Spline interpolation is often preferred over polynomial interpolation, because the interpolation error can be made small even when using low degree polynomials for the spline [1, 2, 3]. Many research results were obtained about spline lacunary interpolation, such as, spline $ (0, 2, 3) $ and $ (0, 2, 4) $ lacunary interpolation [4], Varma's $ (0, 2) $ case of spline [5] and the minimizing error bounds in lacunary interpolation by spline for $ (0, 1, 4) $ and $ (0, 2) $ case given by Saceed [6] and Jawmer [7].

A fractal is a detailed, recursive and infinitely self-similar mathematical set in which Hausdorff dimension strictly exceeds its topological dimension [8]. Fractals exhibit similar patterns at increasingly small scales [9]. If this replication is exactly the same at every scale, it is called a self-similar pattern [10, 11]. Fractal was widely used as a research tool for generating natural-looking shapes such as mountains, trees, clouds, etc. There were increasing researches in fractal functions and their applications over the last three decades. Fractal function is a good choice for modeling natural object [12], and fractal interpolation techniques provide good deterministic representations of complex phenomena. Barnsley [13] and Hutchinson [14] are pioneers in terms of applying fractal function to interpolate sets of data. The fractal interpolation problems based on Hermite functions and cubic spline are solved in ref. [15] and ref. [16]. Viswanathan [17] gave the fractal spline $ (0, 4) $ and $ (0, 2) $ lacunary interpolation, and Viswanathan [17] also did research on the fractal $ (0, 2) $ lacunary interpolation by using the spline function of ref. [5]. Inspired by their research, this paper devoted to research the general fractal $ (0, m) $ lacunary interpolation with the scaling factors based on the extrapolation spline.

The paper is organized as follows. In Section 2, by using extrapolation algorithm we deduce the explicit formulas of generalized $ (0, m) $ lacunary interpolation spline function. The error estimate is also given and numerical example is presented to demonstrate the effectiveness of our proposed method. In Section 3, we use the spline function which has been obtained in Section 2 to interpolate the fractal function. We find that when scaling factors fulfill $ \|\alpha_{k}\|_{r}\leq \frac{1}{(2N)^{m}} $, there is fractal function $ T_{\Delta, b}^{\alpha}\in C^{m}[0, 1] $ satisfying, $ {T_{\Delta, b}^{\alpha}}^{(p)}(x_{k}) = S_{\Delta}^{(p)}(x_{k}), \ p = 0, m; \ {T_{\Delta, b}^{\alpha}}^{(r)}(x_{N}) = S_{\Delta}^{(r)}(x_{N}), \ r = 0, 1, \cdots, m $. Approximation orders for the proposed class of fractal interpolation and their derivatives are discussed. Numerical simulations are also carried out to show the validity and efficiency of our proposed method. The concluding remarks are given in Section 4.

In this section, the spline explicit formula for the $ (0, m) $ lacunary interpolation function is constructed by using extrapolation algorithm. The method we adopt is similar to those given in literature [19]. An example for illuminating the details and efficiency of the procedure is provided, and the error estimate will be shown in Theorem 2.2.

For a given partition $ \Delta: x_{1} < x_2 < \cdots < x_N, x_{k + 1} - x_k = h_{k}, \ I = [x_{1}, x_{N}] $, and real values $ \{y_{1}, y_2, \cdots, y_{N - 1}; y^{(m)}_{1}, y^{(m)}_2, \cdots, y^{(m)}_{N - 1};\ y_{N}, y^{(1)}_{N}, \cdots, y^{(m)}_{N}\} $.

We define the spline function $ S_{\Delta}(x) $ in each subinterval such that

$ \begin{equation} S_{\Delta} \left( x \right) = \left\{ \begin{array}{l} S_1 \left( x \right), x_1 \le x \le x_2 , \\ S_k \left( x \right), x_k \le x \le x_{k + 1} , k = 2, 3, \cdots, N - 2, \\ S_{N - 1} \left( x \right), x_{N - 1} \le x \le x_N , \\ \end{array} \right. \end{equation} $

(2.1)

where

$ \begin{eqnarray} S_{k}(x)& = &y_{k}+y^{(m)}_{k}\frac{(x-x_{k})^{m}}{m!}+\sum\limits_{r = 1}^{m-1}a_{k, r}\frac{(x-x_{k})^{r}}{r!}+a_{k, m+1}\frac{(x-x_{k})^{m+1}}{(m+1)!} \\&&+a_{k, m+2}\frac{(x-x_{k})^{m+2}}{(m+2)!}. \end{eqnarray} $

(2.2)

$ S_{\Delta}(x) $ has the following conditions

$ \begin{equation} \left\{ {\begin{array}{l} {S_{\Delta}(x_k ) = f (x_k ) = y_k, \ k = 1, 2, 3, \cdots, N - 1} \\ {S_{\Delta}^{(m)} (x_k ) = f^{(m)} (x_k ) = y_k^{(m)}, \ k = 1, 2, 3, \cdots, N - 1} \\ {S_{\Delta}^{(r)} (x_N ) = f^{(r)} (x_N ) = y_N^{(r)}, \ r = 0, 1, 2, \cdots, m}. \\ \end{array}} \right. \end{equation} $

