VECTOR GRÜNWALD FORMULA FOR FRACTIONAL DERIVATIVES
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
Abstract. The Grünwald formula is used to numerically estimate fractional derivatives.
It is an extension of the finite difference formula for integer order derivatives. This paper
develops an extension of the Grünwald formula for vector fractional derivatives. This
result should be useful for numerical solution of fractional partial differential equations
where the space variable is a vector.
1. Introduction
Fractional derivatives have been around for centuries [22, 26] but recently they have
found new applications in physics [2, 6, 7, 9, 15, 18, 19, 29], hydrology [1, 4, 5, 10, 14, 28],
and finance [24, 25, 27]. Analytical solutions of ordinary fractional differential equations
[22, 23] and partial fractional differential equations [8, 16] are now available in some
special cases. But the solution to many fractional differential equations will have to rely
on numerical methods, just like their integer-order counterparts. Numerical solutions of
fractional differential equations require a numerical estimate of the fractional derivative.
In one dimension, this estimate is called the Grünwald formula [20, 22, 26]. A variant
of this formula has been used to develop practical numerical methods for solving certain
fractional partial differential equations that model flow in porous media [21, 30]. The
purpose of this paper is to develop a multivariable analogue of the Grünwald formula for
estimating multidimensional fractional derivatives, so that the results in [21, 30] can be
extended to two and three dimensional flow regimes.
The scalar Grünwald formula is a discrete approximation to the fractional derivative
by a fractional difference quotient
∞
X
dα f (x)
α
−α α
−α
m
(1.1)
= lim h ∆h f (x) = lim h
(−1)
f (x − mh)
α
m
h→0
h→0
dx
m=0
where for noninteger α > 0 the binomial coefficient
(−1)m−1 αΓ(m − α)
α
(1.2)
=
,
m
Γ(1 − α)Γ(m + 1)
see for example [20, 26]. When α is a positive integer, the sum terminates at m = α and
equation (1.1) reduces to the usual one-sided difference formula for the derivative. In this
Date: 20 January 2004.
Key words and phrases. Fractional derivatives, numerical analysis, Grünwald approximation formula.
1
2
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
paper, we extend (1.1) to multivariable fractional derivatives, see Theorems 4.1 and 4.3
below.
The multivariable fractional derivative first appeared in [16] in connection with a model
of anomalous
diffusion. Given f : Rd → R and using the Fourier transform convention
R ihk,xi
α
fˆ(k) = e
f (x)dx we specify this operator by requiring that DM
f (x) has Fourier
transform
Z
(1.3)
(−ihk, θi)α M(dθ) fˆ(k)
kθk=1
where M(dθ) is an arbitrary probability measure on the unit sphere S d−1 = {x ∈ Rd :
kxk = 1}. If d = 1 then (1.3) reduces to
p(−ik)α + (1 − p)(ik)α fˆ(k)
so that
dα f (x)
dα f (x)
+
(1
−
p)
.
dxα
d(−x)α
When α = 2 the integral in equation (1.3) reduces to
X
2
Z
d
(1.4)
−
kj θj M(dθ) = (−ik) · A(−ik)
α
DM
f (x) = p
kθk=1
j=1
where the matrix A = (aij ) with aij =
α
f (x)
DM
R
θi θj M(dθ). Then
= ∇ · A∇f (x) =
d X
d
X
aij
i=1 j=1
∂ 2 f (x)
.
∂xi ∂xj
The vector fractional derivative appears in the diffusion/dispersion term of evolution
equations as a generalization of this Laplacian term. The fractional order α speeds up
the diffusion of a cloud of particles as α decreases, and the mixing measure M governs
the direction of large particle jumps, allowing an asymmetric plume.
2. Preliminary Results
α
In this section, we show that the multivariable fractional derivative operator DM
f (x)
defined by the Fourier transform in (1.3) above, for certain functions, can be represented
as a mixture of fractional directional derivatives Dθα f (x). This will enable us in the
next section to apply the scalar Grünwald formula in each radial direction. We assume
d
throughout this section that α > 0 is not an integer and
P that θ is a unit vector in R .
