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Title Introduction to Hyperfunctions and Their Integral Transforms: An Applied and Computational Approach English 2.2 MB 428
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Introduction to Hyperfunctions and Their Integral Transforms: An Applied and Computational Approach
9783034604079
Contents
Preface
1 Introduction to Hyperfunctions
1.1 Generalized Functions
1.2 The Concept of a Hyperfunction
1.3 Properties of Hyperfunctions
1.3.1 Linear Substitution
1.3.2 Hyperfunctions of the Type f(φ(x))
1.3.3 Differentiation
1.3.4 The Shift Operator as a Differential Operator
1.3.5 Parity, Complex Conjugate and Realness
1.3.6 The Equation φ(x)f(x) = h(x)
1.4 Finite Part Hyperfunctions
1.5 Integrals
1.5.1 Integrals with respect to the Independent Variable
1.5.2 Integrals with respect to a Parameter
1.6 More Familiar Hyperfunctions
1.6.1 Unit-Step, Delta Impulses, Sign, Characteristic Hyperfunctions
1.6.2 Integral Powers
1.6.3 Non-integral Powers
1.6.4 Logarithms
1.6.5 Upper and Lower Hyperfunctions
1.6.6 The Normalized Power x^α_+/Γ(α + 1)
1.6.7 Hyperfunctions Concentrated at One Point
2 Analytic Properties
2.1 Sequences, Series, Limits
2.2 Cauchy-type Integrals
2.3 Projections of Functions
2.3.1 Functions Satisfying the H¨older Condition
2.3.2 Projection Theorems
2.3.3 Convergence Factors
2.3.4 Homologous and Standard Hyperfunctions
2.4 Projections of Hyperfunctions
2.4.1 Holomorphic and Meromorphic Hyperfunctions
2.4.2 Standard Defining Functions
Sato’s Decomposition Theorems
Perfect Hyperfunctions and Their Standard Defining Functions
Extension to Non-Perfect Hyperfunctions
Strong Defining Functions
2.4.3 Micro-analytic Hyperfunctions
2.4.4 Support, Singular Support and Singular Spectrum
2.5 Product of Hyperfunctions
2.5.1 Product of Upper or Lower Hyperfunctions
2.5.2 Products in the Case of Disjoint Singular Supports
2.5.3 The Integral of a Product
2.5.4 Hadamard’s Finite Part of an Integral
2.6 Periodic Hyperfunctions and Their Fourier Series
2.7 Convolutions of Hyperfunctions
2.7.1 Definition and Existence of the Convolution
2.7.2 Sufficient Conditions for the Existence of Convolutions
2.7.3 Operational Properties
2.7.4 Principal Value Convolution
2.8 Integral Equations I
3 Laplace Transforms
3.1 Loop Integrals
3.2 The Two-Sided Laplace Transform
3.2.1 The Classical Laplace Transform
3.3 Laplace Transforms of Hyperfunctions
3.4 Transforms of some Familiar Hyperfunctions
3.4.1 Dirac Impulses and their Derivatives
3.4.2 Non-negative Integral Powers
3.4.3 Negative Integral Powers
3.4.4 Non-integral Powers
3.4.5 Powers with Logarithms
3.4.6 Exponential Integrals
3.4.7 Transforms of Finite Part Hyperfunctions
3.5 Operational Properties
3.5.1 Linearity
3.5.2 Image Translation Rule
3.5.3 The Multiplication or Image Differentiation Rule
3.5.4 Similarity Rule
3.5.5 Differentiation Rule
3.5.6 Integration Rule
3.5.7 Original Translation Rule
3.5.8 Linear Substitution Rules
3.6 Inverse Laplace Transforms and Convolutions
3.6.1 Inverse Laplace Transforms
The Inversion Formulas
Laplace hyperfunctions
3.6.2 The Convolution Rule
3.6.3 Fractional Integrals and Derivatives
3.7 Right-sided Laplace Transforms
