Distance function wavelets – Part II: Extended results and conjectures

W. Chen
Simula Research Laboratory, P. O. Box. 134, 1325 Lysaker, Norway
E-mail: wenc@simula.no
(21 May 2002)

Summary
Report II is concerned with the extended results of distance function wavelets (DFW).
The fractional DFW transforms are first addressed relating to the fractal geometry and
fractional derivative, and then, the discrete Helmholtz-Fourier transform is briefly
presented. The Green second identity may be an alternative devise in developing the
theoretical framework of the DFW transform and series. The kernel solutions of the
Winkler plate equation and the Burger’s equation are used to create the DFW transforms
and series. Most interestingly, it is found that the translation invariant monomial solutions
of the high-order Laplace equations can be used to make very simple polynomial DFW
series. In most cases of this study, solid mathematical analysis is missing and results are
obtained intuitively in the conjecture status.
Keywords: distance function wavelets, fractional Helmholtz transform and series,
discrete Helmholtz-Fourier transform, Green second identity, Winkler plate, Burger’s
equation, Kelvin functions, Laplace equation, harmonic function, translation invariant
monomial solution, polynomial DFW series.

1

1. Introduction
This report is the second in series [1,2] about my latest advances on the distance function
wavelets (DFW). Unlike the common distance functions, e.g., MQ and TPS, which have
no provision for scaling and carry out a multiresolution hierarchy by simply dropping or
adding some points [3], the DFW is comprised of both the scale and translation
arguments. To better understand what will be presented here, the readers are advised to
have a look at Report I [1] beforehand. The report is featured with lots of grand
conjectures, where firm mathematical underpinnings are conspicuously lacking in most
cases. But nevertheless the author assumes that many results have certain physical
grounds and are in agreement with the faith that God rules the world with simplicity and
beauty.
The rest of report is organized into six thematic sections. In section 2, the fractional
Helmholtz-Fourier transform (HFT) and series and Helmholtz-Laplace transform (HLT)
are presented in relation to fractal geometries and fractional derivatives. In section 3, we
briefly present the discrete Helmholtz-Fourier transform without mathematical
justifications. Section 4 tries to derive the distance function wavelets by the Green second
identity and the Laplace transform. In section 5, the solutions of the Winkler plate
equation and the Burger’s equation are utilized to create some novel DFW transforms and
series. In section 6, the translation invariant monomial solutions of the high-order
Laplace equations are applied to develop the polynomial DFW series. Finally, section 7
provides a few supplementary results on the DFW and points out some potential uses.

2. Fractional Helmholtz transforms and series
In recent years, much attention has been attracted to the so-called fractional derivative
[4], fractional Fourier [5] and Laplace transforms and fractal geometry. The underlying
relationships between them are also unraveled [5-7]. Note that “fractional” is just
conventional misnomer since it also actually indicates real number relating to the so-

2

called fractional derivative and integral transforms. The HFT and HLT [1] were
recognized as the distance function counterparts of the Fourier and Laplace transforms,
respectively. By analogy with the latter two, this section addresses the fractional DFW
Helmholtz transforms and series.

The HFT is given by

1
Cn

Fn (ω ,ξ ) =

∫

f ( x )E n

−iω ξ − x

IR n

dx

(1a)

and
f (x ) =

1
Cn

+∞

∫ ∫

− ∞ IR

Fn (ω ,ξ )E n

iω x −ξ

n

dξdω ,

(1b)

while the HLT [1] is stated by

Ln (s,ξ ) = ∫

IR n

f ( x )E n

−s ξ −x

dx ,

(2)

where n denotes the dimensionality, and the function E [1] is defined by

E n−λx

E niλx


λ1 2 −λx
e ,
n = 1,

=  n −1 2 2π
λ
(2πλx )−(n 2 )+1 K (n 2 )−1 (λx ), n ≥ 2,
 2π

x≥0,


λ1 2
(cos(λx ) + i sin(λx )),
n = 1,

x≥0,
=  n −1 2 2π
i
λ
− ( n 2 )+1

(J (n 2 )−1 (λx ) + iY(n 2 )−1 (λx )), n ≥ 2,
 4 (2πλx )

3

(3a)

(3b)

where Jn/2 and Yn/2 are respectively the Bessel functions of the first and second order of
the n/2-1 order, and Kn/2 is the modified Bessel function of the second kind. There is a
number of ways to connect the HFT and HLT with the fractional derivative. One simple
way is through frequency domain. Obviously, the HFT hold