The directional derivative Dθ f (x) = hθ, ∇f (x)i = j θj ∂f (x)/∂xj = dg/ds at s = 0
where g(s) = f (x + sθ). Its Fourier transform is
d
X
j=1
θj (−ikj )fˆ(k) = (−ihk, θi)fˆ(k)
VECTOR GRÜNWALD FORMULA
3
since (−ikj )fˆ(k) is the Fourier transform of ∂f (x)/∂xj . The scalar fractional derivative
can be defined by
Z ∞
dn
1
α
D+ g(s) =
rn−α−1 g(s − r)dr
Γ(n − α) dsn 0
where n = 1 + [α] is the smallest integer greater than α, and it is easy to check that
this convolution integral has Fourier transform (−iu)α ĝ(u), see [3, 26]. The fractional
order directional derivative Dθα f (x) is defined by D+α g(s) evaluated at s = 0, where
g(s) = f (x + sθ). Then it is clear that Dθα f (x) has Fourier transform (−ihk, θi)α fˆ(k)
which reduces to the classical case when α = 1. Now (1.3) is revealed as a mixture of
fractional directional derivatives. The next theorem will make this precise.
Definition 2.1. For any positive integer l, let W l,1 (Rd ) denote the collection of functions
f ∈ C l (Rd ) whose partial derivatives up to order l are in L1 (Rd ) and whose partial
derivatives up to order l − 1 vanish at infinity.
Theorem 2.2. For f ∈ W l,1 (Rd ) with l = [α] + 2 we have
Z
α
(2.1)
DM f (x) =
Dθα f (x)M(dθ) a.e.
kθk=1
α
f by (2.1), then the proof below shows
Remark 2.3. If we define DM
Z
α
b
(DM f ) (k) =
(−ihk, θi)α M(dθ)fˆ(k)
kθk=1
so no a.e. is needed in (2.1).
Before we prove Theorem 2.2 we need the following
technical result. Recall that the
R ∞ t−1 −x
Gamma function defined for t > 0 by Γ(t) = 0 x e dx can be extended to the rest
of the complex plane, less the non-positive integers, by analytic continuation, and that
consequently the formula Γ(t+1) = tΓ(t) holds for all real t, less the non-positive integers.
Lemma 2.4. For f ∈ W l,1 (Rd ) with l = [α] + 2 we have
(2.2)
Dθα f (x)
−1
=
Γ(1 − α)
Z
∞
0
[α]
h
i
X
(−r)p p
f (x − rθ) −
Dθ f (x) αr −α−1 dr
p!
p=0
a.e., where [α] denotes the integer part of α. Moreover, there exists a constant C > 0
(independent of θ) such that
Z
(2.3)
|Dθα f (x)|dx ≤ C
d
R
for all θ ∈ S d−1 .
4
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
Proof. Let g(x, θ) denote the right hand side of (2.2). We first show that for some constant
C > 0 (independent of θ) we have
Z
(2.4)
|g(x, θ)|dx ≤ C
Rd
for all θ ∈ S d−1 .
Note that by Taylor expansion with integral form of the remainder we obtain
Z
[α]
X
(−1)[α]+1 r [α]+1
(−r)p p
Dθ
f (x − sθ)(r − s)[α] ds
Dθ f (x) =
(2.5)
f (x − rθ) −
p!
[α]!
0
p=0
Write for any δ > 0
Z
Z Z δ
[α]
X
(−r)p p
−α−1
f
(x
−
rθ)
−
D
|g(x, θ)| dx ≤ |Cα |
f
(x)
dr dx
αr
θ
p!
Rd
Rd 0
p=0
Z
Z
[α]
(2.6)
∞
X
(−r)p p
−α−1
f
(x
−
rθ)
−
+ |Cα |
D
f
(x)
dr dx
αr
θ
p!
Rd δ
p=0
= |Cα |(I1 + I2 )
where Cα = −1/Γ(1 − α). Now by Tonelli we have


Z ∞ Z
[α]
p Z
X
r

I2 ≤
|f (x − rθ)| dx +
|Dθp f (x)| dx αr −α−1 dr
p! Rd
δ
Rd
p=0
(2.7)
Z ∞
Z ∞
[α]
X
1
p
−α−1
kDθ f k1
αr
dr +
αr p−α−1 dr < ∞
= kf k1
p!
δ
δ
p=0
since p − α − 1 ≤ [α] − α − 1 < −1 for all 0 ≤ p ≤ [α].
For I1 we obtain by using (2.5)
Z r
Z Z δ
−α−1
1
[α]+1
[α]
αr
D
f
(x
−
sθ)(r
−
s)
ds
dr dx
I1 ≤
θ
[α]!
d
0
0
R
Z Z δZ r
1
[α]+1
D
≤
f
(x
−
sθ)
θ
(r − s)[α] ds αr−α−1 dr dx
[α]! Rd 0 0
Z δZ rZ 1
[α]+1
(2.8)
=
f (x − sθ) dx(r − s)[α] ds αr−α−1 dr
Dθ
[α]! 0 0 Rd
Z δZ r
1
[α]+1
= kDθ
f k1
(r − s)[α] ds αr−α−1 dr
[α]! 0 0
Z δ
α
[α]+1
= kDθ
f k1
r−α+[α] dr < ∞ (since − α + [α] > −1)
([α] + 1)! 0
VECTOR GRÜNWALD FORMULA
5
Then (2.6) along with (2.7) and (2.8) prove (2.4).
