Numerical Methods Based on Sinc and Analytic Functions

In this chapter we introduce some mathematical concepts which we shall invoke at least once in the remainder of this monograph. Section 1.1 is a self-contained presentation of concepts of analytic function theory which are both useful and powerful in applications. Sections 1.2 through 1.9 introduce some other mathematical concepts which are important to this text, such as Hilbert, Fourier and Laplace transforms, Fourier series, conformal mapping, spaces of analytic functions and representation of classes of functions via the cardinal function.

In our presentation, we saw Fourier series emerge in Corollary 1.1.11, by way of Laurent’s Theorem. In Section 1.6 we studied Fourier exponential, sine and cosine series in greater detail. In Section 2.1, we again turn to complex variables, using Laurent’s Theorem and a transformation of Section 1.7.4 to obtain a simple derivation of the Chebyshev polynomials. We thus witness the explicit orthogonality over [0,2π] of integer powers of e ^ix transcend, upon taking real and imaginary parts to sines, to cosines, and to Chebyshev polynomials. These results have enriched science by making possible a vast number of explicit expressions in analysis, as well as a wide range of applications. That such orthogonality exists in such a discrete form further enriches science, and especially applications, with many beautiful formulas. The powerful Fast Fourier Transform (FFT) algorithm is made possible by the existence of this discrete orthogonality; this algorithm has had a tremendous impact in computing and applications. This discrete Fourier transform (DFT) is also developed in the setting of Laurent’s Theorem, since this is the space of functions where it best illustrates its true power and accuracy.

In this chapter we develop methods to approximate analytic functions via Sinc series. Initially we obtain bounds on the error of approximation of functions that are analytic in the strip D _d of Equation (1.7.9), via the Sinc series derived in Section 1.9. The region D _d is a natural choice, since the modulus of the function sin(πz/h) which we employ to get the error of Sinc approximation is both large and nearly constant on the boundary of D _d . We also make an assumption on the growth of f in D _d , which has far-reaching consequences, and which enables us to replace the infinite Sinc series by a finite one.

The formulas of the previous chapter are all related to the Cardinal function representation ; with \({z_k}\, = \,{\phi ^{ - 1}}(kh)\). We recall here, that\(C(F,h)\, \circ \,\phi \) both interpolates F on Г = φ ⁻¹(R) , and it also accurately approximates it in a suitable space of functions. In the present chapter we examine some related methods of approximation. This chapter may be termed the unfinished chapter, in the sense that many other formulas are possible by each approach presented here, although we do not present or study the majority of these as we have done in the case of the Cardinal expansion; indeed, a thorough study, as done in the case of Sinc methods in this text would require the writing of at least an additional text. We have merely chosen to point out some connections of Sinc formulas with other formulas, thus opening doors to new avenues of research, and to present some esthetically pleasing formulas which may have practical value as well.

Our goal in this chapter is to familiarize the reader with the use of Sinc methods for solving ordinary and partial differential equations. As for the case of integral equations considered in the previous chapter, the reader will find that replacing differential equations by their discrete Sinc approximations is straightforward and simple. Probably the most important advantage of Sinc methods for solving initial value problems over classical methods, such as Adams-type methods, or Runge-Kutta methods, is that Sinc methods are easily implemented and give good accuracy for problems with singularities and boundary layers, which are stiff. Also, they can produce accurate approximate solutions to problems over infinite or semi- infinite intervals. As in the previous chapter, the implementation of Sinc procedures is carried out along with adequate descriptions of the spaces of analytic functions, which are nearly always present in applications, and in which the Sinc methods perform close to optimally. Our presentation for solving differential equations by Sinc methods is cast in the terminology of the properties of solutions which we require, namely, that a solution should be both analytic and also belong to the class Lip _α in the domain of analyticity in order to get accurate error bounds. Solutions of differential equations arising in applications nearly always possess these properties.