[

]

F ∇ 2 f ( x ) = (iω ) F [ f ( x )] = (iω ) F (ω , ξ ) ,
2

2

(4)

and generally, this process may be iterated for the m-th order divergence derivative to

[

]

F ∇ m f ( x ) = (iω ) F [ f ( x )] ,
m

(5)

where m is a integer number in the most stereotyped mathematics. Observing (5), there is
no reason hindering the extension of m to the set of real numbers, and then, comes out the
fractional derivative. If m<0 in (5), then we have fractional integration. Let us advance a
little further. m could be even a complex number corresponding to a complex order
partial differentiation [8], while keeping (5). We could develop a complete theory of
fractional HFT and HLT by analogy with the fractional Fourier and Laplace transforms.

On the other hand, Blu and Unser [3] have pointed out that the self-similarity of fractal
geometry is one fundamental concept to create the distance function wavelet. This
inspired the author to assume that dimensionality n can be a real number. Consequently,
(1) and (2) turn out to be a new type of the fractional HFT and HLT again

E n−λx


λ1 2 −λx
e ,
n = 1,

=  n −1 22π
λ
(
)−(n 2 )+1 K (n 2 )−1 (λx ), n ≠ 1,
 2π 2πλx

4

x≥0,

(6a)

E niλx


λ1 2
(cos(λx ) + i sin(λx )),
n = 1,

x≥0.
=  n −1 2 2π
i
λ
− ( n 2 )+1

(J (n 2 )−1 (λx ) + iY(n 2 )−1 (λx )), n ≠ 1,
 4 (− 2πiλx )

(6b)

The same is for fractional Helmholtz series given in [1]. A fractional n implies fractal
geometry. It is noted that the DFW HFT has a clear edge over the classic Fourier
transform in that the dimensionality explicitly appears in the HFT and thus may be
suitable to serve as an alternative powerful mathematical tool to quantitatively describe
the fractal geometry. In some areas (e.g. control engineering), the complex dimension is a
useful concept. Thus, n even could be a complex number. One thing that deserves more
attention is how to define the Euclidean distance in complex dimension geometry, and
then how about a negative n. The problem is what physical backdrops are behind these
exotic mathematical devices. The author conceives that there is an underlying link
between the complex dimensionality and complex order derivative through the HFT or
the HLT, when n is a complex number in (1), (2) and (3). It is certain that there are lots of
issues unanswered out there.

The above research displays that the dimensionality (fractal), scale (frequency) and
differentiation can be connected through the DFWs. [2] will give a detailed discussion on
these inherent relationships through the research of the frequency dependent attenuation
of acoustics and elastic waves.

3. Discrete Helmholtz-Fourier transform

In terms of continuous Helmholtz-Fourier transform (1), it is easy to attain the 1D
discrete Helmholtz-Fourier transform (DHT) as in the discrete Fourier transform. In
multidimensional cases, the basis functions of the continuous complex HFT, however,

5

encounter the singularity at the origin. It is noted that the basis functions of the
continuous HF J transform

ϕ1 (λrk ) =

ϕ n (λrk ) =

λ1 2
2

cos(λrk ) ,

(7a)

λn −1 2
(2πλrk )−(n 2 )+1 J (n 2 )−1 (λrk ) ,
2π

n≥2

(7b)

has arbitrary degree of continuity, where rk = x − x k . The HF J transforms are stated as

1
CJ

Fn (λ , x k ) =

∫ f (x )ϕ (λ x
n

Ω

k

− x )d x

(8a)

and
f (x ) = ∫

+∞

0

∫ F (λ , x )ϕ (λ x − x )dx dλ .
n

Ω

k

n

k

k

(8b)

The corresponding discrete transforms are

Fˆn (λ j ,ξ l ) = ∑ f ( x k )ϕ n (λ j ξ l − x k ),
N

(9a)

k =1

and
f (xk ) =

1
Ng

∑∑ Fˆ (λ ,ξ )ϕ (λ
M

N

n

j

l

j =1 l =1

We do not give any mathematical justification in this section.

6

n

j

x k − ξ l ).