Recall that the Fourier transform of f (x − a) is eihk,ai fˆ(k). In view of (2.4) we compute
using Fubini’s theorem
Z
eihk,xi g(x, θ)dx
ĝ(k, θ) =
d
R
Z ∞h
[α]
X
(irhk, θi)p i −α−1 ˆ
irhk,θi
e
= Cα
αr
−
drf (k).
p!
0
p=0
By letting t = hk, θi we have to compute
J(α) =
Z
∞
0
[α]
h
X
(irt)p i −α−1
irt
e −
αr
dr.
p!
p=0
The argument is similar to the case 1 < α < 2 proved in [17], Lemma 7.3.8. We only
sketch the argument. For s > 0 let
Z ∞h
[α]
X
((it − s)r)p i −α−1
(it−s)r
e
(2.9)
Js (α) =
αr
−
dr.
p!
0
p=0
If 0 < α < 1 then by (7.27) in [17] we obtain
Z ∞
(it−s)r
Js (α) =
− 1 αr −α−1 dr = −Γ(1 − α)(s − it)α .
e
0
If α > 1 then we integrate by parts to obtain
Z ∞h
[α]−1
X ((it − s)r)p i
(it−s)r
e
r−α dr
−
Js (α) = (it − s)
p!
0
p=0
it − s
Js (α − 1)
α−1
(it − s)[α]
= ··· =
Js (α − [α]).
(α − 1) · · · (α − [α])
=
Now, since 0 < α − [α] < 1, by (7.27) of [17] we have Js (α − [α]) = C1 (s − it)α−[α] where
C1 = −Γ(1 − (α − [α])). Using the property Γ(t + 1) = tΓ(t) we get
−Γ([α] − α + 1)
(it − s)[α] (s − it)α−[α]
(α − 1) · · · (α − [α])
−Γ([α] − α + 1)
(s − it)α
=
(1 − α) · · · ([α] − α)
= −Γ(1 − α)(s − it)α .
Js (α) =
6
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
Note that the absolute value of the integrand in (2.9) is bounded by C2 r[α]−α−1 as
r → ∞ and by C3 r[α]−α as r → 0, so by dominated convergence we get Js (α) → J(α) as
s → 0. Hence J(α) = (−it)α and then
ĝ(k, θ) = (−ihk, θi)α fˆ(k) = (Dθα f ) b (k).
Hence, by the uniqueness of the Fourier transform we have Dθα f (x) = g(x, θ) a.e. and
(2.4) holds true. This concludes the proof.
R
Proof of Theorem 2.2. Let h(x) = kθk=1 Dθα f (x)M(dθ) where Dθα f (x) is given by (2.2)
and hence (Dθα f )b(k) = (−ihk, θi)α fˆ(k). Since by Lemma 2.4
Z
Z
|Dθα f (x)| dx M(dθ) < ∞
kθk=1 Rd
we can apply Fubini’s theorem to obtain
Z
Z
ihk,xi
ĥ(k) =
e
Dθα f (x) M(dθ) dx
d
R
kθk=1
Z
Z
eihk,xi Dθα f (x) dx M(dθ)
=
kθk=1 Rd
Z
(2.10)
=
(Dθα f )b(k) M(dθ)
kθk=1
Z
α
(−ihk, θi)α M(dθ)fˆ(k) = (DM
f )b(k)
=
kθk=1
and the uniqueness of the Fourier transform yields (2.1). This concludes the proof.
3. A multivariable Grünwald formula
In this section we derive, using Theorem 2.2 above, a multivariable analogue of the
scalar Grünwald formula (1.1). Furthermore, generalizing a result in [31], a speed of
convergence result is also obtained. Speed of convergence results are critical for numerical
applications.
Tuan and Gorenflo [31] show that for certain functions f
dα f (x)
= h−α ∆αh f (x) + O(h)
dxα
as h → 0. We now apply a similar argument to the fractional directional derivative.
