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Least Squares Chebyshev and Fourier

If we work with a basis of functions, orthogonal with respect to some inner product, then we approximate any given function via the inner products of the given function with the basis functions. Because of the orthogonality of the basis, the difference between the approximation and the given function is orthogonal to the basis. Thus by construction, we obtain least squares approximations for any given function. The lecture demonstrates this construction for Chebyshev polynomials and for a basis of trigonometric functions. The construction with a trigonometric basis corresponds to taking the Fourier transform of a signal, represented by a sine wave, as the least squares decomposition of sines and cosines running at increasing frequencies. The lecture ends with some suggestion for projects.