() @ Appl. Gen. Topol. 18, no. 2 (2017), 361-375doi:10.4995/agt.2017.7263 c© AGT, UPV, 2017 Uniform reconstruction of continuous functions with the RAFU method Eduardo Corbacho Department of Mathematics, IES Sáenz de Buruaga, 06800 Mérida, Spain. (ecorbachoc@gmail.com) Communicated by E. A. Sánchez-Pérez Abstract The RAFU (radical functions) method can be used to obtain the uni- form reconstruction of a continuous function from its values at some of the points of partitions of a closed interval. In this work we will prove that we can reconstruct a continuous function from average sam- ples of these points, from linear combinations of them and from local average samples given by convolution. A uniform error bound of or- der O ( h 3 2 ) ) + ω (h) with the step size h will be established. If these data are unknown but approximate values of them are known, uni- form reconstruction will be also possible. Error estimates of order O ( h 3 2 ) + ω (h) + η with noise level η will be given. The case of a non-uniform net will be treated. Examples and algorithms will be also shown. 2010 MSC: 41A30; 37L65; 41A65. Keywords: RAFU method; RAFU approximation; uniform approximation. 1. Introduction Suppose that the interval [a, b] is partitioned by the n + 1 equally spaced points a = x0 < x1 < ... < xn = b, such that xi = a + ih, for i = 0, ...,n, with h = b−a n . Consider, for each natural n and k = 1, ..., n − 1 the functions Fn (xk, x) = 2n+1 √ xk−a+ 2n+1 √ x−xk 2n+1 √ b−xk+ 2n+1 √ xk−a defined in [a, b]. Then, given f ∈ C [a, b], the Received 12 February 2017 – Accepted 10 April 2017 http://dx.doi.org/10.4995/agt.2017.7263 E. Corbacho sequence of radical functions (Cn)n defined in [a, b] as (1.1) Cn(x) = f(x0) + n ∑ j=2 [f(xj) − f(xj−1)] · Fn (xj−1, x) converges uniformly to f in [a, b] as n → +∞. We define the RAFU method on approximation to an arbitrary function f to any approximation procedure that uses functions Cn defined as (1.1) to approach the function f. As for the RAFU method, the reader can see [7, 8, 9, 10]. In [8] we proved that the called RAFU linear space is uniformly dense in C [a, b] by using a S-separation condition due to Blasco-Moltó [3] or its equiv- alent S ′ -separation condition due to Garrido-Montalvo [13]. Moreover, this linear space can be used as an example of approximation by series in the work of Gassó-Hernández-Rojas [14]. The main goal of this work is to use this method to approach a continuous function f from average samples of the values f(xj), from linear combinations of f(xj) and f(xj+1) and from local average samples given by ( χ[− h2 , h 2 ] ⋆ f ) (x). In all these cases we will establish a uniform error bound of order O ( h 3 2 ) ) + ω (h). Moreover, if the data f(xj) or linear combinations or average samples or local average samples are unknown, but approximate values of them are known, that is to say, for the case of the noise data, we will prove that it is also possible to obtain the reconstruction of the function f. Error bounds of order O ( h 3 2 ) + ω (h) + η, where η is the noise level, will be given. Such problems often occur in environmental science, mathematical statistics, digital image, mechanics, numerical analysis and electricity ; we refer to [1, 5, 11, 12, 15, 16, 17] for more details. Spline functions have been used to approximate a function f in some of these practical applications by other authors, H. Behforooz [1, 2], E.J.M. Delhez [11], F.G. Lang and X.P. Xu [18] and T. Zhanlav and R. Mijiddorj [19]. In these papers it was necessary