Recall that the Fourier transform of Dθα f (x) is (−ihk, θi)α fˆ(k). Further define
∞
X
α
−α α
−α
m
(−1)
f (x − mhθ)
(3.1)
h ∆h,θ f (x) = h
m
m=0
VECTOR GRÜNWALD FORMULA
7
Theorem 3.1. For f ∈ W l,1 (Rd ) with l = [α] + d + 2
Dθα f (x) = h−α ∆αh,θ f (x) + O(h)
(3.2)
independent of θ ∈ S d−1 and x ∈ Rd .
R
Proof. Let fˆ(k) = eihk,xi f (x)dx be the Fourier transform of f (x) so that eihk,ai fˆ(k) is
the Fourier transform of f (x − a).
Note the well known result that
∞ X
α
α
(3.3)
(1 + z) =
zm
m
m=0
for any complex |z| ≤ 1 and any α > 0. Further, the binomial series is absolutely
convergent (page 180, [13]).
It is readily verified that
Z X
∞
∞
X
α
α
m
m
(−1)
|(−1)
f (x − mhθ) dx ≤ kf k1
|<∞
m
m
Rd m=0
m=0
Consequently, the right hand side of (3.1) defines an element of L1 (Rd ). Thus we can
take Fourier transforms in (3.1) to obtain
∞
X
α ihk,mhθi ˆ
−α α
−α
m
h ∆h,θ f b(k) = h
f (k)
(−1)
e
m
m=0
!
∞ X
α
= h−α
(−eihk,hθi )m fˆ(k)
m
m=0
(3.4)
α
−α
=h
1 − eihk,hθi fˆ(k)
α 1 − eihk,hθi
−α
α
fˆ(k)
=h
(−ihk, hθi)
−ihk, hθi
= (−ihk, θi)α w(−ihk, hθi)fˆ(k)
where
α
1 − e−z
α
w(z) =
= 1 − z + O(|z|2 ).
z
2
Note that |w(−ix) − 1| ≤ C|x| for all x ∈ R for some C > 0. Then
h−α ∆αh,θ f b(k) = (−ihk, θi)α fˆ(k) + (−ihk, θi)α (w(−ihk, hθi) − 1)fˆ(k)
(3.5)
= Dθα f b(k) + ϕ̂(h, k)
where ϕ̂(h, k) = (−ihk, θi)α (w(−ihk, hθi) − 1)fˆ(k). Since f ∈ W l,1 (Rd ) the RiemannLebesgue Lemma allows us to conclude that
|fˆ(k)| ≤ M/(1 + kkk)l
8
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
for some M ([12] Theorem 8.22) and thus
M(1 + kkk)α+1
M
=
(1 + kkk)α+1 |fˆ(k)| ≤
[α]+d+2
(1 + kkk)
(1 + kkk)[α]−α+d+1
which yields the integrability result
Z
(1 + kkk)α+1 |fˆ(k)|dk < ∞.
(3.6)
d
R
Now since
|ϕ̂(h, k)| ≤ kkkα Ckhkk |fˆ(k)| ≤ C h (1 + kkk)α+1 |fˆ(k)|
ϕ̂(h, ·) ∈ L1 (Rd ) for each h. Taking inverse Fourier transforms of
ϕ̂(h, k) = h−α ∆α f b(k) − D α f b(k)
h,θ
we conclude that for some constant cd > 0
|h
−α
∆αh,θ f (x)
−
Dθα f (x)|
(3.7)
θ
Z
−ihk,xi
= cd
e
ϕ̂(h, k) dk
d
ZR
|ϕ̂(h, k)| dk
≤ cd
Rd
≤ Kh
for some constant K independent of θ ∈ S d−1 and x ∈ Rd . This shows that
(3.8)
Dθα f (x) = h−α ∆αh,θ f (x) + O(h)
independent of θ ∈ S d−1 and x ∈ Rd and the theorem is proven.
As a consequence of (2.1) and (3.2) we obtain
Corollary 3.2. For f ∈ W l,1 (Rd ) with l = [α] + d + 2 there exists a constant C > 0
independent of x such that
Z
α
−α α
(3.9)
h ∆h,θ f (x)M(dθ) ≤ Ch.
DM f (x) −
kθk=1
α
Remark 3.3. The vector fractional derivative DM
is the negative generator of a multivarid
able stable semigroup on R . A stable probability distribution ν on Rd has characteristic
function
Z
α
(−ihk, θi) M(dθ)
(3.10)
ν̂(k) = exp −C
kθk=1
for some C > 0. This follows from the Lévy representation by a straightforward but
lengthy calculation, see for example [17] Section 7.3. Then it follows that the generator
α
Aν = −DM
. In one variable this idea has been used to prove (1.1) by discretizing the
Lévy measure of the stable probability distribution, see [20]. The same approach also
works for vector fractional derivatives, but that method makes it more difficult to obtain
the rate of convergence.
VECTOR GRÜNWALD FORMULA
9
4. Vector Grünwald formula on a regular grid
Finite difference algorithms for solving fractional partial differential equations require
a method for estimating fractional derivatives using only the function values on a regular
grid. In this section, we modify the Grünwald approximation formula for vector fractional
derivatives developed in Section 3 to operate on a rectangular grid, and we also establish
order of convergence. In view of Corollary 3.2, for functions f ∈ W [α]+d+2,1 (Rd ), we can
α
approximate the multivariable fractional derivative DM
f (x) uniformly in x ∈ Rd by
Z
∞
X
α
α
−α
m
(4.1)
DM f (x) = h
(−1)
f (x − mhθ)M(dθ) + O(h).
m kθk=1
m=0
R
In order to apply this formula numerically we have to approximate kθk=1 f (x−mhθ)M(dθ)
in such a way that the order O(h) in (4.1) is preserved. We will only consider the case
when M is a discrete measure or when M has a Lipschitz-continuous density with respect
to the surface measure on S d−1 . This is usually sufficient for most practical purposes.
Usually the function f is only known (or stored) on a regular grid Gh = u(h)Zd ⊂ Rd for
some u(h) > 0. Typically one has u(h) = h or u(h) = hβ for some β > 0. Since x − mhθ
is not a grid point, we have to evaluate f on a grid point close to x − mhθ and then we
have to control the error.
Recall that M is a probability measure on S d−1 .
Case I: (discrete M)
Assume that
(4.2)
M=
∞
X
al εθl
l=1
for some al ≥ 0, kθl k = 1, where εa denotes the point mass in a ∈ Rd . In view of Theorem
3.1 and (4.1) we have
∞
∞
X
X
α
α
−α
m
(4.3)
DM f (x) = h
al
(−1)
f (x − mhθl ) + O(h)
m
l=1
m=0
uniformly in x ∈ Rd for f ∈ W [α]+d+2,1 (Rd ).
Now let Gh = h1+α Zd be the grid of mesh size h1+α . Given any vector v ∈ Rd let
g(v) ∈ Gh denote the nearest grid point close to v. Then, for some constant Cd > 0 we
have
(4.4)
kv − g(v)k ≤ Cd h1+α
for all v ∈ Rd .
Ties can be broken arbitrarily, and in fact, all of the ensuing arguments apply equally
well for any function g : Rd → Gh such that (4.4) holds.
10
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
Theorem 4.1. If f ∈ W [α]+d+2,1 (Rd ) and (4.2) holds, then
∞
∞
X
X
α
α
−α
m
(4.5)
DM f (x) = h
al
(−1)
f (x − g(mhθl )) + O(h)
m
l=1
m=0
d
uniformly in x ∈ R , especially in x ∈ Gh .
Proof. Since f ∈ W [α]+d+2,1 (Rd ), f is continuously differentiable and all first order partial
derivatives are bounded. Hence f is Lipschitz-continuous, that is, there exists a constant
K > 0 such that
|f (x) − f (y)| ≤ Kkx − yk for all x, y ∈ Rd .
(4.6)
Hence, by (4.4) and (4.6), for some constant C > 0 and any l ≥ 1 we have
∞
∞
X
X
α
α
−α m
m
h (−1)
(−1)
f (x − mhθl ) −
f (x − g(mhθl ))
m
m
m=0
m=0
∞
X
α −α
m
(−1)
· |f (x − mhθl ) − f (x − g(mhθl ))|
≤h
m
m=0
∞
X
α −α 1+α
m
(−1)
≤KCd h h
m
m=0
=Ch.
Then (4.5) follows from (4.3) and the proof is complete.
Case II: (M has a Lipschitz-continuous density)
Assume that there exists a function ψ : S d−1 → R+ which is Lipschitz-continuous, that
is, for some L > 0
(4.7)
|ψ(θ1 ) − ψ(θ2 )| ≤ Lkθ1 − θ2 k for all θ1 , θ2 ∈ S d−1 ,
such that M(dθ) = ψ(θ)σ(dθ), where σ denotes the surface measure on S d−1 . We introduce standard polar coordinates on S d−1 (see, e.g., [11] p. 218). Let X = [0, π)d−2 ×[0, 2π)
and denote the elements of X by Φ = (φ1 , . . . , φd−1 ). Define the diffeomorphism


cos φ1


sin φ1 cos φ2


.

