A Levinson-Galerkin algorithm for trigonometric approximation

Thomas Strohmer

Trigonometric polynomials are widely used for the approximation of a smooth function from a set of nonuniformly spaced samples. If the samples are perturbed by noise, a good choice for the polynomial degree of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Standard methods for trigonometric least squares approximation assume that the degree for the approximating polynomial is known a priori, which is usually not the case in practice. We derive a multi-level algorithm that recursively adapts to the least squares solution of suitable degree. We analyze under which conditions this multi-approach yields the optimal solution. The proposed algorithm computes the solution in at most $\ord(rM + M^2)$ operations ($M$ being the polynomial degree of the approximation and $r$ being the number of samples) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle of the heart.

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