(2.3)

and satisfying

$ \{S_{\Delta} \left( x \right) \in C^m(I); S \left( x \right) \in P_{\left( {m + 2} \right)}(x) , x \in \left[ {x_k , x_{k + 1} } \right], k = 1, 2, \cdots , N - 1\}. $

The coefficients of these polynomials can be determined by the following conditions

$ \begin{equation} \left\{ {\begin{array}{l} {S_k \left( {x_{k + 1} } \right) = S_{k + 1} \left( {x_{k + 1} } \right) = y_{k + 1} , k = 1, 2, \cdots, N-2}\qquad \\ {S_k^{(m)} (x_{k + 1} ) = S_{k + 1}^{(m)} (x_{k + 1} ) = y_{k + 1}^{(m)} , k = 1, 2, \cdots, N-2, }\qquad \\ {S_k^{(q)} (x_{k + 1} ) = S_{k + 1}^{(q)} (x_{k + 1} ), 1 \le q<m, k = 1, 2, \cdots, N-2, } \\ {S_{N - 1}^{(r)} (x_N ) = y_N^{(r)} , r = 0, 1, \cdots, m.} \qquad \qquad \qquad \qquad \qquad \end{array}} \right. \end{equation} $

(2.4)

For $ k = 1, 2, \cdots, N-1 $, we denote

$ \begin{equation} \left\{ \begin{array}{l} \alpha _{k, 0} = y_{k + 1} - y_k - hy'_k - \frac{{h^2 }}{2}y_k^{\left( 2 \right)} - \frac{{h^3 }}{{3!}}y_k^{\left( 3 \right)} - \cdots - \frac{{h^m }}{{m!}}y_k^{\left( m \right)} , \\ \alpha _{k, 1} = y'_{k + 1} - y'_k - hy''_k - \frac{{h^2 }}{2}y_k^{\left( 3 \right)} - \frac{{h^3 }}{{3!}}y_k^{\left( 4 \right)} - \cdots - \frac{{h^{m - 1} }}{{\left( {m - 1} \right)!}}y_k^{\left( m \right)} , \\ \alpha _{k, 2} = y''_{k + 1} - y''_k - hy_k^{\left( 3 \right)} - \frac{{h^2 }}{2}y_k^{\left( 4 \right)} - \frac{{h^3 }}{{3!}}y_k^{\left( 5 \right)} - \cdots - \frac{{h^{m - 2} }}{{\left( {m - 2} \right)!}}y_k^{\left( m \right)} , \\ \quad \quad \quad \vdots \\ \alpha _{k, m - 1} = y_{k + 1}^{\left( {m - 1} \right)} - y_k^{\left( {m - 1} \right)} - hy_k^{\left( m \right)} , \\ \alpha _{k, m} = y_{k + 1}^{\left( m \right)} - y_k^{\left( m \right)} . \end{array} \right. \end{equation} $

(2.5)

To obtain the unknown coefficients $ a_{k, j}\; (k = 1, 2, \cdots, N-1, j = 1, 2, \cdots, m+2, j \neq m) $, we take the following five steps

Step 1 For $ k = N-1 $, we have

$ \begin{array}{l} \alpha _{N - 1, 0} = h\left( {\alpha _{N - 1, 1} - y'_{N - 1} } \right) + \frac{{h^2 }}{2}\left( {\alpha _{N - 1, 2} - y''_{N - 1} } \right) + \cdots + \frac{{h^{m - 1} }}{{\left( {m - 1} \right)!}}\alpha _{N - 1, m - 1} \\ \qquad \qquad +\frac{{h^{m + 1} }}{{\left( {m + 1} \right)!}}\alpha _{N - 1, m + 1} +\frac{{h^{m + 2} }}{{\left( {m + 2} \right)!}}\alpha _{N - 1, m + 2} ; \\ \alpha _{N - 1, 1} = \left( {\alpha _{N - 1, 1} - y'_{N - 1} } \right) + h\left( {\alpha _{N - 1, 2} - y''_{N - 1} } \right) + \cdots + \frac{{h^{m - 2} }}{{\left( {m - 2} \right)!}}\alpha _{N - 1, m - 1} \\ \qquad\qquad +\frac{{h^m }}{{m!}}\alpha _{N - 1, m + 1} + \frac{{h^{m + 1} }}{{\left( {m + 1} \right)!}}\alpha _{N- 1, m + 2} ; \\ \alpha _{N - 1, 2} = \left( {\alpha _{N - 1, 2} - y''_{N - 1} } \right) + h\left( {\alpha _{N - 1, 2} - y'''_{N - 1} } \right) + \cdots + \frac{{h^{m - 3} }}{{\left( {m - 3} \right)!}}\alpha _{N - 1, m - 1} \\ \qquad\qquad +\frac{{h^{m - 1} }}{{\left( {m - 1} \right)!}}\alpha _{N - 1, m + 1} + \frac{{h^m }}{{m!}}\alpha _{N - 1, m + 2} ; \qquad\qquad \vdots \\ \alpha _{N - 1, m - 1} = \left( {\alpha _{N - 1, m - 1} - y_{N - 1}^{\left( {m - 1} \right)} } \right) + \frac{{h^2 }}{2}\alpha _{N - 1, m + 1} + \frac{{h^3 }}{{3!}}\alpha _{N - 1, m + 2} ; \\ \alpha _{N - 1, m} = h\alpha _{N - 1, m + 1} + \frac{{h^2 }}{2}\alpha _{N - 1, m + 2} . \end{array} $

