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**Integral and discrete transforms with applications and error analysis.**
*(English)*
Zbl 0753.44001

The book deals mainly with the Laplace transform, the Fourier (exponential, sine and cosine) transform and the Bessel (Hankel) transform, the corresponding finite transforms (over finite intervals) in connection with Fourier series and other orthogonal expansions, and the discrete Fourier and orthogonal polynomial transforms. Also the higher- dimensional Fourier transform, the Hilbert and the Mellin transform and the \(z\)-transform are treated.

In the introduction it is shown how to choose the appropriate transform. The transforms are applied to solve initial and boundary value problems of ordinary and partial differential equations, and to solve Volterra integral equations, with emphasis on problems of signal and system theory. The errors arising at the discretization are discussed in detail, and the advantages of the fast Fourier transform are pointed out.

Every chapter ends with many exercises with hints for the solution or with complete answers. An appendix contains basic material on differential equations, and tables of the different transforms.

The book is dedicated as a text to (electrical and computer) engineers and students in science. The mathematical considerations are not always rigorous, e.g. the definition of the null set on p. 104 and of uniform convergence on p. 373, the missing conditions for the inverse Laplace transform on p. 253 and the uncomplete conditions for the coefficients \(p(x)\) and \(q(x)\) on p. 696 and 697. The unprecise definition of the Dirac delta function on p. 188 ff. should be replaced by its simple correct introduction in the quoted book of J. Mikusiński [Operational calculus (1983; Zbl 0532.44003)].

In the introduction it is shown how to choose the appropriate transform. The transforms are applied to solve initial and boundary value problems of ordinary and partial differential equations, and to solve Volterra integral equations, with emphasis on problems of signal and system theory. The errors arising at the discretization are discussed in detail, and the advantages of the fast Fourier transform are pointed out.

Every chapter ends with many exercises with hints for the solution or with complete answers. An appendix contains basic material on differential equations, and tables of the different transforms.

The book is dedicated as a text to (electrical and computer) engineers and students in science. The mathematical considerations are not always rigorous, e.g. the definition of the null set on p. 104 and of uniform convergence on p. 373, the missing conditions for the inverse Laplace transform on p. 253 and the uncomplete conditions for the coefficients \(p(x)\) and \(q(x)\) on p. 696 and 697. The unprecise definition of the Dirac delta function on p. 188 ff. should be replaced by its simple correct introduction in the quoted book of J. Mikusiński [Operational calculus (1983; Zbl 0532.44003)].

Reviewer: L.Berg (Rostock)

### MSC:

44A05 | General integral transforms |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

65T50 | Numerical methods for discrete and fast Fourier transforms |

44-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral transforms |

45D05 | Volterra integral equations |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |