1st Edition

Integral and Discrete Transforms with Applications and Error Analysis

By Abdul Jerri Copyright 1992

    This reference/text desribes the basic elements of the integral, finite, and discrete transforms - emphasizing their use for solving boundary and initial value problems as well as facilitating the representations of signals and systems.;Proceeding to the final solution in the same setting of Fourier analysis without interruption, Integral and Discrete Transforms with Applications and Error Analysis: presents the background of the FFT and explains how to choose the appropriate transform for solving a boundary value problem; discusses modelling of the basic partial differential equations, as well as the solutions in terms of the main special functions; considers the Laplace, Fourier, and Hankel transforms and their variations, offering a more logical continuation of the operational method; covers integral, discrete, and finite transforms and trigonometric Fourier and general orthogonal series expansion, providing an application to signal analysis and boundary-value problems; and examines the practical approximation of computing the resulting Fourier series or integral representation of the final solution and treats the errors incurred.;Containing many detailed examples and numerous end-of-chapter exercises of varying difficulty for each section with answers, Integral and Discrete Transforms with Applications and Error Analysis is a thorough reference for analysts; industrial and applied mathematicians; electrical, electronics, and other engineers; and physicists and an informative text for upper-level undergraduate and graduate students in these disciplines.


    Guide to Course Adoption

    1 Compatible Transforms

    The Method of Separation of Variables and the Integral Transforms

    Integral Transforms

    Compatible Transforms

    Examples of Compatible Transforms

    Nonlinear Terms

    Classification of the Transforms

    Integral Transforms

    Band-Limited Functions (or Transforms)

    Finite Transforms—The Fourier Coefficients

    The Truncation and Discretization (Sampling) Errors

    The Discrete Transforms

    Comments on the Inverse Transforms—Tables of the Transforms

    Integral Equations—Basic Definitions

    The Compatible Transform and the Adjoint Problem

    The Adjoint Differential Operator

    The Two Eigenvalue Problems

    Constructing the Compatible Transforms for Self-Adjoint Problems—Second-Order Differential Equations

    Examples of the Strum-Liouville and Other Transforms—Boundary Value Problems

    The nth-Order Differential Operator

    Relevant References to Chapter 1


    2 Integral Transforms

    Laplace Transforms

    Transform Pairs and Operations

    The Convolution Theorem for Laplace Transforms

    Solution of Initial Value Problems Associated with Ordinary and Partial Differential Equations

    Applications to Volterra Integral Equations with Difference Kernels

    The z-Transform

    Fourier Exponential Transforms

    Existence of the Fourier Transform and Its Inverse—the Fourier Integral Formula

    Basic Properties and the Convolution Theorem

    Boundary and Initial Value Problems—Solutions by Fourier Transforms

    The Heat Equation on an Infinite Domain

    The Wave Equation

    The Schodinger Equation

    The Laplace Equation

    Signals and Linear Systems—Representation in the Fourier (Spectrum) Space

    Linear Systems

    Bandlimited Functions—the Sampling Expansion

    Bandlimited Functions and B-Splines (Hill Functions)

    Fourier Sine and Cosine Transforms

    Compatibility of the Fourier Sine and Cosine Transforms with Even-Order Derivatives

    Applications to Boundary Value Problems on Semi-Infinite Domain

    Higher-Dimensional Fourier Transforms

    Relation Between the Hankel Transform and the Multiple Fourier Transform—Circular Symmetry

    The Double Fourier Transform of Functions with Circular Symmetry—The Jo-Hanckel Transform

    A Double Fourier Transform Convolution Theorem for the Jo-Hankel Transform

    The Hankel (Bessel) Transforms

    Applications of the Hankel Transforms

    Laplace Transform Inversion

    Fourier Transform in the Complex Plane

    The Laplace Transform in the Inversion Formula

    The Numerical Inversion of the Laplace Transform


    Other Important Integral Transforms

    Hilbert Transform

    Mellin Transform

    The z-Transform and the Laplace Transform Relevant for Chapter 2


    3 Finite Transforms—Fourier Series and Coefficients

    Fourier (Trigonometric) Series and General Orthogonal Expansion

    Convergence of the Fourier Series

    Elements of Infinite Series—Convergence Theorems

    The Orthogonal Expansions—Bessel’s Inequality and Fourier Series

    Fourier Sine and Cosine Transforms

    Fourier (Exponential) Transforms

    The Finite Fourier Exponential Transform and the Sampling Expansion

    Hankel (Bessel) Transforms

    Another Finite Hankel Transform

    Classical Orthogonal Polynomial Transforms

    Legendre Transforms

    Laguerre Transform

    Hermite Transforms

    Tchebychev Transforms

    The Generalized Sampling Expansion

    Generalized Translation and Convolution Products

    Impulse Train for Bessel Orthogonal Series Expansion for a (New) Bessel-Type Possion Summation Formula

    A Remark on the Transform Methods and Nonlinear Problems

    Relevant References to Chapter 3


    4 Discrete Transforms

    Discrete Fourier Transforms

    Fourier Integrals, Series, and the Discrete Transforms

    Computing for Complex- Valued Functions

    The Fast Fourier Transform

    Construction and Basic Properties of the Discrete Transforms

    Opertational Difference Calculus for the DFT and the z-Transform

    Approximating Fourier Integrals and Series by Discrete Fourier Transforms

    Examples of Computing Fourier Integrals and Series

    Discrete Orthogonal Polynomial Transforms

    Basic Properties and Illustrations

    Properties of the Discrete Legendre Transforms

    The Use of the Orthogonal Polynomial Transforms

    Bessel-Type Possion Summation Formula (for the Bessel-Fourier Series and the Hankel Transforms)

    Relevant References for Chapter 4


    Appendix A Basic Second-Order Differential Equations and Their (Series) Solutions—Special Functions


    Method of Variation of Parameters

    Power Series Method of Solution

    Frobenius Method of Solution- Power Series Expansion About a Regular Singular Point

    Special Differential Equations and Their Soultions

    Bessel’s Equation

    Legendre’s Equation

    Other Special Equations


    Appendix B Mathematical Modeling of Partial Differential Equations—Boundary and Initial Value Problems

    Partial Differential Equations for Vibrating Systems

    Diffusion (or Heat Conduction) Equation


    Appendix C Tables of Transforms

    Laplace Transforms

    Fourier Exponential Transforms

    Fourier Sine Transforms

    Fourier Cosine Transforms

    Hankel Transforms

    Mellin Transforms

    Hilbert Transforms

    Finite Exponential Transforms

    Finite Since Transforms

    Finite Cosine Transforms

    Finite (First) Hankel Transforms, Jn(Ka)=0

    Finite (Second) Hankel Transforms, kjn(ka)+hJn(ka)=0

    Finite Legendre Transforms

    Finite Tchebychev Transforms

    Finite Hermite Transforms



    Index of Notations

    Subject Index



    Abdul Jerri