Step 2 Solving the equation systems of Step 1, we obtain

$ \begin{equation} a_{N - 1, r} - y_{N - 1}^{\left( r \right)} = A_{0r} \alpha _{N - 1, 0} + A_{1r} \alpha _{N - 1, 1} + \cdots A_{mr} \alpha _{N - 1, m}, \end{equation} $

(2.6)

$ \begin{equation} a_{N - 1, s} = B_{0s} \alpha _{N - 1, 0} + B_{1s} \alpha _{N - 1, 1} + \cdots B_{ms}\alpha _{N - 1, m}, \qquad\quad \end{equation} $

(2.7)

where $ r = 1, 2, \cdots, m-1;s = m+1, m+2. $

Step 3 For $ k = 1, 2, \cdots, N-2 $, establishing the equations systems

$ \begin{array}{l} \alpha _{k, 0} = h( {a_{k, 1} - y'_k } ) + \frac{{h^2 }}{2}( {a_{k, 2} - y''_k } ) + \cdots + \frac{{h^{m - 1} }}{{( {m - 1} )!}}( {a_{k, m - 1} - y_k^{( {m - 1} )} } ) \\ \qquad \qquad +\frac{{h^{m + 1} }}{{( {m + 1} )!}}a_{k, m + 1}+\frac{{h^{m+ 2} }}{{( {m + 2} )!}}a_{k, m + 2} , \\ \alpha _{k, 1} = ( {a_{k, 1} - y'_k } ) - ( {a_{k + 1, 1} - y'_{k + 1} }) + h( {a_{k, 2} - y''_k } ) + \frac{{h^2 }}{2}( {a_{k, 3} - y^{(3)}_k } ) \\ {\rm{ }} \qquad \qquad + \cdots +\frac{{h^{m - 2} }}{{( {m - 2} )!}}( {a_{k, m - 1} - y_k^{( {m - 1} )} } )+\frac{{h^m }}{{m!}}a_{k, m + 1} + \frac{{h^{m + 1} }}{{( {m + 1} )!}}a_{k, m + 2} , \\ \alpha _{k, 2} = ( {a_{k, 2} - y''_k } ) - ( {a_{k + 1, 1} - y''_{k + 1} } ) + h( {a_{k, 3} - y^{(3)}_k } ) + \frac{{h^2 }}{2}( {a_{k, 4} - y_k^{( 4 )} } ) \\ {\rm{ }} \qquad \qquad +\cdots +\frac{{h^{m - 3} }}{{( {m - 3} )!}}( {a_{k, m - 1} - y_k^{( {m - 1} )} } )+ \frac{{h^{m - 1} }}{{( {m - 1} )!}}a_{k, m + 1} + \frac{{h^m }}{{m!}}a_{k, m + 2} , \\ \qquad \qquad \vdots \\ \alpha _{k, m - 1} = ( {\alpha _{k, m - 1} - y_k^{(m - 1)} } ) - ( {\alpha _{k + 1, m - 1} - y_k^{(m - 1)} } ) + \frac{{h^2 }}{2}a_{k, m + 1} + \frac{{h^3 }}{3}a_{k, m + 2} , \\ \alpha _{k, m} = ha_{k, m + 1} + \frac{{h^2 }}{2}a_{k, m + 2} . \\ \end{array} $

Step 4 Combinating the equations of $ \alpha _{k, 0} $ and $ \alpha _{k, m} $ in Step 3, we obtain

$ \begin{array}{l} a_{k, m + 1} = C_{1, m + 1} ( {a_{k, 1} - y'_k } ) + \cdots + C_{m - 1, m + 1} ( {a_{k, m - 1} - y_k^{( {m - 1} )} } ), \\ a_{k, m + 2} = D_{1, m + 1} ( {a_{k, 1} - y'_k } ) + \cdots + D_{m - 1, m + 1} ( {a_{k, m - 1} - y_k^{( {m - 1} )} } ). \\ \end{array} $

Step 5 Solving the rest parts of Step 3.