3.8 Integral Equations II
3.8.1 Volterra Integral Equations of Convolution Type
3.8.2 Convolution Integral Equations over an Infinite Range
4 Fourier Transforms
4.1 Fourier Transforms of Hyperfunctions
4.1.1 Basic Definitions
4.1.2 Connection to Laplace Transformation
4.2 Fourier Transforms of Some Familiar Hyperfunctions
4.3 Inverse Fourier Transforms
4.3.1 Reciprocity
4.4 Operational Properties
4.4.1 Linear Substitution Rule
4.4.2 Shift-Rules
4.4.3 Complex Conjugation and Realness
4.4.4 Differentiation and Multiplication Rule
4.4.5 Convolution Rules
4.5 Further Examples
4.6 Poisson’s Summation Formula
4.7 Application to Integral and Differential Equations
4.7.1 Integral Equations III
4.7.2 Heat Equation and Weierstrass Transformation
5 Hilbert Transforms
5.1 Hilbert Transforms of Hyperfunctions
5.1.1 Definition and Basic Properties
5.1.2 Operational Properties
5.1.3 Using Fourier Transforms
5.2 Analytic Signals and Conjugate Hyperfunctions
5.3 Integral Equations IV
6 Mellin Transforms
6.1 The Classical Mellin Transformation
6.2 Mellin Transforms of Hyperfunctions
6.3 Operational Properties
6.3.1 Linearity
6.3.2 Scale Changes
6.3.3 Multiplication by (log x)^n
6.3.4 Multiplication by x^μ, μ ∈ \mathbb{C}
6.3.5 Reflection
6.3.6 Differentiation Rules
6.3.7 Integration Rules
6.4 Inverse Mellin Transformation
6.5 \mathcal{M}-Convolutions
6.5.1 Reciprocal Integral Transforms
6.5.2 Transform of a Product and Parseval’s Formula
6.6 Applications
6.6.1 Dirichlet’s Problem in a Wedge-shaped Domain
6.6.2 Euler’s Differential Equation
6.6.3 Integral Equations V
6.6.4 Summation of Series
7 Hankel Transforms
7.1 Hankel Transforms of Ordinary Functions
7.1.1 Genesis of the Hankel Transform
7.1.2 Cylinder Functions
7.1.3 Lommel’s Integral
7.1.4 MacRobert’s Proof
7.1.5 Some Hankel Transforms of Ordinary Functions
7.1.6 Operational Properties
Parseval’s Relation
The Similarity and Division Rule
Differentiation Rules
7.2 Hankel Transforms of Hyperfunctions
7.2.1 Basic Definitions
7.2.2 Transforms of some Familiar Hyperfunctions
7.2.3 Operational Properties
The Similarity Rule
The Division Rule
The Differentiation Rule
7.3 Applications
A Complements
A.1 Physical Interpretation of Hyperfunctions
A.1.1 Flow Fields and Holomorphic Functions
A.1.2 Polya fields and Defining Functions
A.2 Laplace Transforms in the Complex Plane
A.2.1 Functions of Exponential Type
A.2.2 Laplace Hyperfunctions and their Transforms
A.3 Some Basic Theorems of Function Theory
A.3.1 Interchanging Infinite Series with Improper Integrals
A.3.2 Reversing the Order of Integration
A.3.3 Defining Holomorphic Functions by Series and Integrals
B Tables
Convolution Properties of Hyperfunctions
Operational Rules for the Laplace Transformation
Some Laplace Transforms of Hyperfunctions
Operational Rules for the Fourier Transformation
Some Fourier Transforms of Hyperfunctions
Operational Rules for the Hilbert Transformation
Some Hilbert Transforms of Hyperfunctions
Operational Rules for the Mellin Transformation
Some Mellin Transforms of Hyperfunctions
Operational Rules for the Hankel Transformation
Some Hankel Transforms of order ν of Hyperfunctions
Bibliography
List of Symbols
Index

