1.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Numerical Fourier Transform Based on
Hyperfunction Theory
Hidenori Ogata
Dept. Computer and Network Engineering,
The Graduate School of Informatics and Engineering,
The Univerisity of Electro-Communications, Tokyo, Japan
21 June, 2018
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

2.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Aim of the study
Aim of the study
Numerical Fourier transform based on hyperfunction theory
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

3.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Aim of the study
Aim of the study
Numerical Fourier transform based on hyperfunction theory
Fourier transform
F[f ](ξ) =
∞
−∞
f (x) exp(−2πiξx) dx.
It is very familiar in science and engineering.
But, it is difficult to compute it by conventional methods,
especially, if f (x) decays slowly as x → ±∞.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

4.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Fourier transforms are difficult to compute!
Numerical integration by the DE rule.
(1)
∞
0
dx
1 + x2
, (2)
∞
0
cos x
1 + x2
dx.
-16
-14
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60 70
log10(error)
N
(1)
(2)
vertical axis: log10(error)
horizontal axis: number
of sampling points N
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

5.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Aim of the study
Aim of the study
Numerical Fourier transform based on hyperfunction theory
Fourier transform
F[f ](ξ) =
∞
−∞
f (x) exp(−2πiξx) dx.
It is very familiar in science and engineering.
But, it is difficult to compute it by conventional methods,
especially, if f (x) decays slowly as x → ±∞.
It is easy to compute it using hyperfunction theory.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

6.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Contents
Contents
1 Basis of hyperfunction theory
2 Fourier transform in hyperfunction theory
3 Numerical Fourier transform
4 Numerical examples
5 Summary
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

7.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
1. Hyperfunction theory
M. Sato, 1958
a theory of generalized functions
based on complex function theory
Hyperfunction
the difference between the boundary values
of an analytic function F(z)
f (x) = [F(z)] ≡ F(x + i0) − F(x − i0).
F(z) : defining function of the hyperfunction f (x)
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

8.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
1. Hyperfunction: f (x) = F(x + i0) − F(x − i0)
Examples
δ(x) = −
1
2πi
1
x + i0
−
1
x − i0
= lim
ǫ↓0
1
π
ǫ
x2 + ǫ2
,
χ(−1,1)(x) =
1 x ∈ (−1, 1)
0 x ∈ [−1, 1]
= −
1
2πi
log
(x + i0) + 1
(x + i0) − 1
− log
(x − i0) + 1
(x − i0) − 1
.
log z: the principal value, i.e., log x ∈ R for x > 0.
-1 -0.5 0 0.5 1Re z -1
-0.5
0
0.5
1
Im z
-6
-4
-2
0
2
4
6
Re F(z)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Re z -2-1.5-1-0.500.511.52
Im z
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Re F(z)
defining function F(z) of δ(x) defining function F(z) of χ(−1,1)(x)
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

9.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
2. Fourier transform in hyperfunction theory
How to treat Fourier transforms in hyperfunction theory.
F[f ](ξ) =
∞
−∞
f (x)e−2πiξx
dx.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

10.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
2. Fourier transform in hyperfunction theory
How to treat Fourier transforms in hyperfunction theory.
F[f ](ξ) =
0
−∞
f (x)e−2πi(ξ+i0)x
dx +
+∞
0
f (x)e−2πi(ξ−i0)x
dx .
We can define F[f ](ξ) thanks to e−2πǫ|x|
even if the integral is not convergent in the conventional sense.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

11.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
2. Fourier transform in hyperfunction theory
How to treat Fourier transforms in hyperfunction theory.
F[f ](ξ) =
0
−∞
f (x)e−2πi(ξ+i0)x
dx +
+∞
0
f (x)e−2πi(ξ−i0)x
dx .
We can define F[f ](ξ) thanks to e−2πǫ|x|
even if the integral is not convergent in the conventional sense.
(Example)
F[1](ξ) =
0
−∞
e−2πi(ξ+i0)x
dx +
∞
0
e−2πi(ξ−i0)x
dx
= −
1
2πi
1
ξ + i0
−
1
ξ − i0
= δ(ξ).
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

