##### 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

Hadamard’s finite part, 126

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

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

Hadamard’s finite part, 126

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