 ## Doctoral Dissertations

Winter 2008

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

Eric Grinbergl

#### Abstract

In this dissertation we present and solve an ill-posed inverse problem which involves reproducing a function f(x) or its Fourier coefficients from the observed values of the function. The observations of the f(x) are made at n equidistant points on the unit interval with p observations being made at each point. The observations are effected by a random error with a known distribution.

First of all we present a very simple estimator for the Fourier coefficients of f(x). Then we present an iteration algorithm for improving the estimator for the Fourier coefficients. We show that the improved estimator we use is a simplified and improved version of the Maximum Likelihood Estimator.

Second, we introduce the mean squared error (MSE) for the estimators, which is the main measure of estimator performance. We show that a singly iterated estimator has a smaller MSE then a non-iterated estimator and a multiply iterated estimator has a smaller MSE then a singly iterated estimator. We also prove that the errors in estimating the Fourier Coefficients by the singly and multiply improved methods are normally distributed.

Third, we prove a theorem showing that as the sample size goes to infinity, the MSE of our estimator asymptotically approaches the theoretical minimum. That shows that our results are theoretically the "best possible" results.

Fourth, we perform simulations which numerically approximate MSE for a given set of f, error distributions, as well as the number of observation points. We approximate the MSE for the non-iterated error coefficient approximation as well as the singly iterated and multiply iterated ones. We show that indeed the MSE decreases with each iteration. We also plot an error histogram in each case showing that the errors are normally distributed.

Finally, we look at some ways in which our problem can be expanded. Possible expansions include working on the problem in multiple dimensions, taking measurements of f at random points, or both of the above.

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