to suppose that the function f had several derivatives and error estimations were not given in some of them. Given approximate integral values of a function f belongs to H1(a, b), the usual Sobolev space, over subintervals [xj, xj+1], J. Huang and Y. Chen [16] studied the problem of reconstructing the function f from these data. In this work a regularization method was required and the error bound was established in L2 norm. In [4] J. Bustamante, R.C. Castillo and A.F. Collar studied a polynomial approximation of functions from their approximate values at nodes. In this case a regularization method was also required. In this paper, with the only condition that f ∈ C [a, b], our purpose will be to employ the RAFU method to demonstrate that it is possible its reconstruction in all the mentioned cases. Moreover, the computational methods involved will be very easy to implement. The paper does not impress with the difficulties it overcomes. It does not contain complicated calculations or reasonings, but c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 362 Uniform reconstruction of continuous functions with RAFU method we think that the importance of this technique to solve all these problems will balance the deficiency of difficulties. This approximation method can rather apply to functions with low smoothness. The uniform stability of this approx- imation method improves the instability of the interpolation by polynomials; we refer to [7, 8, 9, 10] for more details. Until now the main drawback of the RAFU method on approximation has been its low accuracy for smooth functions. In Section 2 we will improve the degree of uniform approximation given in [7] and this is an important contribution of this work. In fact, the uniform error estimates that RAFU approximation provides can be better than ω ( f, π n+1 ) which is, as far as we know, the best uniform error bound known until now in order to approximate continuous functions by algebraic polynomials in [−1, 1] ([6] p. 147). Moreover, in the case of RAFU approximation, the approximating continuous functions are always known. In Section 3, as elementary corollaries, we will solve all our main purposes. By using 4.1.0.0 Mathematica program, we will give in Section 4 some concise algorithms used in this paper. This approximation procedurre can also be used when the set of the points that define the subdivisions of the interval [a, b] is not a uniform net. In Section 5 we will study this case. 2. Improvement of the degree of uniform approximation with the RAFU Method Maybe, until now the main drawback of the RAFU method on approxima- tion has been the order of the convergence of the sequence (Cn)n to the function f. Here, we will improve it by using a subsequence of the sequence (2n + 1) n of the index of the roots of the funcions Fn (xk, x) which appear in (1.1). In this section we will consider partitions Pn = {x0 = a, x1, ..., xn = b} of [a, b] with xj = a + j · b−an , j = 0,...,n. Moreover, each interval [xk−1, xk] of length b−a n will be divided into three equal parts of length b−a 3n : [ xk−1, xk−1 + b−a 3n ] , [ xk−1 + b−a 3n , xk − b−a3n ] , [ xk − b−a3n , xk ] Lemma 2.1. For n ≥ 2, it follows that: 1. Let 1 ≤ p ≤ n − 1 be, p integer. Then ∣ ∣ 2n2+1 √ p n − 1 ∣ ∣ ≤ 1 n √ n 2. ∣ ∣ ∣ 2n2+1 √ 1 3 − 1 ∣ ∣ ∣ ≤ 1 n √ n 3. ∣ ∣ ∣ 2n2+1 √ 1 3n − 1 ∣ ∣ ∣ ≤ 1 n √ n 4. Let 1 ≤ p ≤ n − 1 be, p integer. Then ∣ ∣ 2n2+1 √ n − p − 1 ∣ ∣ ≤ 1 n √ n Lemma 2.2. Let Pn a partition of [a, b]. For each natural n and k = 1, ..., n − 1, we define in [a, b] the function Fn,2 (xk, x) = 2n2+1 √ xk − a + 2n 2+1 √ x − xk 2n2+1 √ b − xk + 2n 2+1 √ xk − a Then, it satisfies that 0 ≤ Fn,2 (xk, x) ≤ 1. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 363 E. Corbacho The values of the functions Fn,2 (xk, x), for any k, do not depend on a and b. In fact, considering x = a + αx b−a n for an αx, it verifies that Fn,2 (xk, x) = 2n2+1 √ ( a + k b−a n ) − a + 2n2+1 √ ( a + αx b−a n ) − ( a + k b−a n ) 2n2+1 √ ( a + n b−a n ) − ( a + k b−a n ) + 2n 2+1 √ ( a + k b−a n ) − a = = 2n2+1 √ k b−a n + 2n 2+1 √ (αx − k) b−an 2n2+1 √ (n − k) b−a n + 2n 2+1 √ k b−a n = 2n2+1 √ k + 2n 2+1 √ (αx − k) 2n2+1 √ (n − k) + 2n 2+1 √ k Lemma 2.3. Let Pn be a partition of [a, b] and x ∈ [ xk−1 + b−a 3n , xk − b−a3n ] . Then, for any k = 1, ..., n − 1, it follows that 1. If x − xk > 0 then 2n2+1 √ 1 n + 2n 2+1 √ 1 3n 2 ≤ Fn,2 (xk, x) ≤ 1 2. If x − xk < 0 then 0 ≤ Fn,2 (xk, x) ≤ 2n2+1 √ n−1− 2n 2+1 √ 1 3 2 Moreover, these bounds are valid as x ∈ [ a, x1 − b−a3n ] , x ∈ [ xn−1 + b−a 3n , b ] and x ∈ ( xj − b−a3n , xj + b−a 3n ) with j 6= k. Lemma 2.4. Let Pn be a partition of [a, b]. If x ∈ [ xk−1 + b−a 3n , xk − b−a3n ] with k = 1, ..., n − 1, x ∈ [ a, x1 − b−a3n ] , x ∈ [ xn−1 + b−a 3n , b ] , or x ∈ ( xj − b−a3n , xj + b−a 3n ) where j 6= k then for all n ≥ 2 it follows that 1. ∣ ∣ ∣ ∣ 2n2+1 √ 1 n + 2n 2+1 √ 1 3n 2 − 1 ∣ ∣ ∣ ∣ ≤ 1 n √ n 2. ∣ ∣ ∣ ∣ 2n2+1 √ n−1− 2n 2+1 √ 1 3 2 − 0 ∣ ∣ ∣ ∣ ≤ 1 n √ n Proofs of Lemmas 2.1, 2.2, 2.3 and 2.4 can be obtained by elementary esti- mates. Proposition 2.5. Let Pn be a partition of [a, b] and En the step function defined by (2.1) En(x) = k1 · χ[a,x1] + n−1 ∑ p=2 kp · χ(xp−1,xp] + kn · χ(xn−1,b] Let Cn be the radical function associated to En defined by (2.2) Cn(x) = k1 + n ∑ j=2 [kj − kj−1] · Fn,2 (xj−1, x) Then, for all n ≥ 2 it follows that: (1) |Cn(x) − En(x)| ≤ 2(Mn−mn)n√n , x ∈ [a, b] \ ∪ n−1 k=1 ( xk − b−a3n , xk + b−a 3n ) (2) |Cn(x) − [kj(1 − αx) + kj+1αx]| ≤ 2(Mn−mn)n√n , x ∈ ( xj − b−a3n , xj + b−a 3n ) , j = 1,..., n − 1 where Mn and mn are the maximum and the minimum of the kj and αx ∈ (0, 1) is a number which depends upon x. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 364 Uniform reconstruction of continuous functions with RAFU method Proof. The proof is simlar to the proof given in [7]. Part 1. This part is proved considering three possible cases. Case 1. Suppose that x ∈ [ xj−1 + b−a 3n , xj − b−a3n ] , j = 2, ...,n − 1 then |Cn(x) − En(x)| = |Cn(x) − kj| = ∣ ∣ ∣ ∣ ∣ Cn(x) − ( k1 + j ∑ p=2 [kp − kp−1] ) ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] [1 − Fn,2 (xp, x)] + n ∑ p=j+1 [kp − kp−1] [0 − Fn,2 (xp, x)] ∣ ∣ ∣ ∣ ∣ ∣ ≤ by Lemmas 2.3 and 2.4 ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] · 1 n √ n + n ∑ p=j+1 [kp − kp−1] · − 1 n √ n ∣ ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] · 1 n √ n + n ∑ p=j+1 [kp−1 − kp] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ∣ ≤ 1 n √ n |[kj − k1] + [kj − kn]| ≤ 2 (Mn − mn)√ n Case 2. Suppose that x ∈ [ a, x1 − b−a3n ] . Then x − xp < 0, p = 1, ..., n − 1 and proceeding as in Case 1 and by using Lemmas 2.3 and 2.4, we obtain |Cn(x) − En(x)| = |Cn(x) − k1| = ∣ ∣ ∣ ∣ ∣ Cn(x) − ( k1 + j ∑ p=2 [kp − kp−1] ) ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] [0 − Fn,2 (xp, x)] ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] · − 1 n √ n ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp−1 − kp] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ≤ 2 (Mn − mn) n √ n Case 3. Suppose that x ∈ [ xn−1 + b−a 3n , b ] . Then