Joseph Fourier was a young scientist in the employ of Napoleon. He went with Napoleon on his ill-fated expedition into Egypt. It was here that he recorded many observations that lead him to his work in heat transport theory. Later in life - he lived between 1768 and 1830 - he was the prefect in Grenoble and here he wrote his seminal work "Memoire sur la Chaleur" which expounded upon his ideas of heat transfer and outlined his new method of mathematical analysis which we now call Fourier Analysis.
The basic thesis of his theory was that if one had a complete set of functions, any arbitrary function could be accurately described by a linear combination of the various members of this complete set of functions. He provided his proof of this work as his submission for the Paris Prize competition one year. The judges (who that year were Lagrange, Laplace, and Legendre) argued that though his work proved that a complete set of functions would accurately describe many functions, he had not proven that they could describe ANY function. They were right. He had not proven it. But Fourier was also right. However, he did not win the Prize that year and it was many years later that others were finally able to complete the proof. Fourier analysis is extremely important in modern mathematics and the ideas are the basis upon which will describe the wavefunction of complex systems such as molecule and solids.