..
T (Φ) = T (φ1 , . . . , φd−1 ) = 


sin φ sin φ · · · sin φ cos φ 
1
2
d−2
d−1
sin φ1 sin φ2 · · · sin φd−2 sin φd−1
and note that σ(dθ) = λd−1 (T −1 (dθ)) where λd−1 is the Lebesgue product measure on X.
Further we define the standard metric on X by
d(Φ, Φ̄) =
d−1
X
j=1
|φj − φ̄j |2
1/2
.
VECTOR GRÜNWALD FORMULA
11
Then it is easy to see that for some constant C > 0
kT (Φ) − T (Φ̄)k ≤ Cd(Φ, Φ̄) for all Φ, Φ̄ ∈ X.
Hence, in view of (4.7) we get for some constant C > 0 that
(4.8)
|ψ(T (Φ)) − ψ(T (φ̄))| ≤ Cd(Φ, Φ̄) for all Φ, Φ̄ ∈ X.
R
Note that in kθk=1 f (x − mhθ)M(dθ) in (4.1) we integrate f over a sphere with radius
mh around x. In order to get an approximation error that is O(h) we need to approximate
M in a way that depends on both m and h. When m is larger, the sphere of radius mh is
larger, and so we need to use a finer discretization of the density to get the same accuracy.
Now we define the grid points we will use to approximate the measure M. Given any
m ≥ 1 and h > 0 define for 1 ≤ i ≤ d − 2
(h,m,j)
φi
=j
h1+α
m
for 0 ≤ j ≤ Ji (h, m) =
πm (h,m,Ji (h,m)+1)
=π
and φi
h1+α
and
(h,m,j)
φd−1
=j
h1+α
m
for 0 ≤ j ≤ Jd−1 (h, m) =
2πm (h,m,Jd−1 (h,m)+1)
= 2π
and φi
1+α
h
and set
(h,m)
(h,m,j1 )
θj1 ,...,jd−1 = T φ1
(h,m,j2 )
, φ2
(h,m,jd−1 ) , . . . , φd−1
(h,m,j)
(h,m,j+1)
(h,m,j)
for 0 ≤ ji ≤ Ji (h, m) + 1 and 1 ≤ i ≤ d − 1. Note that ∆φi
= φi
− φi
(h,m,j)
1+α
1+α
h /m for 0 ≤ j < Ji (h, m) while 0 ≤ ∆φi
≤ h /m for j = Ji (h, m). Define
Jd−1 (h,m)
J1 (h,m)
(4.9)
Mh,m =
X
j1 =0
···
X
(h,m,j1 )
(h,m,j
)
(h,m)
ψ θj1 ,...,jd−1 ∆φ1
· · · ∆φd−1 d−1 εθ(h,m)
jd−1 =0
=
.
j1 ,...,jd−1
Then we have
Theorem 4.2. Assume M has a density such that (4.7) holds. Then for any f ∈
W [α]+d+2,1 (Rd ) there exists a constant C > 0 (independent of x and h) such that
(4.10)
∞
X
α
DM f (x) − h−α
(−1)m
m=0
for all x ∈ Rd and h > 0.
Z
α
f (x − mhθ)Mh,m (dθ) ≤ Ch
m kθk=1
12
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
Proof. In view of Theorem 2.2 we write
Z
∞
X
α
α
m
D f (x) − h−α
(−1)
f (x − mhθ)Mh,m (θ)
M
m kθk=1
m=0
Z
Z
≤
Dθα f (x)M(dθ) −
h−α ∆αh,θ f (x)M(dθ)
kθk=1
Z
+
kθk=1
h
−α
∆αh,θ f (x)M(dθ)
−h
−α
kθk=1
∞
X
m=0
Z
α
(−1)
f (x − mhθ)Mh,m (dθ)
m kθk=1
m
=E1 (h) + E2 (h).