(9b)

4. Distance function wavelets generated by Green’s function

[9] derives various coordinate variable transforms from the Green’s function. This section
tries to follow the same strategy to develop the distance function transforms. Consider the
partial differential equation

i

∂φ ( x, t )
= ℜ{φ ( x, t )}
∂t

(10)

with the initial condition

φ ( x,0) = f ( x )

(11)

and boundary conditions
 u ( x, t ) = 0,

 ∂u ( x, t ) = 0,
 ∂n

x ∈ Su ,
x ∈ ST ,

t ≥ 0,

(12)

where ℜ is the spatial partial differential operator, and x is a multidimensional variable.
In terms of the Laplace transform with respect to time t, we have

[ℜ − is]φ (x, s ) = −if (x ) .

(13)

The Green function solution of (13) is

φ ( x, s ) = −i ∫ g ( x − ξ ;−is ) f (ξ )dξ ,
Ωn

(14)

where g(x-ξ,-is) is the Green function of the operator ℜ-is satisfying the boundary
condition (12). Applying the inverse Laplace transform to (14) and let t=0, we attain

7

f (x ) = −

1
2π

γ +i∞

∫γ

−i∞

ds ∫ g (x − ξ ;−is ) f (ξ )dξ .
Ωn

(15)

Let s=iλ, where λ are the eigenvalues of the operator ℜ+λ, we have
f (x ) =

1 −ic +∞
dλ g (x − ξ ; λ ) f (ξ )dξ ,
2πi ∫−ic−∞ ∫Ωn

(16)

If ℜ is a self-adjoint operator, all eigenvalues λ are real. (16) is restated as

f (x ) =

1 +∞
dλ g ( x − ξ ; λ ) f (ξ )dξ .
2πi ∫−∞ ∫Ωn

(17)

It is known that the Green functions of many PDEs can be expressed by the
corresponding eigenfunctions, i.e.
∞

g ( x − ξ ; λ ) = ∑αψ m ( x )ψ m (ξ ) ,

(18)

m =0

where ψm is the eigenfunctions, and the index m is related with different eigenvalues λ.
To proceed further, we need to postulate that due to the translation or rotational invariant,
(18) may be restated as

g (x − ξ , λ ) =

1
λ2 n −1ψ ( x − ξ , λ )ψ (ξ − x, λ )dx ,
∫
Ω
C n

(19)

where the upper bar denotes the complex conjugate. (20) is a DFW expression of f(x).
Now substituting (19) into (17) produces the DFW transform

f (x ) =

1
Cψ

∫

+∞

−∞

dλ ∫ ψ (ξ − x, λ )dξ ∫ λ2 n −1ψ ( x − ξ , λ ) f (ξ )dx .
Ωn

Ωn

8

(20)

On the other hand, it is observed that (17) can also be seen as an anomalous wavelet
representation. Namely,
F ( x, λ ) = ∫ g ( x − ξ ; λ ) f (ξ )dξ ,

(21a)

1 +∞
F ( x, λ )dλ .
2πi ∫−∞

(21b)

Ωn

and
f (x ) =

For instance, in terms of the free Green function of the Helmholtz equation in 1D infinite
domain, the Fourier transform can be restated
F ( x, λ ) = ∫

+∞

−∞

f (ξ )e iλ ( x −ξ )dξ ,

(22a)

and
f (x ) =

1 +∞
F ( x, λ )dλ .
2πi ∫−∞

(22b)

The wavelets transforms (21) and (22) have exotic expressions compared with the
standard wavelet formalism. But nevertheless they have most properties of the wavelets.

5. Some distance function wavelets

Except the distance function wavelets developed [1], there are plenty of the solutions of
various PDEs eligible to create the DFW. This section is dedicated to developing a few
new types of the DFW.
5.1. DFWs involving Kelvin functions

The deflection of thin elastic plates resting on a Winkler elastic foundation with stiffness

κ is governed by

9

− ∆
∇4u + κ 2u =  i
 0

(23)

where ∆i represents the Dirac delta function at a source point i corresponding to the
fundamental solution (vs. zero for general solution); domain Ω can be unbounded or
bounded with or without boundary conditions. The plate is a two dimensions problem.
However, in this study, we extend equation (23) up to five dimensions since we found
their fundamental solution [10]

ϕ n* (rk ) =

(

i
rk κ
2π

)

− n 2 +1

(ker

n 2 −1

)

( rk κ ) + ikein 2−1 ( rk κ ) ,

2≤ n ≤5,

(24)

2≤ n ≤5,

(25)

and general solution

(

ϕ n# (rk ) = rk κ

)

− n 2 +1

(ber

n 2 −1

)