x− xp > 0, p = 1, ..., n− 1 and proceeding as in Case 1, we can put |Cn(x) − En(x)| = |Cn(x) − kn| = ∣ ∣ ∣ ∣ ∣ Cn(x) − ( k1 + j ∑ p=2 [kp − kp−1] ) ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] [1 − Fn,2 (xp, x)] ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∣ n ∑ p=2 [kp − kp−1] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ≤ 2 (Mn − mn) n √ n taking into account Lemmas 2.3 and 2.4. Part 2. Suppose that x ∈ ( xj − b−a3n , xj + b−a 3n ) , j = 1,..., n − 1, then c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 365 E. Corbacho [kj(1 − αx) + kj+1αx] − Cn(x) = [kj + (kj+1 − kj) αx] − Cn(x) = k1 − k1 + j ∑ p=2 [kp − kp−1] [1 − Fn,2 (xp−1, x)] + [kj+1 − kj] [αx − Fn,2 (xj, x)] + n ∑ p=j+1 [kp+1 − kp] [0 − Fn,2 (xp, x)] Since for x ∈ ( xj − b−a3n , xj + b−a 3n ) it follows that 0 < Fn,2 (xj, x) < 1 we can put αx = Fn,2 (xj, x). So that, from Lemmas 2.3 and 2.4, taking absolute value and proceeding as in Case 1, |Cn(x) − [kj(1 − αx) + kj+1αx]| ≤ ∣ ∣ ∣ ∣ ∣ ∣ j ∑ p=2 [kp − kp−1] · 1 n √ n + n ∑ p=j+1 [kp − kp+1] · 1 n √ n ∣ ∣ ∣ ∣ ∣ ∣ = 1 n √ n |[kj − k1] + [kj+1 − kn]| ≤ 2 (Mn − mn) n √ n � Theorem 2.6. Let f be a continuous function defined in [a, b]. Then there exists a sequence of radical functions (Cn)n defined in [a, b] as in (2.2) such that |Cn(x) − f(x)| ≤ 2 (M − m) n √ n + ω ( b − a n ) for all n ≥ 2 and x ∈ [a, b] being M and m the maximum and the minimum of f in [a, b] respectively and ω ( b−a n ) its modulus of continuty. Proof. For each n ≥ 2, let Pn be a partition of [a, b], let En be the step function defined by En(x) =          f(a) x ∈ [a, x1] f(x2) x ∈ (x1, x2] ... f(b) x ∈ (xn−1, b] and let Cn be the corresponding radical function defined from En as (2.2). If x ∈ [a, b] \ ∪n−1 k=1 ( xk − b−a3n , xk + b−a 3n ) then, |Cn(x) − f(x)| = |Cn(x) − En(x) + En(x) − f(x)| ≤ 2 (M − m) n √ n + |En(x) − f(x)| = 2 (M − m) n √ n + |f(xj) − f(x)| ≤ c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 366 Uniform reconstruction of continuous functions with RAFU method 2 (M − m) n √ n + ω ( b − a n ) taking into account that En(x) = f(xj) for some j and Proposition 2.5. If x ∈ ∪n−1 k=1 ( xk − b−a3n , xk + b−a 3n ) , Proposition 2.5 applies and we can choose an appropiate index j to obtain |Cn(x) − f(x)| ≤ |Cn(x) − [f(xj) (1 − αx) + f(xj+1)αx]| + |[f(xj) (1 − αx) + f(xj+1)αx] − f(x)| ≤ 2 (M − m) n √ n + |[f(xj) (1 − αx) + f(xj+1)αx] − [f(x) (1 − αx) + f(x)αx]| ≤ 2 (M − m) n √ n + |f(xj) − f(x)| (1 − αx) + |f(xj+1) − f(x)| (1 − αx) ≤ 2 (M − m) n √ n + ω ( b − a n ) (1 − αx + αx) = 2 (M − m) n √ n + ω ( b − a n ) � Remark 2.7. It is well-known (see for instance [6] pp. 147) that if f ∈ C [−1, 1], then there exist an algebraic polynomial Pn of degree ≤ n such that for all x ∈ C [−1, 1], |Pn(x) − f(x)| ≤ ω ( π n + 1 ) As far as we know, this error estimate is the best possible currently known. By means of Theorem 2.6, we have proved an analogous result by using radical continuous functions. In case of the interval [−1, 1], the error bound becomes 2(M−m) n √ n + ω ( 2 n ) . So, depending on the function, this error estimate can be better than error bound in algebraic polynomial approximation. Moreover, RAFU method provides the explicit form of the function which approximate to the funtion f for each n. However in the case of algebraic polynomials this does not happen. Therefore, this is an important contribution of this work. 3. Main results 3.1. Uniform reconstruction of f from average samples. The following corollary provides a sequence uniformly convergent to the original function f and a uniform error bound. Observe that the uniform error bound is the same as Theorem 2.6. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 367 E. Corbacho Corollary 3.1. Under the hypothesis of Theorem 2.6, if the data ki of the step function (2.1) are substituted by ki = f(xi1)n1+...+f(xip)np n1+...+np , x1q ∈ [a, x1] or xiq ∈ (xi−1, xi], i = 2, ..., n, q = 1, ..., p, n1 + ... + nq 6= 0 then |Cn(x) − f(x)| ≤ 2 (M − m) n √ n + ω ( b − a n ) for all x ∈ [a, b], n ≥ 2 and where Cn(x) is defined as 2.2 but from the new data ki. Proof. In the proof of Proposition 2.5 we can put ki = f(xi1)n1+...+f(xip)np n1+...+np , i = 1, ..., n and the same result holds for M and m. Moreover, if we define the functions En in Proposition 2.5 from ki = f(xi1)n1+...+f(xip)np n1+...+np , i = 1, ..., n and we put that f(x) = f(x)n1+...+f(x)np n1+...+np , then we can easily check that Corollary 3.1 is true considering now Cn defined as (2.2) but from ki, i = 1, ..., n. � Example 3.2. In Figure 1 we show the approximation to the piecewise contin- uous function f(x) defined by 0.5 if x ∈ [0, 0.39), 0.5x−0.185 0.02 if x ∈ [0.39, 0.41), 1 if x ∈ [0.41, 0.69), −0.5x+0.365 0.02 if x ∈ [0.69, 0.71) and 0.5 if x ∈ [0.71, 1] from ki = f(xi1)+...+f(xi15) 15 , i = 1, ..., 200 considering C200(x). (a) Approximating function and f (b) Approximation error Figure 1. Uniform reconstruction from average samples. Remark 3.3. If ni = 1, we have the usual average values. 3.2. Uniform reconstruction of f from approximate values. In [4] J. Bustamante, R. C. Castillo and A. F. Collar solved this problem by means of a regularization method. In [7] we studied this case but here we give a uniform error bound. The reader can compare our error bound with the estimation of the error shown in [4]. When we do not know the values f(xi) but the data f(xi)+ηi, with |ηi| < η for a fixed η > 0 are known, then the following result can be useful to obtain an approximation of the function f. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 368 Uniform reconstruction of continuous functions with RAFU method Corollary 3.4. Under the hypothesis of Theorem 2.6, if the data ki of the step function (2.1) are changed for ki = f(xi) + ηi, being |ηi| < η, i = 1,...,n then |Cn(x) − f(x)| ≤ 2 (M − m + η) n √ n + ω ( b − a n ) + η for all x ∈ [a, b], n ≥ 2 and where Cn(x) is defined as (2.2) but from the new data ki. Proof. With these data ki, i = 1,..., n we can obtain the error bound 2(M−m+η) n √ n in Proposition 2.5. Moreover, if we change f(xi) for ki = f(xi)+ηi, i = 1, ..., n in the proof of Theorem 2.6 then the new error bound becomes 2(M−m+η) n √ n + ω ( b−a n ) + η. � Example 3.5. Approximation to the piecewise continuous function function f(x) defined by 4x if x ∈ [0, 0.25), 1 if x ∈ [0.25, 0.5), −0.5x+0.5 0.02 if x ∈ [0.5, 0.75) and 0.5 if x ∈ [0.71, 1] using the data ki = f(xi)+ηi, i = 1, ..., n, with |ηi| ≤ 1100 ( ηi = 1 100 sin4πxi ) and considering C180(x) (Figure 2). (a) Approximating function and f (b) Approximation error Figure 2. Uniform reconstruction from approximate values. 