The solutions from step 4 will be substituted by the remaining equations $ a_{k, m+1}, a_{k, m+2} $, starting with $ ({a_{k, j} - y_k^{(j)} }) - ({a_{k + 1, j} - y_{k + 1}^{(j)} }) + \cdots $. In view of the coefficient matrix of the equation systems (2.6) is non-singular, thus the unknowns $ a_{N-1, 1}, a_{N-1, 2}, \cdots, $ $ a_{N-1, m-1} $ be solved and obtained. Repeating above iteration, all undetermined coefficients of $ S_{\Delta}(x) $ can be calculated. Therefore, we obtain the following conclusion for spline $ (0, m) $ Birkhoff interpolation problem.

Theorem 2.1 Assume that $ f(x)\in C^{m}(I) $. For a uniform partition $ \Delta: = \{x_1, x_2, \cdots, x_N:x_k < x_{k+1}, \ k = 1, 2, \cdots, N-1\} $ of the compact interval $ I = [x_{1}, x_{N}] $, there is a unique spline function $ S_{\Delta}(x)\in S_{(m+2)}^{m}(\Delta) $, satisfying the $ (0, m) $ lacunary interpolation conditions

$ S_{\Delta}(x_{k}) = f(x_{k}), \ S_{\Delta}^{(m)}(x_{k}) = f^{(m)}(x_{k}) \ (1\leq k\leq N-1); $

$ S_{\Delta}^{(r)}(x_{N}) = f^{(r)}(x_{N}), \ r = 0, 1, \cdots, m. $

From the Theorem 2 of ref. [20], we can get the following approximation theory.

Theorem 2.2 Assume that $ f(x)\in C^{m}[-1, 1] $, $ S_{\Delta}(x) $ is the $ (0, m) $ lacunary spline function defined by eq.(2.1). When $ 0\leq r\leq m $ and $ -1\leq x\leq 1 $, we have

$ \|f(x)-S_{\Delta}(x)\|_{\infty}\leq C\Delta^{m}_{m+2}(x)w(f^{(m)}, \Delta_{m+2}(x)), $

where $ C $ is constant independent of $ x $, $ \Delta_{m+2}(x) = \frac{\sqrt[]{1-x^2}}{m+2}+\frac{1}{(m+2)^2} $, $ \|f\|_{\infty} = \max\{|f(x)|:x\in [-1, 1]\} $, $ w(f^{(r)}, \delta) = \max\{|f^{(r)}(x_1)-f^{(r)}(x_2)|:|x_1-x_2|\leq\delta\} $ is the maximum norm modulus of continuity of $ f^{(r)}(x) $ on the interval $ [-1, 1] $.

Assume $ f(x) = x*\cos(x)-\sin(x) $. For a uniform partition $ \Delta: = \{x_1, x_2, \cdots, x_N:x_k < x_{k+1}, k = 1, 2, \cdots, N\} $ of the interval $ I = [-1, 1] $, applying the result of spline $ (0, m) $ Birkhoff interpolation, we take $ m = 2 $, and consider the absolute error $ e_{k, p} = |a_{k, p}-y_{k}^{(p)}| $ and the error curve $ R(x) = |f(x)-S_{\Delta}(x)| $ for the nodes $ x_{k} $. We list a table and present error curve graphs. The numerical results are listed in Table 1, and the function fitting graphs are listed in Fig. 1-2. It is observed that our proposed spline function is effective and stable.

For positive integer $ N > 2 $, consider a data set $ \{(x_n, y_n)\in \mathbb{R}^2:n = 0, 1, 2, ..., N\} $, where $ x_{0} < x_{1} < x_{2} < \cdots < x_{N} $. Let $ I = [x_{0}, x_{N}] $, $ I_{n} = [x_{n-1}, x_{n}] $, $ n\in J = \{1, 2, \cdots, N\} $ and $ L_{n}:I\rightarrow I_{n} $ be homeomorphic affinities such as

$ |L_{n}(x)-L_{n}(x^{*})|\leq l_{n}|x-x^{*}| $

for all $ x, x^{*} \in I $ and $ 0\leq l_{n} < 1 $.

Consider $ N-1 $ continuous maps $ F_{n}:I\times \mathbb{R}\rightarrow \mathbb{R} $ satisfying the following conditions

$ |F_{n}(x, y)-F_{n}(x, y^{*})|\leq r_{n}|y-y^{*}|; \ F_{n}(x_{0}, \ y_{0}) = y_{n-1}, \ F_{n}(x_{N}, \ y_{N}) = y_{n}( n\in J) $

for all $ x\in I, y, y^{*}\in \mathbb{R} $, and for some $ 0\leq r_{n} < 1 $.

