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Numerical methods based on sinc and analytic functions. (English) Zbl 0803.65141
Springer Series in Computational Mathematics. 20. New York, NY: Springer- Verlag. xv, 565 p. (1993).
This excellent monograph offers a self-contained presentation of the sinc method and its application to the numerical solution of integral and differential equations. This book will be the standard reference for the sinc method. It is of interest for mathematicians, computational scientists and graduate students.
Let $$h>0$$ and $$\text{sinc} (x) : = (\pi x)^{-1} \sin (\pi x)$$. Using the basis functions $S(k,h) (x) : = \text{sinc} \bigl( (x - kh)/h \bigr),$ a given function $$f$$ bounded on the real line is approximated by the cardinal function $C(f,h) (x) : = \sum^ \infty_{k = - \infty} f(kh) S(k,h) (x).$ First, the approximation of $$f$$ by means of $$C(f,h)$$ was studied by de la Vallée Poussin and Whittaker. Later, Shannon’s sampling theorem gave an essential impulse to application of this theory in signal processing. The author has special merits in this topic, since he has studied the sinc method over 30 years intensively. Thus, many results presented in this book are new. Note that the sinc method is closely related to the approximation by translates, wavelet theory, and multiscale technique.
Basic facts on analytic functions, polynomial approximation, and Fourier technique are presented in the first two chapters. Chapter 3 deals with the approximation of $$f$$ by $$C(f,h)$$, where $$f$$ is analytic on a strip containing the real line. Interpolation, quadrature, Fourier and Hilbert transforms, derivatives, and indefinite integrals are determined approximately. All of these procedures converge at exponential and close to optimal rate. Using a conformal mapping, the results of Chapter 3 are extended in Chapter 4 to approximations over a contour such that a finite or semi-infinite interval is a special case.
In Chapter 5, procedures related to sinc methods are discussed. Chapter 6 illustrates the application of sinc methods to the approximate solution of integral equations. The author considers nonlinear Volterra integral equations, Cauchy singular integral equations, convolution equations, Wiener-Hopf integral equations, and the inversion of Laplace transform. If there exists an analytic solution, then it is shown that an exponential convergence rate is reachable by sinc methods.
Finally, Chapter 7 demonstrates the use of sinc methods to obtain approximate solutions of ordinary and partial differential equations for both initial and boundary value problems. It is pointed out that Galerkin, finite element, spectral, and collocation methods are essential the same for the sinc methods, since they all yield nearly the same system of linear equations, whose solutions have the same order of accuracy.
Each section ends with some problems. Each chapter closes with historical remarks. This book is completed by a detailed list of references containing 296 items.
Reviewer: M.Tasche (Rostock)

##### MSC:
 65T40 Numerical methods for trigonometric approximation and interpolation 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65R20 Numerical methods for integral equations 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 65Dxx Numerical approximation and computational geometry (primarily algorithms) 44A10 Laplace transform 45Exx Singular integral equations 45G10 Other nonlinear integral equations