Integral and Discrete Transforms with Applications and Error Analysis Book

	Preface
	Guide to Course Adoption
Ch. 1	Compatible Transforms	1
1.1	The Method of Separation of Variables and the Integral Transforms	2
1.1.1	Integral Transforms	6
1.2	Compatible Transforms	7
1.2.1	Examples of Compatible Transforms	10
1.2.2	Nonlinear Terms	20
1.3	Classification of the Transforms	20
1.3.1	Integral Transforms	21
1.3.2	Band-Limited Functions (or Transforms)	21
1.3.3	Finite Transforms - the Fourier Coefficients	23
1.3.4	The Truncation and Discretization (Sampling) Errors	26
1.3.5	The Discrete Transforms	27
1.4	Comments on the Inverse Transforms - Tables of the Transforms	29
1.4.1	Integral Equations - Basic Definitions	30
1.5	The Compatible Transform and the Adjoint Problem	30
1.5.1	The Adjoint Differential Operator	32
1.5.2	The Two Eigenvalue Problems	36
1.6	Constructing the Compatible Transforms for Self-Adjoint Problems - Second-Order Differential Equations	49
1.6.1	Examples of the Sturm-Liouville and Other Transforms - Boundary Value Problems	52
1.6.2	A Remark Concerning Initial Value Problems	56
1.7	The nth-Order Differential Operator	58
	Relevant References to Chapter 1	63
	Exercises	63
Ch. 2	Integral Transforms	81
2.1	Laplace Transforms	82
2.1.1	Transform Pairs and Operations	91
2.1.2	The Convolution Theorem for Laplace Transforms	104
2.1.3	Solution of Initial Value Problems Associated with Ordinary and Partial Differential Equations	111
2.1.4	Applications to Volterra Integral Equations with Difference Kernels	121
2.1.5	The z-Transform	127
2.2	Fourier Exponential Transforms	128
2.2.1	Existence of the Fourier Transform and Its Inverse - the Fourier Integral Formula	132
2.2.2	Basic Properties and the Convolution Theorem	157
2.3	Boundary and Initial Value Problems - Solutions by Fourier Transforms	171
2.3.1	The Heat Equation on an Infinite Domain	171
2.3.2	The Wave Equation	174
2.3.3	The Schrodinger Equation	176
2.3.4	The Laplace Equation	181
2.4	Signals and Linear Systems Representation in the Fourier (Spectrum) Space	183
2.4.1	Linear Systems	184
2.4.2	Bandlimited Functions - the Sampling Expansion	204
2.4.3	Bandlimited Functions and B-Splines (Hill Functions)	212
2.5	Fourier Sine and Cosine Transforms	213
2.5.1	Compatibility of the Fourier Sine and Cosine Transforms with Even-Order Derivatives	216
2.5.2	Applications to Boundary Value Problems on Semi-Infinite Domain	219
2.6	Higher-Dimensional Fourier Transforms	224
2.6.1	Relation Between the Hankel Transform and the Multiple Fourier Transform - Circular Symmetry	231
2.6.2	The Double Fourier Transform of Functions with Circular Symmetry-The J[subscript 0]-Hankel Transform	233
2.6.3	A Double Fourier Transform Convolution Theorem for the J[subscript 0]-Hankel Transform	236
2.7	The Hankel (Bessel) Transforms
2.7.1	Applications of the Hankel Transforms
2.8	Laplace Transform Inversion	251
2.8.1	Fourier Transform in the Complex Plane	251
2.8.2	The Laplace Transform Inversion Formula	253
2.8.3	The Numerical Inversion of the Laplace Transform	254
2.8.4	Applications	255
2.9	Other Important Integral Transforms	257
2.9.1	Hilbert Transform	257
2.9.2	Mellin Transform	257
2.9.3	The z-Transform and the Laplace Transform	258
	Relevant References for Chapter 2	261
	Exercises	261
Ch. 3	Finite Transforms - Fourier Series and Coefficients	329
3.1	Fourier (Trigonometric) Series and General Orthogonal Expansion	332
3.1.1	Convergence of the Fourier Series	343
3.1.2	Elements of Infinite Series - Convergence Theorems	372
3.1.3	The Orthogonal Expansions - Bessel's Inequality and Fourier Series	384
3.2	Fourier Sine and Cosine Transforms	421
3.3	Fourier (Exponential) Transforms	427
3.3.1	The Finite Fourier Exponential Transform and the Sampling Expansion	429
3.4	Hankel (Bessel) Transforms	433
3.4.1	Another Finite Hankel Transform	437
3.5	Classical Orthogonal Polynomial Transforms	440
3.5.1	Legendre Transforms	441
3.5.2	Laguerre Transform	445
3.5.3	Hermite Transforms	446
3.5.4