Abstract
This chapter discusses orthogonality, orthogonalization, and series of orthogonal functions. The chapter presents the notion of orthogonality by means of the Stieltjes—Lebesgue integral. The Riesz—Fischer theorem is discussed. The chapter discusses the completeness of an orthogonal system, and the completeness of the trigonometrical system. Orthogonal polynomials, and the Christoffel—Darboux formula are discussed. The chapter reviews convergence theorem for expansions in orthogonal polynomials. The Christoffel—Darboux formula indicates a certain similarity of expansions in orthogonal polynomials with Fourier expansions. From this formula it is possible to deduce a convergence theorem allowing several applications and having a classical analogue in the theory of Fourier series. The Jacobi polynomials are discussed. Among all the Jacobi polynomials, the Chebysheff polynomials are perhaps of the greatest importance. The chapter also reviews the bounds for general orthonormal systems and orthonormal systems of polynomials, and discusses the Haar's orthogonal system.
Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
