Multilevel Approximation of Univariate Periodic Functions with Gaussians

In this thesis, we provide the error analysis of an algorithm for approximating the cosine frequencies for even  functions (the approach being feasible for adaptation to odd functions). We extend these estimates to the multilevel context, both for quasi-interpolation in Chapter 3 and interpolation in Chapter 4. We look at periodic functions in one dimension, i.e., those defined on the unit circle T, where C1(T) denote the linear space of 1-periodic continuous functions.

Our analysis is based on tools provided by Fourier analysis. Our research shows that interpolation is easier to analyse since the convergence is much faster. The numerical experiments for quasi-interpolation in Chapter 3 show that the error for approximating the frequencies goes to 10^-7 accuracy, while for interpolation in Chapter 4 goes  to 10^-13 accuracy. Overall, our multilevel interpolation algorithm has a faster convergence rate than the multilevel quasi-interpolation algorithm so we can provide error estimates with greater precision. Various theoretical results for upper bounds are confirmed by numerical simulations throughout the thesis.