12.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
2. Fourier transform in hyperfunction theory
How to treat Fourier transforms in hyperfunction theory.
F[f ](ξ) =
0
−∞
f (x)e−2πi(ξ+i0)x
dx +
+∞
0
f (x)e−2πi(ξ−i0)x
dx .
We can define F[f ](ξ) thanks to e−2πǫ|x|
even if the integral is not convergent in the conventional sense.
Fourier transform F[f ](ξ)
Hyperfunction with the defining functions F±(ζ)
F[f ](ξ) = F+(ξ + i0) − F−(ξ − i0),
F+(ζ) =
0
−∞
f (x)e−2πiζx
dx ( Im ζ > 0 ),
F−(ζ) = −
∞
0
f (x)e−2πiζx
dx ( Im ζ < 0 ).
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

13.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
3. Numerical Fourier transform: strategy
We use the definition of Fourier transforms in hyperfunction theory
for numerical Fourier transforms.
F[f ](ξ) =
∞
−∞
f (x)e−2πiξx
dx = F+(ξ + i0) − F−(ξ − i0),
where F+(ζ) =
0
−∞
f (x)e−2πiζx
dx ( Im ζ > 0 ),
F−(ζ) = −
∞
0
f (x)e−2πiζx
dx ( Im ζ < 0 ).
We can compute F±(ζ) in C because the integrands include
the exponentially decaying factor exp(−2π| Im ζ||x|).
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

14.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
3. Numerical Fourier transform: strategy
We use the definition of Fourier transforms in hyperfunction theory
for numerical Fourier transforms.
F[f ](ξ) =
∞
−∞
f (x)e−2πiξx
dx = F+(ξ + i0) − F−(ξ − i0),
where F+(ζ) =
0
−∞
f (x)e−2πiζx
dx ( Im ζ > 0 ),
F−(ζ) = −
∞
0
f (x)e−2πiζx
dx ( Im ζ < 0 ).
Strategy
1 Get F±(ζ) in { ζ ∈ C | ± Im ζ > 0 }.
2 Get F±(ξ ± i0) on R by analytic continuation.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

15.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
3. Numerical Fourier transform: strategy 1/2
1. Get F±(ζ) in { ζ ∈ C | ± Im ζ > 0 }.
F+(ζ) =
0
−∞
f (x)e−2πiζx
dx ( Im ζ > 0 ),
F−(ζ) = −
∞
0
f (x)e−2πiζx
dx ( Im ζ < 0 ).
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

16.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
3. Numerical Fourier transform: strategy 1/2
1. Get F±(ζ) in { ζ ∈ C | ± Im ζ > 0 }.
F±(ζ) = ±
∞
0
f (∓x)e±2πiζx
dx ( ± Im ζ > 0 ).
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

17.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
3. Numerical Fourier transform: strategy 1/2
1. Get F±(ζ) in { ζ ∈ C | ± Im ζ > 0 }.
F±(ζ) = ±
∞
0
f (∓x)e±2πiζx
dx ( ± Im ζ > 0 ).
We get F±(ζ) in Taylor series.
F±(ζ) =
∞
n=0
c
(±)
n (ζ − ζ
(±)
0 )n
( ± Im ζ
(±)
0 > 0 ),
c
(±)
n =
1
n!
F
(n)
± (ζ
(±)
0 ) = ±
1
n!
∞
0
(±2πix)n
f (∓x)e±2πiζ
(±)
0 x
exponential decay
dx.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

18.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
3. Numerical Fourier transform: strategy 2/2
2. Get F±(ξ ± i0) on R by analytic continuation.
Taylor series → continued fraction
F±(ζ) =
∞
n=0
cn(ζ − ζ
(±)
0 )n
=
a
(±)
0
1 +
a
(±)
1 (ζ − ζ
(±)
0 )
1 +
a
(±)
2 (ζ − ζ
(±)
0 )
1 +
...
R
R
ζ
(±)
0
convergence region
c
(±)
n → a
(±)
n by quotient difference (QD) algorithm
The QD algorithm is unstable. → multiple precision arithmetic
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

19.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
3. Numerical Fourier transform: strategy 2/2
2. Get F±(ξ ± i0) on R by analytic continuation.
Taylor series → continued fraction
F±(ζ) =
∞
n=0
cn(ζ − ζ
(±)
0 )n
=
a
(±)
0
1 +
a
(±)
1 (ζ − ζ
(±)
0 )
1 +
a
(±)
2 (ζ − ζ
(±)
0 )
1 +
...
R
R
ζ
(±)
0
convergence region
F[f ](ξ) = F+(ξ + i0) − F−(ξ − i0).
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

