Fractal Jacobi Systems and Convergence of Fourier–Jacobi Expansions of Fractal Interpolation Functions

Abstract

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of ${\mathbb{R}}$ interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in ${\mathcal{L}^{2}(-1,1)}$ with respect to the weight function ${\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}$, ${r > -1}$ and ${s > -1}$. In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of ${\alpha}$-fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions ${\mathcal{L}_{\rho}^{2}(-1,1)}$ is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.