3.3. Uniform reconstruction of f from local average samples. In many applications it is more realistic to assume that the available samples are local average samples near a certain x. We consider the special case in which we know data of the type (3.1) ( χ[−h,h] ⋆ f ) (x) = ∫ +∞ −∞ χ[−h,h](y)f(x − y)dy = ∫ x+h x−h f(z)dz where ⋆ denotes the convolution of the functions χ[−h,h] and f. Sometimes we deal with phenomena which involve a function and its integral. For example, in mechanics, the velocity v(t) and the displacement s(t), or the acceleration a(t) and the velocity v(t); in statistics, the probability density function and the cumulative distribution function and in electricity, the current function I(t) and the charge function q(t) are some real examples about this consideration. The tasks are to approximate the function f from integral values as (3.1) and c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 369 E. Corbacho to give error bounds for this aproximation. There have been only a few research papers to deal with these problems; see for example, H. Behforooz [1, 2], E.J.M. Delhez [11], F.G. Lang and X.P. Xu [18] and T. Zhanlav and R. Mijiddorj [19]. In these papers it was necessary to suppose that the function f had several derivatives and error estimations were not given in some of them. Here, with Corollary 3.6, RAFU method solves easily the problem of the reconstruction of the function from the integral values and provides a uniform error bound for this reconstruction with the only condition that f ∈ C [a, b]. On the other hand, let △ be a subdivision of the interval [a, b] with grids a = x0 < x1 < ... < xn = b whose mesh size is denoted by h = max1≤i≤nhi, hi = xi − xi−1, 1 ≤ i ≤ n and Mi(f) = 1hi ∫ xi xi−1 f(x)dx. In practice, due to the measurement error, the exact values Mi(f) are unknown but we know approximate average values ui, 1 ≤ i ≤ n such that |ui − Mi(f)| < δ where δ is a positive constant describing the level of error of the data. In [16] J. Huang and Y. Chen proposed a regularization method for solving the problem (P): given the approximate values ui, 1 ≤ i ≤ n satisfying the previous condition how does one reconstruct the original function f efficiently? They established the rigorous error estimates in L2 norm for functions f ∈ H1 (a, b) where H1 (a, b) is the usual Sobolev space consisting of all L2 (a, b)-integrable functions whose 1-order weak derivative are also L2 (a, b)-integrable. For f ∈ H1 (a, b). They solved this problem in terms of the Tikhonov regularization method. In this work, by means of Corollaries 3.4 and 3.6, we establish another solution of problem (P) in the uniform norm for all f ∈ C [a, b]. Note that H1 (a, b) is continuously embedded in C [a, b]. Our solution does not need regularization. See Algorithm 4.2 and Figure 5. Corollary 3.6. With the hypothesis of Theorem 2.6, if the data ki of the step function (2.1) are defined by ki = ∫ x̃i+h x̃i−h f(z)dz 2h , with [x̃1 − h, x̃1 + h] ⊆ [a, x1] or [x̃i − h, x̃i + h] ⊆ (xi−1, xi], i = 2, ..., n, then |Cn(x) − f(x)| ≤ 2 (M − m) n √ n + ω ( b − a n ) for all x ∈ [a, b], n ≥ 2 and where Cn(x) is defined as (2.2) but from the new data ki. Proof. We can put that ∫ x̃i+h x̃i−h f(z)dz = f(zi)2h for some value zi ∈ [x̃i − h, x̃i + h] by the integral properties because f is continuous. Then, ki = f(zi) for all i and we finish with the same proof of Theorem 2.6. � Example 3.7. Consider the special case given by x̃i = xi−1+xi 2 , i = 1, ..., n and h = b−a n to approximate the continuous function f(x) defined by |sin8πx| if x ∈ [0, 0.5) and x − 0.5 if x ∈ [0.5, 1] from local average samples ki = ∫ x̃i+h x̃i−h f(z)dz 2h , i = 1, ..., 180 with C144(x) (Figure 3). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 370 Uniform reconstruction of continuous functions with RAFU method (a) Approximating function and f (b) Approximation error Figure 3. Uniform reconstruction from local average samples. 