20.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples
(1) F[tanh(πx)](ξ) = −i cosech(πξ),
(2) F[(1 + x2
)−ν−1/2
] =
2πν+1/2
Γ(ν + 1/2)
|ξ|ν
Kν(2π|ξ|) ( ν = 1.5 ),
(3) F[log |x|](ξ) = −γδ(ξ) −
1
2|ξ|
.
The center of the Taylor series of F±(ζ): ζ
(±)
0 = ±i.
C++ programs,
multiple precision arithmetic (100 decimal digits).
exflib: multiple precision arithmetic library.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

21.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: errors of our method
-60
-50
-40
-30
-20
-10
0
-4 -2 0 2 4
log10(error)
xi
(1)
(2)
(3)
vertical axis: log10(error)
horizontal axis: ξ
(1) F[tanh(πx)](ξ), (2) F[(1+x2
)−ν−1/2
](ξ), (3) F[log |x|].
The number of sampling points for numerical integration (the DE rule)
(1) 1330 (2) 1416 (3) 1430.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

22.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: an interesting fact
We do not need to compute oscillatory integrals
for Fourier transforms.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

23.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: an interesting fact
We do not need to compute oscillatory integrals
for Fourier transforms.
The defining function ( F[f ](ξ) = F+(ξ + i0) − F−(ξ − i0) )
F±(ζ) = ±
∞
0
f (∓x)e±2πiζx
dx
=
∞
n=0
c
(±)
n (ζ ∓ i)n
( ± Im ζ > 0 ),
c
(±)
n = ±
1
n!
∞
0
(±2πix)n
f (∓x) exp(−2πx)dx.
including no oscillatory function.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

24.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: comparisons
Comparison with the previous methods
1 DE rule & Richardson extrapolation
M. Sugihara, J. Comp. Appl. Math., 17 (1987) 47–68.
F[f ](ξ) = lim
n→∞
∞
−∞
f (x) exp(−2πiξx)exp(−2−n
x2
)dx.
2 DE-type rule for oscillatory integrals
T. Ooura & M. Mori,
J. Comp. Appl. Math., 38 (1991) 353–360.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

25.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: comparisons
F[tanh(πx)](ξ) = −i cosech(πξ), ξ = 1.
multiple precision arithmetic (100 decimal digits, exflib)
the center of the Taylor series ζ
(±)
0 = 1 ± i.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

26.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: comparisons
F[tanh(πx)](ξ) = −i cosech(πξ), ξ = 1.
multiple precision arithmetic (100 decimal digits, exflib)
the center of the Taylor series ζ
(±)
0 = 1 ± i.
number of sampling points
for integration error
our method 2610 2.8 × 10−28
DE & Richardson 10060 9.1 × 10−20
DE for
oscillatory integrals 948 5.0 × 10−25
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

27.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: comparisons
number of sampling points
for integration error
our method 2610 2.8 × 10−28
DE & Richardson 10060 9.1 × 10−20
DE for
oscillatory integrals 948 5.0 × 10−25
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

28.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
4. Numerical examples: comparisons
number of sampling points
for integration error
our method 2610 2.8 × 10−28
DE & Richardson 10060 9.1 × 10−20
DE for
oscillatory integrals 948 5.0 × 10−25
Our method > DE & Richardson
Our method < DE for oscillatory integrals
Our method computes F[f ](ξ) as a function, i.e.,
once we obtain the coefficients a
(±)
n of the continued fraction,
we can compute F[f ](ξ)’s for many ξ’s.
The conventional methods compute F[f ](ξ) as an integrals.
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

29.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Summary
1 Numerical Fourier transform based on hyperfunction theory
2 Fourier transform as a hyperfunction
F[f ](ξ) =
0
−∞
f (x)e−2πi(ξ+i0)x
dx +
∞
0
f (x)e−2πi(ξ−i0)x
dx.
defining functions F±(ζ)
Get F±(ζ) in { ζ ∈ C | ± Im ζ > 0 }.
Get F±(ξ ± i0) on R by analytic continuation
(by the continued fraction).
3 Numerical examples shows the effectiveness of our method.
Problems for future studies
1 Theoretical error estimate
2 Analytic continuation by continued fractions
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory

30.
Hyperfunction theory
Fourier transform in hyperfunction theory
Numerical Fourier transform
Numerical examples
Summary
Thank you very much!
Hidenori Ogata Numerical Fourier Transform Based on Hyperfunction Theory