Orthogonal Polynomials: Essential Tools and Applications in Modern Mathematics

Emmanuel A. Ikekwere

Abstract


Orthogonal Polynomials are classes of polynomials pn(x) defined over a range [a,b] that obey an orthogonality relation. Where w(x) is a weight function and δmn is the Kronecker delta. If cn = 1, the polynomials are orthogonal and orthonormal. They are also defined as polynomials whose inner product equals zero. The basic properties, recurrence relations, distributions of zeros and other characterizations of such polynomials are included. The very classical orthogonal polynomials and some basic concepts of orthogonal polynomials were investigated with major forms of orthogonality on Legendre, Bessel, and Chebyshev’s polynomials. Gram Schmidt and Sturn-Liouville’s approach was used to show orthogonality in the three poly-nomials reviewed and illustrative examples of each were stated. Finally, some applications in numerical integration; interpolation, linear spaces, vector spaces and unit circles were studied.


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