Defining functions $ w_{n}:\ I\times \mathbb{R}\rightarrow I_{n}\times \mathbb{R}\subseteq I\times \mathbb{R}, \ w_{n}(x, y) = (L_{n}(x), \ F_{n}(x, y)), \ n\in J $. The $ \{I\times \mathbb{R}, \ w_n: \ n = 1, 2, ..., N\} $ is called an Iterated Function System (IFS) [13]. From ref. [17], we know that the IFS has a unique attractor $ G(g) $ which is the graph of a continuous function $ g:I\rightarrow \mathbb{R} $ satisfying $ g(x_{n}) = y_{n} \ (n = 0, 1, 2, \cdots, N) $, and function $ g $ is the fixed point of the Read-Bajraktarević (RB) operator $ T:C_{y_{0}, y_{N}}(I)\rightarrow C_{y_{0}, y_{N}}(I) $ defined

$ T(g)(x) = F_{n}(L_{n}^{-1}(x), \ g(L_{n}^{-1}(x))), \ x\in I_{n}, \ n = 1, 2, \cdots, N. $

The above function $ g $ is called Fractal Interpolation Function(FIF) corresponding to the data $ \{(x_{n}, y_{n}):\ n = 0, 1, 2, \cdots, N\} $ and it satisfies

$ \begin{equation} g(x) = F_{n}(L_{n}^{-1}(x), \ g(L_{n}^{-1}(x))), \ x\in I_{n}, \ n\in J \end{equation} $

(3.1)

For a partition $ \Delta: = \{x_{0}, x_{1}, x_{2}, \cdots, x_{N}:x_{n-1} < x_{n}, n\in J\} $ of $ I = [x_{0}, x_{N}] $, $ x_{n}-x_{n-1}: = h_{n} $, and the data set $ \{(x_{n}, f(x_{n})), \ n = 0, 1, 2, \cdots, N\} $, suppose that $ F_{n}(x, y) = \alpha_{n}(y-b(x))+f(L_{n}(x)) $, where $ \alpha_{n} $ is called scaling factors, satisfying $ |\alpha_{n}| < 1 $, and $ b:I\rightarrow \mathbb{R} $ is a continuous function, such as $ b\neq f $, $ b(x_{0}) = f(x_{0}) $, $ b(x_{N}) = f(x_{N}) $.

Thus, we ensure the existence of a fractal function $ (Tf)(x_{n}) = f(x_{n}) \ (n = 0, 1, 2, \cdots, N) $. From (3.1), we can obtain

$ (Tf)(x) = f(x)+\alpha_{n}(L_{n}^{-1}(x))\cdot(f-b)(L_{n}^{-1}(x)), x\in I_{n}, n\in J. $

The most widely studied functions $ L_n(x) $ so far are defined by the following,

$ L_n(x) = d_{n}x+b_{n} $

with

$ d_{n} = \frac{x_{n}-x_{n-1}}{x_{N}-x_{0}}, \quad b_{n} = \frac{x_{N}x_{n-1}-x_{n}x_{0}}{x_{N}-x_{0}} $

In many cases, the data are evenly sampled $ h = x_{n}-x_{n-1}, x_{N}-x_{0} = Nh $.

Example 1 Consider function $ f(x) = (2x^{2}-5x+3)\sin(10x) $. For a uniform partition $ \Delta: = \{x_{0}, x_{1}, \cdots, x_{6} :x_{n-1} < x_{n}, n = 1, 2, \cdots, 5\} $ of $ [0, 1] $ with step size $ h = \frac{1}{5} $, $ b(x) = x^{2}f(x) $. The left graph of Figure 3 shows the fractal function with $ \alpha_{n} = 0.3 \ (n = 1, 2, \cdots, 5) $. The right graph presents the fractal function with scaling variable $ \alpha_{1} = 0.4+\frac{x}{8}, \ \alpha_{2} = 0.3-\frac{\sin(x)}{4}, \ \alpha_{3} = 0.6-\frac{x^{2}}{4}, \ \alpha_{4} = 0.3+\frac{\cos(x)}{4}, \ \alpha_{5} = 0.3+\frac{x}{5}. $

We denote $ \alpha = (\alpha_{1}, \alpha_{2}, \cdots, \alpha_{N}) $, $ \|\alpha\|_{r} = \max\{\|\alpha_{k}^{(p)}\|_{\infty}:p = 0, 1, r\} $.

Theorem 3.1 Assume that $ f\in C^{m}(I) $, $ \Delta: = \{x_{0}, x_{1}, \cdots, x_{N} :x_{n-1} < x_{n}, n\in J\} $ be an arbitrary partition of $ I = [x_{0}, x_{N}] $. There are suitable smooth functions $ b $ and $ \alpha_{n} $, when scaling factors $ \alpha_{n} $ fulfill $ \|\alpha_{n}\|_{\infty} < \max\limits_{1 \leq n \leq N}\{|\frac{{d_{n}}}{2}|^{m}\} $, the corresponding fractal function $ T_{\Delta, b}^{\alpha}\in C^{m}(I) $ satisfies

$ {T_{\Delta, b}^{\alpha}}(x_{n}) = S_{\Delta}(x_{n});\ {T_{\Delta, b}^{\alpha}}^{(m)}(x_{n}) = S_{\Delta}^{(m)}(x_{n}), \ 1\leq n \leq N-1 $

with boundary conditions,

$ {T_{\Delta, b}^{\alpha}}^{(p)}(x_{0}) = S_{\Delta}^{(p)}(x_{0}) \ (p = 0, m);\ {T_{\Delta, b}^{\alpha}}^{(r)}(x_{N}) = S_{\Delta}^{(r)}(x_{N}), \ r = 0, 1, \cdots, m. $

