψ(u)dµ(u) (73) ≤ CUBO 20, 1 (2018) Approximation by Shift Invariant Univariate . . . 15 [ N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + ψ ] (N∗) ∫a −a ψ(u)dµ(u) (15) = [ N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + ψ ] · 1 = N∑ i=1 ∣ ∣f(i) (x) ∣ ∣ i! (a + 1) i 2ki + (a + 1) N 2kNN! ω1 ( f(N), a + 1 2k ) , (75) proving the claim. We finish with Corollary 4.5. Let f ∈ CN (R,R+), N ≥ 1, f (i) (x) = 0, i = 1, ...,N. Then i) |(Bk (f)) (x) − f(x)| ≤ aN 2kNN! ω1 ( f(N), a 2k ) , (76) and ii) |(Γk (f)) (x) − f(x)| ≤ (a + 1) N N!2kN ω1 ( f(N), a + 1 2k ) , (77) ∀ k ∈ Z, ∀ x ∈ R. Proof. By (47) and (61). Corollary 4.6. Let f ∈ C1 (R,R+), f ′ (x) = 0. Then i) |(Bk (f)) (x) − f(x)| ≤ a 2k ω1 ( f′, a 2k ) , (78) and ii) |(Γk (f)) (x) − f(x)| ≤ ( a + 1 2k ) ω1 ( f′, a + 1 2k ) , (79) ∀ k ∈ Z, ∀ x ∈ R. Proof. By (59) and (62). In inequalities (76)-(79) observe the high speed of convergence and approximation. 16 George A. Anastassiou CUBO 20, 1 (2018) 5 Appendix Let f ∈ CU (R,R+), and the positive sublinear Shilkret operator (M(f)) (x) := (N∗) ∫a −a f(x + u)ψ(u)dµ(u) , ∀ x ∈ R. (80) We observe the following (for any x,y ∈ R): |(M(f)) (x) − (M(f)) (y)| = ∣ ∣ ∣ ∣ (N∗) ∫a −a f(x + u)ψ(u)dµ(u) − (N∗) ∫a −a f(y + u)ψ(u)dµ(u) ∣ ∣ ∣ ∣ (20) ≤ (N∗) ∫a −a |f(x + u) − f(y + u)|ψ(u)dµ(u) ≤ ω1 (f, |x − y|) ( (N∗) ∫a −a ψ(u)dµ(u) ) (15) = ω1 (f, |x − y|) · 1 = ω1 (f, |x − y|) . (81) Therefore it holds the global smoothness preservation property: ω1 (M(f) ,δ) ≤ ω1 (f,δ) , ∀ δ > 0. (82) References [1] G.A. Anastassiou, High order Approximation by univariate shift-invariant integral operators, in: R. Agarwal, D. O’Regan (eds.), Nonlinear Analysis and Applications, 2 volumes, vol. I, pp. 141-164, Kluwer, Dordrecht, (2003). [2] G.A. Anastassiou, Intelligent Mathematics: Computational Analysis, Springer, Heidelberg, New York, 2011. [3] G.A. Anastassiou, S. Gal, Approximation Theory, Birkhauser, Boston, Basel, Berlin, 2000. [4] G.A. Anastassiou, H.H. Gonska, On some shift invariant integral operators, univariate case, Ann. Polon. Math., LXI, 3, (1995), 225-243. [5] Niel Shilkret, Maxitive measure and integration, Indagationes Mathematicae, 33 (1971), 109- 116. Introduction Background Univariate Theory Higher order of Approximation Appendix