Construction of orthogonal functions in Hilbert space

2018-06-26A major problem in mathematical analysis: Whether the sum of a column function can be used to approximate the given function. For example, Taylor expansion, Fourier series, and so on. The orthogonality in Hilbert spaces can promote the concept of Fourier series, thus, a more general orthogonal function system is obtained. This paper studies the origin and theoretical background of orthogonal functions and attempts to construct orthogonal functions. The first part of this paper introduces the Hilbert space, relevant theoretical background of L² spaces and a brief sketch of the Sturm–Liouville problem and its solutions. The second part present the construction of Legendre functions, Hermite functions and Bessel functions