Chebyshev Methods and Fast DCT Algorithms

Abstract

This chapter is concerned with Chebyshev methods and fast algorithms for the discrete cosine transform (DCT). Chebyshev methods are fundamental for the approximation and integration of real-valued functions defined on a compact interval. In Sect. 6.1, we introduce the Chebyshev polynomials of first kind and study their properties. Further, we consider the close connection between Chebyshev expansions and Fourier expansions of even 2π-periodic functions, the convergence of Chebyshev series, and the properties of Chebyshev coefficients. Section 6.2 addresses the efficient evaluation of polynomials, which are given in the orthogonal basis of Chebyshev polynomials. We present fast DCT algorithms in Sect. 6.3. These fast DCT algorithms are based either on the FFT or on the orthogonal factorization of the related cosine matrix.