( rk κ ) + ibein 2−1 ( rk κ ) ,

where ker and kei are respectively the Kelvin and modified Kelvin functions of the
second kind, and ber and bei respectively represent the Kelvin and modified Kelvin
functions of the first kind. Note that ber and bei have arbitrarily degree of differential
continuity, kei has the second differential continuity, but ker encounters a singularity at
the origin.
Steady Schrodinger’s equation in a radial domain
x 2 w′′ + xw′ − (ix 2 + ν 2 ) w = 0

(26)

also has the solutions (24) and (25). By using the fundamental solution (24), we can
construct the distance function transform for a suitable function f(x)

10

f (η )ϕ n* ( ξ − η )κ n −1dη , 2≤ n ≤5,

(27a)

F (κ , ξ )ϕ n* ( x − ξ )dξdκ , 2≤ n ≤5,

(27b)

F (κ , ξ ) = ∫

IR n

and
f ( x ) = Cϕ−1 ∫

+∞

0

∫

IR n

The solution (25) is also capable to construct the DFW. By analogy with the HelmholtzFourier series [1], we have expansion series

∞

∞

(

f ( x ) = f 0 (x ) + ∑∑ rk κ j
j =1 k =1

)

− n 2 +1

(α

jk

)

bern 2−1 ( rk κ j ) + β jk bein 2−1 ( rk κ j ) ,

2≤ n ≤5,

(28)

where αjk and βjk are the expansion coefficients. Since f0(x) is related to zero value of κ,
equation (23) degenerates into a biharmonic equation

∇ 4 f 0 (x ) = 0

(29)

with the boundary data. As we did for the Helmholtz-Fourier series [1], f0(x) can be
evaluated by the boundary element method or the boundary knot method.
5.2. DFW transform and series with solutions of Burger’s equation

The Berger’s equation [11] for large deflections of plate is
− ∆
∇4u − α 2∇2u =  i
 0

11

(30)

where the Berger parameter α is constant over the domain but nonlinearly depends on the
lateral load. The nonlinear relation between the external load and the deformation is
represented by the second right-hand term [11]. We have the fundamental solution [12]

ψ 2* (rk ) =

−1
2πα 2

(ln(rk ) + K 0 (αrk ) ) ,

 α
rk2 −n
+ 
ψ (rk ) =
(n − 2 )S n (1)  2πrk
*
n





(31a)

− n 2 +1

K n 2−1 (αrk ) ,

3≤ n, (31b)

and general solution [11]
  α
ψ (rk ) = 1 + 
  2πrk

#
n





− n 2 +1


I n 2−1 (αrk )  ,



2≤ n,

(32)

where Kn/2-1 and In/2-1 are respectively the modified Bessel functions of the first and
second kinds, and Sn(1) is the surface size of unit n-dimensional sphere. By analogy with
the Helmholtz-Laplace transform, we have

B (α , ξ ) = ∫

IR n

f (x ) =

1
Cψ

f (η )ψ n* (α ξ − η )α 2 n −1dη .

γ + i∞

∫γ ∫

−i∞ IR n

B (α , ξ )ψ n# (µ x − ξ )dξdα .

(33a)

(33b)

The general solution (32) can also be used to construct the DFW series for representing
the functions bounded within finite domains.

12

6. DFW series with monomial Laplacian solutions

There are numerous solutions of the high-order Laplace equations satisfying rotational or
translation invariant. This section will develop the distance function wavelets series by
using the translation invariant monomial solutions. In [2], the rotational invariant
solutions are employed to create the DFW.
A translation in the plane is a transform
xi = x + a,

yi = y + b .

(34)

Translation invariant with the Laplacian [13] means
∇ 2 (xi + yi ) = ∇ 2 ( x + y ) .

(35)

For p-order Laplacian, invariance holds for m-order translation when m<2p, i.e. under
xim = ( x + a ) ,
m

xim y im = ( x + a ) ( y + b ) ,
m

m

y im = ( y + b ) ,
m

(36)

we have

∇ 2 p (xim + xim y im + y im ) = ∇ 2 p (x m + x m y m + y m ) .

