Fractal interpolation functions for random data sets

Let x(0) < x(1) < x(2) < ... < x(N) and I = [x(0,) x(N)]. Let u be a continuous function defined on I and let Delta(mu) = {(x(k), mu(k)) : k = 0, 1,..., N}, where mu(k) = u(x(k)). We establish a fractal interpolation function f((T mu)) on I corresponding to the set of points Delta(mu). Let Y-k be a random perturbation of mu(k) and set Delta(Y) = {(x(k), Y-k) : k = 0, 1,..., N}. By a similar way, we construct a fractal interpolation function f((Ty)) on I corresponding to the set Delta(Y). f((Ty)) (x) is a random variable for any x is an element of I, and the function f((Ty)) can be treated as a fractal perturbation of u under some random noise in the set of interpolation points Delta mu. In this article we investigate some statistical properties of f((Ty)) and give estimations of the difference between f((Ty)) and u. (C) 2018 Elsevier Ltd. All rights reserved.