Proof For convenience, in the following $ k = 0, 1, 2, 3, \cdots, N-1 $. Let $ S_{\Delta}(x) $ be prescribed in eq.(2.1), we choose smooth function $ b\in C^{m}(I) $ to satisfy

$ b^{(r)}(x_{0}) = S_{\Delta}^{(r)}(x_{0}), \ b^{(r)}(x_{N}) = S_{\Delta}^{(r)}(x_{N}), \ r = 0, 1, \cdots, m. $

Consider the operator $ T:C^{m}(I)\rightarrow C^{m}(I) $

$ \begin{equation} (Tf)(x) = S_{\Delta}(x)+\alpha_{n}(L_{n} ^{-1}(x))[f(L_{n} ^{-1}(x))-b(L_{n} ^{-1}(x))], \ x\in I_{n}, \ n\in J. \end{equation} $

(3.2)

We can deduce that

$ \begin{equation} (Tf)^{(r)}(x) = S_{\Delta}^{(r)}(x)+d_{n}^{-r}\sum\limits_{i = 0}^{r}\binom{r}{i}(f-b)^{(i)}(L_{n} ^{-1}(x))\alpha_{n}^{(r-i)}(L_{n} ^{-1}(x)). \end{equation} $

(3.3)

Using the conditions on $ b $ and the properties $ L_{n}(x_{N}) = L_{n+1}(x_{0}) = x_{n}(n\in J) $ of $ L_{n} $, we obtain $ (Tf)^{(r)}(x_{n}^{+}) = (Tf)^{(r)}(x_{n}^{-}) = S_{\Delta}^{(r)}(x_{n}) $ for $ r = 0, 1, \cdots, m. $ From (3.2) and (3.3), we get

$ (Tf)^{(r)}(x_{n}) = S_{\Delta}^{(r)}(x_{n}), \ (Tf)^{(r)}(x_{N}) = S_{\Delta}^{(r)}(x_{N});\ r = 0, 1, \cdots, m. $

For $ f, g\in C^{m}(I) $, $ r = 1, 2, \cdots, m $, $ x\in I_{n} $, we have

$ \begin{equation} \begin{aligned} |(Tf)^{(r)}-(Tg)^{(r)}|& = d_{n}^{-r}|\sum\limits_{i = 0}^{r}\binom{r}{i}(f-g)^{(i)}(L_{n}^{-1}(x))\alpha_{n}^{(r-i)}(L_{i}^{-1}(x))|\\ &\leq |d_{n}^{-r}|\|\alpha\|_{r}\|f-g\|_{r}\sum\limits_{i = 0}^{r}\binom{r}{i}. \end{aligned} \end{equation} $

(3.4)

When $ r = 0 $,

$ \|(Tf)(x)-(Tg)(x)\|_{\infty}\leq \|\alpha\|_{\infty}\|f-g\|_{\infty}, $

Therefore, when $ \|\alpha\|_{\infty} < \max\{|\frac{d_{n}}{2}|^{m}:n\in J\} $, $ T $ is a contraction, by the Banach fixed point theorem, $ T $ has a unique fixed point $ T_{\Delta, b}^{\alpha} $ that satisfies

$ (T_{\Delta, b}^{\alpha})(x) = S_{\Delta}(x)+\alpha_{n}(L_{n} ^{-1}(x))[T_{\Delta, b}^{\alpha}(L_{n} ^{-1}(x))-b(L_{n} ^{-1}(x))], \ x\in I_{n}, \ n\in J. $

It is apparently seen from the above discussion that

$ {T_{\Delta, b}^{\alpha}}^{(p)}(x_{n}) = S_{\Delta}^{(p)}(x_{n}), \ p = 0, m;\ {T_{\Delta, b}^{\alpha}}^{(r)}(x_{N}) = S_{\Delta}^{(r)}(x_{N}), $

$ r = 0, 1, \cdots, m; \ n = 0, 1, 2, \cdots, N-1. $

we will give error estimates for $ \|(T_{\Delta, b}^{\alpha}-S_{\Delta})^{r}\|_{\infty} $ and obtain convergence results while choosing suitable perturbation parameter $ \alpha_{n} $. Consider $ I = [0, 1] $, $ 0 = x_{0} < x_{1} < \cdots < x_{N} = 1 $, and $ h = x_{n}-x_{n-1} = \frac{1}{N} $, $ N\geq 3, n = 0, 1, 2, \cdots, N $. $ I_{n} = [\frac{n-1}{N}, \frac{n}{N}] $, $ n\in \{1, 2, \cdots, N\} $. Let homeomorphic affinities $ L_{n} $ be $ L_{n}(x) = \frac{x+(n-1)}{N} $. We have the following convergence estimates.