3.4. Uniform reconstruction of f from linear combinations. Corollary 3.8. Under the hypothesis of Theorem 2.6, if the values ki of the step function (2.1) are defined by ki = f(x̃i)−f(x̃i−1) x̃i−x̃i−1 ·(x ′ i − x̃i−1)+f(x̃i−1) with x′1 ∈ [x̃0, x̃1] ⊆ [a, x1] or x′i ∈ [x̃i−1, x̃i] ⊆ (xi−1, xi], i = 2, ..., n, then |Cn(x) − f(x)| ≤ 2 (M − m) n √ n + ω ( b − a n ) for all x ∈ [a, b], n ≥ 2 and where Cn(x) is defined as (2.2) but from the new data ki. Proof. Since f ∈ C [a, b], there exists a point x′′i in each interval [x̃i−1, x̃i] such that ki = f(x ′′ i ) for all i = 1, ..., n. Then, this proof becomes the proof of Theorem 2.6. � Example 3.9. Consider the special case in which x̃i = xi for all i to ap- proximate the piecewise continuous function f(x) defined by sin4πx if x ∈ [0, 1 20 ) ∪ [1 5 , 3 10 ) ∪ [ 9 20 , 1 2 ), sinπ 5 if x ∈ [ 1 20 , 1 5 ), sin2π 5 if x ∈ [ 3 10 , 9 20 ) and |sin4πx| if x ∈ [1 2 , 1] from the data ki = f(xi)−f(xi−1) xi−xi−1 · (x ′ i − xi−1) + f(xi−1) and the values x′i = xi−i+xi 2 by using C150(x) (Figure 4). (a) Approximating function and f (b) Approximation error Figure 4. Uniform reconstruction from linear combinations. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 371 E. Corbacho 4. Algorithms We show three algorithms by using the 4.1.0.0 Mathematica program. Algorithm 4.1. Uniform reconstruction from average samples. f[x−] := As Example 3.2; a = 0; b = 1; n = 3000; h = b−a n ; v = n 15 ; t = T able[a + 15 · h · i, {i, 0, v}]; d = T able[f[a + j · h], {j, 0, n − 1}]; For[i = 1, i + +, Di = ∑ 15 m=1 dm+15∗(i−1) 15 ]; k = T able[ ∑ 15 m=1 dm+15∗(i−1) 15 , {i, 1, v}];tt = Length[t]; kk = Length[k]; For[i = 2, i ≤ kk, i + +, Mi = (ki−ki−1)· 2n2+1 √ ti−t1 2n2+1 √ ttt−ti+ 2n 2+1 √ ti−t1 ]; For[i = 2, i ≤ kk, i + +, Ni = (ki−ki−1)2n2+1√ttt−ti+ 2n2+1√ti−t1 ]; g[x−] = k1 + ∑ kk i=2 ( Mi + Ni · 2n 2+1 √ Abs[x − ti] · Sign (x − ti) ) ; Plot[{f[x], g[x]}, {x, t1, ttt}] Plot[Abs[f[x] − g[x]], {x, t1, ttt}] Corollary 3.4 can be used together with Corollaries 3.1, 3.6 or 3.8. For instance, in Algorithm 4.2, we use Corollaries 3.4 and 3.6 to reconstruct uni- formly an irregular function f from approximate integral values (Figure 5). Here, Random denotes a random number with uniform distribution on [−1, 1] and 0.01 is the considered relative error level of the data. RAFU method pro- vides this easy solution to the Problem (P) suggested by J. Huang and Y. Chen in [16]. (a) Approximating function and f (b) Approximation error Figure 5. Uniform reconstruction from approximate inte- gral values. Algorithm 4.2. Uniform approximation from approximate integral values. f[x−] := If[0 ≤ x < 0.25, x, If[0.25 ≤ x < 0.5, −x + 0.5, If[0.5 ≤ x < 0.75, x − 0, 5, If[0.75 ≤ x ≤ 1, −x + 1]]]]; a = 0; b = 1; n = 100; h = b−a n ; hh = b−a 2·n ; t = T able[a + j · h, {j, 0, n}]; c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 372 Uniform reconstruction of continuous functions with RAFU method k = T able[ N[Integrate[f[x],{x,a+j·h+a+(j+1)·h 2 −hh, a+j·h+a+(j+1)·h 2 +hh}]] 2·hh ·(1 + 0.01 · Random[Real, {−1, 1}]), {j, 0, n − 1}]; tt = Length[t]; kk = Length[k]; For[i = 2, i ≤ kk, i + +, Mi = (ki−ki−1)· 2n2+1 √ ti−t1 2n2+1 √ ttt−ti+ 2n 2+1 √ ti−t1 ]; For[i = 2, i ≤ kk, i + +, Ni = (ki−ki−1)2n2+1√ttt−ti+ 2n2+1√ti−t1 ]; g[x−] = k1 + ∑ kk i=2 ( Mi + Ni · 2n 2+1 √ Abs[x − ti] · Sign (x − ti) ) ; Plot[{f[x], g[x]}, {x, t1, ttt}] Plot[Abs[f[x] − g[x]], {x, t1, ttt}] Algorithm 4.3. Uniform approximation from linear combinations. f[x−] := As Example 3.9; a = 0; b = 1; n = 150; h = b−a n ; t = T