##### Document Text Contents
Page 214

3.5. Operational Properties 201

If f(t) = [F+(z), F−(z)] is a given hyperfunction of bounded exponential
growth having a Laplace transform f̂(s) for σ− <
s < σ+, the integration rule is

L[−∞D−1f(t)](s) =
1

s
L[f(t)](s), max[σ−, 0] <
s < σ+ (3.121)

provided σ+ > 0, and

L[∞D−1f(t)](s) = −
1

s
L[f(t)](s), σ− <
s < min[σ+, 0] (3.122)

provided σ− < 0. Details are left to the reader.

3.5.7 Original Translation Rule

Proposition 3.13. Let f(t) = [F (z)] ◦−• f̂(s), with σ−(f) <
s < σ+(f), then for
c ∈ R we have

f(t− c) ◦−• e−c s f̂(s) (3.123)
with σ−(f) <
s < σ+(f).
Proof. By canonical splitting of f(t) into f1(t) + f2(t) we arrive at f(t − c) =
f1(t− c) + f2(t− c) and the two integration loops now turn around the point c:

f2(t− c) ◦−• −
∫ (c+)

e−sz F2(z − c) dz = −
∫ (0+)

e−s(ζ+c) F2(ζ) dζ

= e−cs(−1)
∫ (0+)

e−sζ F2(ζ) dζ = e
−cs f̂2(s).

Similarly,

f1(t− c) ◦−• −
∫ (c+)
−∞

e−sz F1(z − c) dz = −
∫ (0+)
−∞

e−s(ζ+c) F1(ζ) dζ

= e−cs(−1)
∫ (0+)
−∞

e−sζ F1(ζ) dζ = e
−cs f̂1(s).

Thus, we obtain

f(t− c) = f1(t− c) + f2(t− c) ◦−• e−cs{f̂1(s) + f̂2(s)} = e−cs f̂(s). �

3.5.8 Linear Substitution Rules

If we combine the similarity rule and original translation rule we obtain

Proposition 3.14. Let f(t) = [F (z)] ◦−• f̂(s), with σ−(f) <
s < σ+(f), then for
a, b ∈ R, a �= 0, we have

f(a t+ b) ◦−• 1|a| e
(b/a) s f̂(

s

a
), (3.124)

with a σ−(f) <
s < aσ+(f) for a > 0, and a σ+(f) <
s < aσ−(f) for a < 0.

Page 215

202 Chapter 3. Laplace Transforms

We conclude this section with the special case where a linear change of
variables is made in a right-sided original. Assume the correspondence f(t) =

[F (z)] ∈ O(R+) ◦−• f̂(s),
s > σ−(f). We are looking for the Laplace transform
of f(a t− b) where both a and b are assumed to be positive. The defining function
F (z) is now real analytic on the negative real axis. Because f(a t−b) = [F (az − b)],
we have that G(z) = F (az − b) is real analytic on (−∞, b/a). That means that
g(t) = f(a t− b) is a hyperfunction that vanishes on (−∞, b/a) and therefore is in
O(R+). Thus, its Laplace transform is

ĝ(s) = −
∫ (0+)

e−szF (az − b) dz.

A change of variables leads to

ĝ(s) = −e
−sb/a

a

∫ (−b+)

e−s/a zF (z) dz

where the integration loop now turns around the point −b. Since F (z) is real
analytic for x < 0 we can replace the new loop by the old one and obtain

ĝ(s) = −e
−sb/a

a

∫ (0+)

e−s/a zF (z) dz

which equals e−sb/af̂(s/a)/a. By keeping track of the growth index σ−(f(t)) we
see that σ−(f(a t− b)) = a σ−(f(t)). Because f(at− b) vanishes on (−∞, b/a) we
may write u(t− b/a)f(at− b) for it. We have proved
Proposition 3.15. Let f(t)∈O(R+) and a and b positive constants. If f(t)◦−•f̂(s),
for
s>σ−, then f(a t− b) ◦−• e−sb/af̂(s/a)/a, for
s>aσ−, where the original
f(a t − b) vanishes on (−∞, b/a). For a = 1, we obtain the translation rule that
can be written as

f(t) ◦−• f̂(s) =⇒ u(t− b)f(t− b) ◦−• e−sbf̂(s). (3.125)

A similar statement holds if f(t) ∈ O(R−), a > 0, b > 0, and f(t) ◦−• f̂(s)
for
s < σ+, then

f(a t+ b)u(−b/a− t) ◦−• e
sb/a

a
f̂(
s

a
),
s < aσ+, (3.126)

i.e., f(at+ b) vanishes on (−b/a,∞).
Example 3.22. Because

fp
u(t)J0(t)

t
◦−• − log(s+

1 + s2

2
)− γ

we obtain

fp
u(t− b)J0(t− b)

t− b ◦−• − e
−bs {log(s+

1 + s2

2
)− γ}.