Theorem 3.2 For the uniform equidistance partition of $ I = [0, 1] $, the following bounds for the fractal function and its derivatives hold:

$ \|T_{\Delta, b}^{\alpha}-S_{\Delta}\|_{\infty} \leq \frac{\|\alpha\|_{\infty}}{1 -\|\alpha\|_{\infty}}\|S_{\Delta}-b\|_{\infty}, $

$ \|((T_{\Delta, b}^{\alpha})-S_{\Delta})^{(r)}\|_{\infty}\leq \frac{(2^{r+1}-1)N^{r}\|\alpha\|_{r}}{1 -N^{r}\|\alpha\|_{\infty}}\|S_{\Delta}-b\|_{r}, \ r = 1, 2, \cdots, m. $

Proof From the equation $ (T_{\Delta, b}^{\alpha})(x) = S_{\Delta}(x)+\alpha_{k}(L_{n} ^{-1}(x))[T_{\Delta, b}^{\alpha}(L_{n} ^{-1}(x))-b(L_{n} ^{-1}(x))], \ x\in I_{n}, \ n\in J, $ we have

$ |(T_{\Delta, b}^{\alpha})-S_{\Delta}| \leq\|\alpha\|_{\infty}[\|T_{\Delta, b}^{\alpha}-S_{\Delta}]\|_{\infty}+\|S_{\Delta}-b\|_{\infty}]. $

Then

$ \begin{equation} \begin{aligned} \|(T_{\Delta, b}^{\alpha})-S_{\Delta}\|_{\infty} & \leq\frac{\|\alpha\|_{\infty}}{1-\|\alpha\|_{\infty}}\|S_{\Delta}-b\|_{\infty}. \end{aligned} \end{equation} $

(3.5)

It's evident that the following equality holds

$ (T_{\Delta, b}^{\alpha})^{(r)}(x) = S_{\Delta}^{(r)}(x)+N^{r}\sum\limits_{i = 0}^{r}\binom{r}{i}((T_{\Delta, b}^{\alpha})-b)^{(i)}(L_{n} ^{-1}(x))\alpha_{n}^{(r-i)}(L_{n} ^{-1}(x)), \ r = 0, 1, \cdots, m. $

Therefore, we have

$ \begin{eqnarray} |(T_{\Delta, b}^{\alpha})^{(r)}-S_{\Delta}^{(r)}| &\leq&\{N^{r}(\|\alpha_{n}\|_{r}\|T_{\Delta, b}^{\alpha}-b\|_{r-1}(2^{r}-1)+ \|\alpha_{n}\|_{\infty}\|\|(T_{\Delta, b}^{\alpha})^{(r)}-b^{(r)}\|_{\infty})\\ & \leq& N^{r}(\|\alpha\|_{r}[\|T_{\Delta, b}^{\alpha}-S_{\Delta}\|_{r-1}+\|S_{\Delta}-b\|_{r-1}](2^{r}-1)+ \|\alpha\|_{\infty}\|[\|(T_{\Delta, b}^{\alpha})^{(r)}\\ &&-S_{\Delta}^{(r)}\|_{\infty}+\|S_{\Delta}^{(r)}-b^{(r)}\|_{\infty}]\}. \end{eqnarray} $

(3.6)

It can be easy deduced that

$ \begin{equation} \begin{aligned} |(T_{\Delta, b}^{\alpha})^{(r)}-S_{\Delta}^{(r)}| &\leq \frac{N^{r}\|\alpha\|_{r}}{1 -N^{r}\|\alpha\|_{\infty}}[(2^{r}-1)\|T_{\Delta, b}^{\alpha}-S_{\Delta}\|_{r-1}+2^{r}\|S_{\Delta}-b\|_{r}]. \end{aligned} \end{equation} $

(3.7)

When $ r = 1 $, using (3.7) and (3.5), we have

$ \begin{multline} |(T_{\Delta, b}^{\alpha})^{(1)}-S_{\Delta}^{(1)}|\leq \frac{N\|\alpha\|_{1}}{1 -N\|\alpha\|_{\infty}} [\|T_{\Delta, b}^{\alpha}-S_{\Delta}\|_{\infty}+2\|S_{\Delta}-b\|_{1}] \leq \frac{3N\|\alpha\|_{1}}{1 -N\|\alpha\|_{\infty}}\|S_{\Delta}-b\|_{1}. \end{multline} $

(3.8)

On account of $ \|\alpha\|_{\infty}\leq 3N\|\alpha\|_{1} $, it follows that

$ \|T_{\Delta, b}^{\alpha}-S_{\Delta}\|_{1} = \max\{\|(T_{\Delta, b}^{\alpha})^{(1)}-S_{\Delta}^{(1)}\|_{\infty}, \|T_{\Delta, b}^{\alpha}-S_{\Delta}\|_{\infty}\}\leq \frac{3N\|\alpha\|_{1}}{1 -N\|\alpha\|_{\infty}}\|S_{\Delta}-b\|_{1}. $

