This book contains a very comprehensive treatment of most of the author's original results in the theory of complex integral representation of cardinal spline functions. The basic idea of the book is to use a suitable inverse integral transform instead of the direct transform itself and then to have recourse to the methods of complex analysis applied to cardinal exponential splines and cardinal logarithmic splines. The method of complex contour integral representation yields a unified treatment of both cases. Besides presenting an outline of inverse integral transform technique, the book investigates several related topics. These include: (1)~various complex integral representations of the basis spline functions, (2)~a useful complex contour integral representation of the Euler-Frobenius polynomials and its consequences, and (3)~the classical Méray-Runge phenomenon. This approach to cardinal spline functions provides a very instructive illustration of the application of inverse integral transform techniques combined with complex variable methods to recent problems arising in approximation theory. Each section of the book ends with a few references and comments.

In the reviewer's opinion, this book will be very useful to a broad audience, interested in present developments of approximation theory. -- Mathematical Reviews