Note that by (3.9) we already have E1 (h) = O(h) uniformly in x ∈ Rd . Moreover
hZ
Z
∞
i
α
−α X
m
E2 (h) = h
(−1)
f (x − mhθ)M(dθ) −
f (x − mhθ)Mh,m (dθ) m
kθk=1
kθk=1
m=0
Z
Z
∞
X
α −α
m
f (x − mhθ)M(dθ) −
f (x − mhθ)Mh,m (dθ)
(−1)
≤h
m
kθk=1
kθk=1
m=0
Recall that σ(dθ) = λd−1 (T −1 (dθ)). Now substitute θ = T (Φ) and apply the mean value
theorem for integrals d − 1 times to obtain
Z
Z
f (x − mhθ)M(dθ) =
f x − mhT (Φ) ψ T (Φ) dΦ
=
Z
kθk=1
π
···
0
J1 (h,m)
=
X
Z
π
0
···
Z
X
2π
f x − mhT (φ1 , . . . , φd−1 ) ψ T (φ1 , . . . , φd−1 ) dφd−1 dφd−2 . . . dφ1
0
Jd−1 (h,m) Z φ(h,m,j1 +1)
X
1
j1 =0
jd−1 =0
j1 =0
jd−1 =0
J1 (h,m)
Jd−1 (h,m)
···
Z
(h,m,j1 )
φ1
(h,m,jd−1 +1)
φd−1
(h,m,jd−1 )
φd−1
f x − mhT (φ1 , . . . , φd−1 ) ψ T (φ1 , . . . , φd−1 ) dφd−1 . . . dφ1
Z φ(h,m,jd−2 +1)
Z φ(h,m,j
Jd−1 (h,m)
J1 (h,m)
1 +1)
X
X
1
d−2
(h,m,jd−1 )
=
···
∆φd−1
· · · (h,m,j )
(h,m,j1 )
φ1
φd−2
d−2
(jd−1 ) (jd−1 ) f x − mhT (φ1 , . . . , φd−2 , φ̄d−1
) ψ T (φ1 , . . . , φd−2 , φ̄d−1
) dφd−2 . . . dφ1
..
.
=
X
j1 =0
···
X
jd−1 =0
(h,m,j1 )
∆φ1
(h,m,jd−1 )
· · · ∆φd−1
(jd−1 ) (jd−1 ) (j )
(j )
f x − mhT (φ̄1 1 , . . . , φ̄d−1
) ψ T (φ̄1 1 , . . . , φ̄d−1
)
VECTOR GRÜNWALD FORMULA
(h,m,ji )
for some φi
Z
(ji )
≤ φ̄i
(h,m,ji +1)
≤ φi
13
and 1 ≤ i ≤ d − 1. Moreover, by (4.9)
f (x − mhθ)Mh,m (dθ)
kθk=1
Jd−1 (h,m)
J1 (h,m)
X
=
X
···
j1 =0
(h,m,j1 )
∆φ1
jd−1 =0
(m,h,j1 )
f x − mhT (φ1
Hence
Z
(4.11)
(m,h,jd−1 )
, . . . , φd−1
f (x − mhθ)M(dθ) −
kθk=1
X
Z
kθk=1
(m,h,j
) (m,h,j1 )
) ψ T (φ1
, . . . , φd−1 d−1 ) .
f (x − mhθ)Mh,m (dθ)
Jd−1 (h,m)
J1 (h,m)
≤
(h,m,jd−1 )
· · · ∆φd−1
···
j1 =0
X
(h,m,j1 )
∆φ1
(h,m,jd−1 )
· · · ∆φd−1
jd−1 =0
(jd−1 ) (jd−1 ) (j )
(j )
) ψ T (φ̄1 1 , . . . , φ̄d−1
)
f x − mhT (φ̄1 1 , . . . , φ̄d−1
(m,h,j
) (m,h,j
) (m,h,j1 )
(m,h,j1 )
, . . . , φd−1 d−1 ) ψ T (φ1
, . . . , φd−1 d−1 ) − f x − mhT (φ1
It follows as in the proof of Theorem 4.1 that f is Lipschitz continuous, that is (4.6) holds.
Moreover kψk∞ < ∞ and kf k∞ < ∞ and (4.8) holds. Using the inequality
|f (u)ψ(u) − f (v)ψ(v)| ≤ kψk∞ |f (u) − f (v)| + kf k∞ |ψ(u) − ψ(v)|
we obtain for some constants C1 , C2 , C > 0 that
(jd−1 ) (jd−1 ) (j )
(j )
) ψ T (φ̄1 1 , . . . , φ̄d−1
)
f x − mhT (φ̄1 1 , . . . , φ̄d−1
(m,h,j
) (m,h,j
) (m,h,j1 )
(m,h,j1 )
, . . . , φd−1 d−1 ) ψ T (φ1
, . . . , φd−1 d−1 ) − f x − mhT (φ1
(j
(j )
)
(m,h,j1 )
d−1
≤C1 mhkT (φ̄1 1 , . . . , φ̄d−1
)) − T (φ1
(m,h,jd−1 )
, . . . , φd−1
)k + C2
h1+α
m
h1+α
h1+α
+ C2
≤ Ch1+α
m
m
for all m ≥ 1, x ∈ Rd and 0 < h ≤ 1. Then the right hand side of (4.11) is bounded from
above by
≤C1 mh
Jd−1 (h,m)
J1 (h,m)
1+α
Ch
X
j1 =0
···
X
(h,m,j1 )
∆φ1
(h,m,jd−1 )
· · · ∆φd−1
= 2π d−1 Ch1+α
jd−1 =0
for all x ∈ Rd and 0 < h ≤ 1. Finally, we obtain for some constant C̄ > 0 that
∞
X
α d−1 1+α
−α
m
E2 (h) ≤ h
= C̄h
(−1)
2π Ch
m
m=0
14
MARK M. MEERSCHAERT, JEFF MORTENSEN, AND HANS-PETER SCHEFFLER
for all x ∈ Rd and 0 < h ≤ 1. This concludes the proof.