Page 427

414 Index

general substitution, 16
generalized delta-hyperfunction, 28
generalized derivatives, 18
Green’s function, 199
growth index, 164

Hölder condition, 77
half-plane functions (upper and

lower), 74
Hankel functions, 345
Hankel transform

by using Laplace
transformation, 351

different definitions, 341
Mac Robert’s proof of the inver-

sion formula, 350
of a hyperfunction, 359
Parseval’s relation, 354

heat equation, 275
Heaviside function, 1
Hermite’s function, 172, 245
Hermite’s polynomial, 172
Hilbert transform

and Fourier transform, 295
and integral equations, 302
and strong defining function,

279
classical definition, 277
generalized definition, 278
iterated, 282
of periodic hyperfunctions, 301
of upper and lower hyperfunc-

tions, 286
holomorphic functions and Pólya

fields, 378
Hurwitz’s zeta function, 335
hyperfunction

analytic continuation, 113
characteristic, 45
complex-conjugate , 27
concentrated at one point, 62
defined by one global analytic

function, 95
definition, 4
depending on a parameter, 44
differentiation, 18
even / odd, 25
finite part, 33

Heisenberg’s, 35
holomorphic, 92
homologous, 90
identity theorem, 113
imaginary part of a, 27
integral power, 46
meromorphic, 39, 92
non-integral power , 49
of bounded exponential growth,

167
of slow growth, 245
perfect, 100
periodic, 129
physical interpretation, 379
primitive, 201
product, 23, 116
of lower and upper, 115

pure imaginary, 27
real, 27
real part of a, 27
real type, 287
singular spectrum, 111
singular support, 111
standard, 90
subclass B1(R), 281
support, 111
upper and lower, 56, 285
with finite moments, 289
with logarithms, 52

indicator function, 380
integral equation

Abel’s, 232
Fredholm, 152
of Cauchy’s type, 272
of convolution type, 153
over infinite range, 235
singular, 153
Volterra, 153, 229

integral of a hyperfunction, 37
integral over a product of hyperfunc-

tions, 122
integral transform

reciprocal, 327
integrals with respect to a parameter,

44
interpretation of an ordinary func-

tion as a hyperfunction, 6

Page 428

Index 415

intuitive picture of a hyperfunction,
7

Jordan’s lemma, 75, 163

Laplace hyperfunctions, 217, 388
Laplace transform

arbitrary support, 170
inversion formula, 390
of a holomorphic function, 381
of a Laplace hyperfunction, 388
of a left-sided original, 166
of a right-sided Dirac comb, 174
of a right-sided original, 165
Pincherle’s Theorem, 388
right-sided, 226

left-sided original, 166
Legendre polynomial, 103
linear substitution, 13
Lipschitz condition, 77
Lommel’s integral, 349
lower component, 3
lower half-neighborhood, 3

Mellin ∗convolution, 325
Mellin transform

∗convolution rule, 326
◦convolution, 327
classical, 311
inversion formula, 312, 324
of a product, 329
of hyperfunctions, 315
Parseval’s formula, 329

microanalytic
from above, 111
from below, 111

ordinary derivatives, 18
ordinary function, 5, 93

Pólya field, 376
Pochhammer’s symbol, 49
Poisson’s summation formula, 270
product rule, 20
projection

of a hyperfunction, 95
of an ordinary function, 76

pv-convolution, 150

real analytic function, 10

real neighborhood, 92
regular point, 5
Riemann’s zeta function, 335
right-sided original, 164

Sato, Mikio, 2
scalar product of hyperfunctions, 39
Schwartz, Laurent, 2
sequential approach, 2
shift operator as differential opera-

tor, 25
sign-hyperfunction, 10
singular point, 5
Sokhotski formulas, 81
standard defining function, 100, 104
strong defining function, 106

uniform convergence in the interior
of, 63

unit-step function, 1
unit-step hyperfunction, 8
upper component, 3
upper half-neighborhood, 3
upper half-plane function, 285

Weierstrass transform, 275
Wiener-Hopf equations, 236
Wiener-Hopf technique, 236

zero hyperfunction, 10