Polynomial Approximation

Abstract

In our presentation, we saw Fourier series emerge in Corollary 1.1.11, by way of Laurent’s Theorem. In Section 1.6 we studied Fourier exponential, sine and cosine series in greater detail. In Section 2.1, we again turn to complex variables, using Laurent’s Theorem and a transformation of Section 1.7.4 to obtain a simple derivation of the Chebyshev polynomials. We thus witness the explicit orthogonality over [0,2π] of integer powers of e^ix transcend, upon taking real and imaginary parts to sines, to cosines, and to Chebyshev polynomials. These results have enriched science by making possible a vast number of explicit expressions in analysis, as well as a wide range of applications. That such orthogonality exists in such a discrete form further enriches science, and especially applications, with many beautiful formulas. The powerful Fast Fourier Transform (FFT) algorithm is made possible by the existence of this discrete orthogonality; this algorithm has had a tremendous impact in computing and applications. This discrete Fourier transform (DFT) is also developed in the setting of Laurent’s Theorem, since this is the space of functions where it best illustrates its true power and accuracy.