As in Case I we now move the evaluation of f to grid points. Recall that g(v) denotes
the nearest grid point close to v ∈ Rd and that (4.4) holds.
Theorem 4.3. Under the assumptions of Theorem 4.2 we have
α
f (x)
DM
J1X
Jd−1 (h,m)
(h,m)
X
α
=h
(−1)
···
m
m=0
j1 =0
jd−1 =0
(h,m,j1 )
(h,m,j
)
(h,m)
(h,m)
ψ θj1 ,...,jd−1 f x − g(mhθj1 ,...,jd−1 ) ∆φ1
· · · ∆φd−1 d−1 + O(h)
−α
∞
X
m
uniformly in x ∈ Rd (and especially in x ∈ Gh ).
Proof. Using (4.4) and (4.6) we have
(h,m)
≤ Ch1+α
)
−
f
(x
−
g(mhθ
))
j1 ,...,jd−1
j1 ,...,jd−1
f (x − mhθ (h,m)
and then for some C4 > 0
Z
∞
α
−α X
m
(−1)
f (x − mhθ)Mh,m (dθ)
h
m kθk=1
m=0
−h
−α
∞
X
m=0
ψ
J1X
Jd−1 (h,m)
(h,m)
X
α
(−1)
···
m
m
(h,m)
θj1 ,...,jd−1 f
j1 =0
x−
jd−1 =0
(h,m,j1 )
(h,m)
g(mhθj1 ,...,jd−1 ) ∆φ1
(h,m,jd−1 ) · · · ∆φd−1
J1X
Jd−1 (h,m)
(h,m)
∞
X
α
−α X
(h,m,j
)
(h,m,j1 )
m
= h
(−1)
···
∆φ1
· · · ∆φd−1 d−1
m
m=0
j1 =0
jd−1 =0
(h,m)
(h,m)
(h,m)
ψ θj1 ,...,jd−1 f x − mhθj1 ,...,jd−1 − f x − g(mhθj1 ,...,jd−1 ) J1X
Jd−1 (h,m)
(h,m)
∞
X
X
α (h,m,j
)
(h,m,j1 )
−α m
≤h (−1)
···
∆φ1
· · · ∆φd−1 d−1 kψk∞ Ch1+α
m
m=0
j1 =0
jd−1 =0
∞
X
α d−1
= h−α (−1)m
2π kψk∞ Ch1+α ≤ C4 h.
m m=0
Now Theorem 4.2 together with (4.9) gives the desired result.
VECTOR GRÜNWALD FORMULA
15
Remark 4.4. If the step size h is chosen so that π/h1+α is an integer, then the Grünwald
approximation formula in Theorem 4.2 becomes simpler. In this case, we have
∞
X
1
α
α
(1+α)(d−1)−α
m
(−1)
DM f (x) = h
m md−1
m=0
Jd−1 (h,m)−1
J1 (h,m)−1
X
j1 =0
···
X
(h,m)
(h,m)
ψ θj1 ,...,jd−1 f x − g(mhθj1 ,...,jd−1 ) + O(h)
jd−1 =0
uniformly in x ∈ Rd (and especially in x ∈ Gh ).
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Mark M. Meerschaert, Department of Mathematics and Statistics, University of
Nevada, Reno, USA
E-mail address: mcubed@unr.edu
Jeff Mortensen, Department of Mathematics and Statistics, University of Nevada,
Reno NV 89557-0084 USA
E-mail address: jm@unr.edu
Hans-Peter Scheffler, Fachbereich Mathematik, University of Siegen, 57068 Siegen,
Germany
E-mail address: hps@math.uni-dortmund.de