The Applications of Cardinal Trigonometric Splines in Solving Nonlinear Integral Equations

The cardinal trigonometric splines on small compact supports are employed to solve integral equations. The unknown function is expressed as a linear combination of cardinal trigonometric splines functions. Then a simple system of equations on the coefficients is deducted. When solving the Volterra integral equations, the system is triangular, so it is relatively straight forward to solve the nonlinear system of the coefficients and a good approximation of the original solution is obtained. The sufficient condition for the existence of the solution is discussed and the convergence rate is investigated. 1. Introduction Trigonometric splines were introduced by Schoenberg in [1]. Univariate trigonometric splines are piecewise trigonometric polynomials of the form (where are real numbers) in each interval and they are nature extensions of polynomial splines. Needless to say, trigonometric splines have their own advantages. A number of papers have appeared to study the properties of the trigonometric splines and trigonometric B-splines (cf. [2–4]) since then. In my previous papers (cf. [5–7]), low degree orthonormal spline and cardinal spline functions with small compact supports were constructed. The method can be extended to construct higher degree orthonormal or cardinal splines. Unlike in the book (cf. [1]), by the cardinal splines we mean the specific splines satisfying cardinal interpolation conditions, which means that the cardinal function has the value one at one interpolation point and value zero at all other interpolation points. Cardinal splines are not only useful in interpolation problems, but they are also useful in deduction of numerical integration formulas [6] and in solving integral equations. Integral equations appear in many fields, including dynamic systems, mathematical applications in economics, communication theory, optimization and optimal control systems, biology and population growth, continuum and quantum mechanics, kinetic theory of gases, electricity and magnetism, potential theory, and geophysics. Many differential equations with boundary value can be reformulated as integral equations. There are also some problems that can be expressed only in terms of integral equations. In this paper we focus on the Volterra integral equations of the second kind: where is a complex number, the kernel , , and are known functions, and is an unknown function to be determined. This paper has six sections. In Section 2, a univariate trigonometric cardinal spline on a small compact support is constructed and properties are studied. In Section 3,