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Fourier decomposition is a well established technique used in stellarpulsation. However the quality of reconstructed light curves using this methodis reduced when the observed data have uneven phase coverage. We use simulatedannealing techniques together with Fourier decomposition to improve the qualityof the Fourier decomposition for many OGLE LMC fundamental mode Cepheids. Thismethod restricts the range that Fourier amplitudes can take. The ranges arespecified by well sampled Cepheids in the Galaxy and Magellanic Clouds.  We also apply this method to reconstructing Cepheid light curves observed bythe HST. These typically consist of 12 V band and 4 I band points. We employ adirect Fourier fit to the 12 V band points using the simulated annealing methodmentioned above and explicitly derive and use Fourier interrelations toreconstruct the I band light curve. We discuss advantages and drawbacks of thismethod when applied to HST Cepheid data over existing template methods.Application of this method to reconstruct the light curves of Cepheids observedin NGC 4258 shows that the derived Cepheid distance ($\mu_0=29.38\pm0.06$,random error) is consistent with its geometrical distance($\mu_0=29.28\pm0.09$) derived from observations of water maser.Comment: LaTex, 52 pages, 8 tables, 24 figures, some figures are in .jpg, ApJ  preprint style, ApJ accepted (April 2003

Reconstructing a Cepheid Light Curve with Fourier Techniques. I. The Fourier Expansion and Interrelations