able[a + j · h, {j, 0, n}]; k = T able[ f[a+(j+1)·h]−f[a+j·h] h · h 2 + f[a + j · h], {j, 0, n − 1}]; tt = Length[t]; kk = Length[k]; For[i = 2, i ≤ kk, i + +, Mi = (ki−ki−1)· 2n2+1 √ ti−t1 2n2+1 √ ttt−ti+ 2n 2+1 √ ti−t1 ]; For[i = 2, i ≤ kk, i + +, Ni = (ki−ki−1)2n2+1√ttt−ti+ 2n2+1√ti−t1 ]; g[x−] = k1 + ∑ kk i=2 ( Mi + Ni · 2n 2+1 √ Abs[x − ti] · Sign (x − ti) ) ; Plot[{f[x], g[x]}, {x, t1, ttt}] Plot[Abs[f[x] − g[x]], {x, t1, ttt}] 5. Uniform reconstruction of f from a non-uniform net From now on, we will consider partitions Pn = {x0 = a, x1, ..., xs = b} of [a, b] with non-uniformly spaced data. Lemma 5.1. Let K be a positive integer. Then, for n ≥ 2 it verifies that ∣ ∣ ∣ 2n2+1 √ nK − 1 ∣ ∣ ∣ ≤ 2 K −1 n √ n and ∣ ∣ ∣ 2n2+1 √ 1 nK − 1 ∣ ∣ ∣ ≤ K n √ n Proof. By induction on K. Case K = 1 can be obtained by elementary esti- mates. Then, we finishes taking into account that ∣ ∣ ∣ 2n2+1 √ n±K − 1 ∣ ∣ ∣ = ∣ ∣ ∣ 2n2+1 √ n±K − 2n 2+1 √ n±1 + 2n2+1 √ n±1 − 1 ∣ ∣ ∣ � Lemma 5.2. Let Pn = {a = x0, x1, ..., xs = b} be a partition of [a, b] with δ (s) = min1≤j≤s |xj − xj−1|. Then, for any k = 1, ..., s − 1 and x ∈ [a, b] \ ( xk − δ(s)3 , xk + δ(s) 3 ) it follows that: (1) 2n 2+1 √ δ(s) b−a 1+ 2n 2+1 √ 1 3 2 ≤ Fn,2 (xk, x) ≤ 1 if x − xk > 0 (2) 0 ≤ Fn,2 (xk, x) ≤ 2n2+1 √ b−a δ(s) − 2n 2+1 √ 1 3 2 if x − xk < 0 c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 373 E. Corbacho The proof can be obtained by elementary estimates. Lemma 5.3. Let K ≥ 2 be a positive integer such that 3(b−a) nK ≤ δ (s). Then, for all n ≥ 2, it verifies that (1) ∣ ∣ ∣ ∣ 1 − 2n2+1 √ δ(s) b−a 1+ 2n 2+1 √ 1 3 2 ∣ ∣ ∣ ∣ ≤ K n √ n (2) ∣ ∣ ∣ ∣ ∣ 2n2+1 √ b−a δ(s) − 2n 2+1 √ 1 3 2 − 0 ∣ ∣ ∣ ∣ ∣ ≤ 2 K−1 n √ n The proof can be obtained easily from Lemmas 2.1 and 5.1. Proposition 5.4. Let Ps = {a = x0, x1, ..., xs = b} be a partition of [a, b] and let Es be a step function defined in [a, b] by Es(x) = k1 · χ[x0, x1] + s ∑ i=2 ki · χ(xi−1, xi], ki real numbers If 3(b−a) nK ≤ δ (s), being δ (s) = min1≤j≤s |xj − xj−1|and K ≥ 2 a positive integer, then for all n ≥ 2 it follows that: (1) |Cn(x) − Es(x)| ≤ 2 K(Ms−ms) n √ n if x ∈ [a, b] \ ∪s−1j=1 ( xj − δ(s)3 , xj + δ(s) 3 ) (2) |Cn(x) − [kj(1 − αx) + kj+1αx]| ≤ 2 K(Ms−ms) n √ n if j = 1, ..., s − 1 and x ∈ ( xj − δ(s)3 , xj + δ(s) 3 ) . where Ms and ms are the maximum and the minimum of the kj, αx ∈ (0, 1) is a number which depends only on x and (Cn)n is the sequence of radical functions associated to Es defined as in (2.2). Proof. It is analogous to the proof of Proposition 2.5 but now we use Lemmas 5.1, 5.2 and 5.3. � Theorem 5.5. Let Pn = {a = x0, x1, ..., xsn = b} be a partition of [a, b] with δ (sn) = min1≤j≤sn |xj − xj−1| and ∆ (sn) = max1≤j≤sn |xj − xj−1| such that 3(b−a) nK ≤ δ (sn) ≤ ∆ (sn) ≤ h being h = b−an and K ≥ 2 a positive integer. Let f be a continuous function defined in [a, b]. Then there exists a sequence (Cn)n defined in [a, b] as in (2.2) such that |Cn(x) − f(x)| ≤ 2K (M − m) n √ n + ω ( b − a n ) for all n ≥ 2 and x ∈ [a, b], being M, m and ω ( b−a n ) as usual. Proof. Similiar to the proof of Proposition 2.5. Here Proposition 5.4 applies. � In the same way the results in Section 3 have been obtained from Theorem 2.6, similar results to Section 3 can be derived from Theorem 5.5 for the case of non-uniform net and this is another important contribution of this work. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 374 Uniform reconstruction of continuous functions with RAFU method Acknowledgements. 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