Fractal interpolant curve fitting and reproducing kernel Hilbert spaces

Abstract In this paper,the linear space $\mathcal F$ of a special type of fractal interpolation functions (FIFs) on an interval I is considered. Each FIF in $\mathcal F$ is established from a continuous function on I . We show that, for a finite set of linearly independent continuous functions on I , we get linearly independent FIFs. Then we study a finite-dimensional reproducing kernel Hilbert space (RKHS) $\mathcal F_{\mathcal B}\subset\mathcal F$ , and the reproducing kernel $\mathbf k$ for $\mathcal F_{\mathcal B}$ is defined by a basis of $\mathcal F_{\mathcal B}$ . For a given data set $\mathcal D=\{(t_k, y_k) : k=0,1,\ldots,N\}$ , we apply our results to curve fitting problems of minimizing the regularized empirical error based on functions of the form $f_{\mathcal V}+f_{\mathcal B}$ , where $f_{\mathcal V}\in C_{\mathcal V}$ and $f_{\mathcal B}\in \mathcal F_{\mathcal B}$ . Here $C_{\mathcal V}$ is another finite-dimensional RKHS of some classes of regular continuous functions with the reproducing kernel $\mathbf k^*$ . We show that the solution function can be written in the form $f_{\mathcal V}+f_{\mathcal B}=\sum_{m=0}^N\gamma_m\mathbf k^*_{t_m} +\sum_{j=0}^N \alpha_j\mathbf k_{t_j}$ , where ${\mathbf k}_{t_m}^\ast(\cdot)={\mathbf k}^\ast(\cdot,t_m)$ and $\mathbf k_{t_j}(\cdot)=\mathbf k(\cdot,t_j)$ , and the coefficients γ m and α j can be solved by a system of linear equations.