Similarly, when $ r = 2 $, we obtain

$ |(T_{\Delta, b}^{\alpha})^{(2)}-S_{\Delta}^{(2)}|\leq \frac{N^{2}\|\alpha\|_{2}}{1 -N^{2}\|\alpha\|_{\infty}} [3\|T_{\Delta, b}^{\alpha}-S_{\Delta}\|_{1}+4\|S_{\Delta}-b\|_{2}] \leq \frac{7N^{2}\|\alpha\|_{2}}{1 -N^{2}\|\alpha\|_{\infty}}\|S_{\Delta}-b\|_{2}. $

$ \|(T_{\Delta, b}^{\alpha})-S_{\Delta}\|_{2}\leq \frac{7N^{2}\|\alpha\|_{2}}{1 -N^{2}\|\alpha\|_{\infty}}\|S_{\Delta}-b\|_{2}. $

Summarizing the above process, we can obtain the result.

Theorem 3.3 Assume that $ f\in C^{m}([0, 1]) $. $ T_{\Delta, b}^{\alpha} $ is the fractal interpolation function given in Theorem 3.1. For uniform equidistance partition on $ [0, 1] $, when the scaling factor $ \alpha_{n} $ satisfies $ \|\alpha_{n}^{(r)}\|_{\infty}\leq \frac{1}{(2N)^m}, $ $ r = 0, 1, \cdots, m $, we have

$ \|(T_{\Delta, b}^{\alpha}-f)^{(r)}\|_{\infty}\leq C_{r}N^{-(m-r)}+C_{r}\triangle^{m-r}_{m+2}(x)\omega(f^{(m)}, \triangle_{m+2}(x)) $

as $ N\rightarrow \infty, r = 0, 1, \cdots, m. $

Proof Using the triangle inequality $ \|(T_{\Delta, b}^{\alpha}-f)^{(r)}\|_{\infty}\leq \|(T_{\Delta, b}^{\alpha}-S_{\triangle})^{(r)}\|_{\infty}+\|(S_{\Delta}-f)^{(r)}\|_{\infty}, $ and from result of ref. [17], for $ n\geq j $, $ k = 0, 1, 2, \cdots, j $, $ |f^{(k)}(x)-P^{(k)}_{n}(x)|\leq C_{j}\triangle^{j-k}_{n}(x)\omega(f^{(j)}, \triangle_{n}(x)), $ we have

$ |f^{(r)}(x)-S_{\triangle}^{(r)}(x)|\leq C\triangle^{m-r}_{m+2}(x)\omega(f^{(m)}, \triangle_{m+2}(x)), $

where $ \triangle_{m+2}(x) = \frac{\sqrt{1-x^{2}}}{m+2}+\frac{1}{(m+2)^2} $. When $ \|\alpha_{n}^{(r)}\|_{\infty} < \frac{1}{(2N)^{m}}(r = 0, 1, \cdots, m) $, from Theorem 3.2, we get

$ \|(T_{\Delta, b}^{\alpha}-S_{\triangle})^{(r)}\|_{\infty}\leq C_{r}\frac{1}{N^{m-r}}, $

here $ C_{r} $ is the constant only dependent on $ r $.

Example 2 Taking $ m = 2 $, consider function $ f(x) = (2x^2-4x+3)^N(\sin(x))^N $, $ b(x) = \cos(x)f(x) $, and the uniform partition of $ [0, 1] $, $ \Delta:0 = x_0 < x_1 < \cdots < x_{N} = 1 $ with step $ h = x_n-x_{n-1} = 1/N, n = 1, 2, \cdots, N $, $ S_\Delta(x) $ is the spline function defined by eq.(2.1). We only consider $ N = 5 $, and take scaling vector $ \alpha_{1} = 0.01x $, $ \alpha_{2} = 0.005x^{2} $, $ \alpha_{3} = 0.01(x-\sin(x)) $, $ \alpha_{4} = 0.01\sin(x) $, $ \alpha_{5} = 0.01(x-\frac{x^{2}}{2}) $. Obviously, $ \alpha = (\alpha_{1}, \alpha_{2}, \cdots, \alpha_{5}) $ satisfies Theorem 3.2. The left graph of Figure 4 is spline function $ S_\Delta(x) $, and the right graph is the fractal function $ T_{\Delta, b}^{\alpha}(x) $.

In this paper, we use spline function to solve the fractal $ (0, m) $ lacunary interpolation problem which is the general case of $ (0, 2) $, $ (0, 4) $ and so on, the explicit formulas of spline function are deduced by extrapolation algorithm. We find that there is fractal lacunary interpolation function with proper perturbation parameters, satisfying interpolation problem. The error estimates and convergence analysis were presented. The numerical examples demonstrate that our proposed method is efficient and viable. A more general theory of fractal Birkhoff interpolation and numerical simulations appear in our later work. The nonconstant case of scaling function is expected to be resolved in future researches.