(37)

Any real function satisfying the Laplacian is called a harmonic function. In this study, we
call the function, which satisfies the high-order Laplacian, the high-order harmonic
function. The harmonic function is also called the potential function which includes the
scalar and vector potential functions in engineering. Thus, the high-order harmonic
function is also often called the high-order scalar or vector potential functions. A linear
combination of translates of the monomial high-order harmonic functions can
approximate many smooth functions under the harmonic basis function space. As an

13

illustrative example, let us consider the two-dimension translation-invariant monomial
solutions of high order Laplace equations

{1, x − x , y − y , (x − x ) , (x − x )( y − y ), ( y − y ) K},
2

k

k

k

2

k

k

k

(38)

we can construct the DFW series to approximate continuously differential function Q(x,y)

∞

[

Q (x, y ) = c0 + ∑ c1k ( x − x k ) + c2 k ( y − y k ) + c3k ( x − x k )
k =1

2

]

(39)

+ c4 k ( x − x k )( y − y k ) + c5k ( y − y k ) K ,
2

where cjk are the expansion coefficients. Note that since the high order Laplacian is scale
invariant operator, the power exponents of monomials are considered the “scale
parameter” here. For higher dimension problems, the similar DFW expansion series can
be constructed. (39) is absolutely not a Taylor expansion although it may look like the
latter. (39) can be restated as

Nx N y

M

Q (x, y ) ≅ ∑∑∑ cij ,k ( x − xk ) ( y − y k ) .
i

j

(40)

i =0 j =0 k =1

Note (40) in fact have only one expansion coefficient for the constant term, i.e.
M

c0 = ∑ coo ,k . In terms of the finite difference, least square, collocation or Galerkin
k =1

schemes, the polynomial DFW expansion (40) can be simply used to the function
approximation and the numerical integration and the solution of partial differential
equation under arbitrary domain geometry in an explicit multiscale and meshfree fashion
[14]. As in the wavelets, we can truncate the scales and translates locally and get a sparse
DFW interpolation matrix in approximating Q(x,y). The completeness, accuracy and
numerical tests of the presented polynomial DFW are under study and will be reported in
a subsequent paper.

14

The Laplace equation is typically an elliptical equation. The present polynomial DFW
series is thus expected to perform well for a variety of elliptical equations. When applied
to parabolic and hyperbolic equations, it would be more efficient and reliable to modify
this DFW series (40) via simple function transform to reflect the features of those
equations. For example, we can transfer the convection-diffusion equation
v
D∇ 2 u + v • ∇u − ku = b( x, y , t )

(41)

into the elliptic modified Helmholtz equation
∇2w −τ 2w = 0

(42)

by an exponential variable transformation [15,16]
v v
 − ν • rk 
u ( x ) = exp 
w( x ) ,
 2 D 

(43)

v
where rk is the distance vector between the source and field points, and

[

]

1

2
v
τ = ( v 2D) + κ D 2 .

(44)

The nonsingular general solution of the convection-diffusion equation (41) is given by
vr
v ⋅r

r
τ n −1 2 − 2 Dk
(2πτrk )−(n 2 )+1 I (n 2 )−1 (τrk ) ,
u (τ , rk ) =
e
2π
#
n

n≥2,

|(45)

where n is the dimensionality and rk is the Euclidean distance as defined with (7b).
Accordingly, we construct the following modified polynomial DFW expansions

15

Nx Ny M
 − ν x ( x − x k ) − ν y ( y − y k )
(x − x k )i ( y − y k ) j (46a)
u ( x, y , t ) ≅ ∑∑∑ a ij ,k (t )exp 

2D
i = 0 j = 0 k =1



and
N

Nx
y M
v
i
j
u ( x, y , t ) ≅ ∑∑∑ d ij ,k (t )u n# (τ , rk )(x − x k ) ( y − y k ) .

(46b)

i = 0 j = 0 k =1

The approximation series (46) embeds the velocity direction.
We also can simplify (40) in some way. For example, the normal multiquadratic radial
basis function approximation
M

Q ( x , y ) ≅ ∑ hk
k =1

(x − x k )2 + ( y − y k )2 + sk2

(47)

can be understood a simplified version of (40) with a single scale parameter (shape
parameter sk). Determining sk, however, is often tricky and problem dependent.
In addition, the Laplacian DFW series can be constructed under the polar coordinates, i.e.
Q (x, y ) ≅ B0 + ∑∑ rk j [A jk sin( j (θ − θ k )) + B jk cos( j (θ − θ k ))],
N

M

(48)

j =1 k =1

where θ-θk is the angle between vectors (x,y) and (xk,yk). j indicates the scale. (48) is not
easy to use in general because of the difficulty in the evaluation of the angle.
It is feasible to create similar translation invariant DFW series via the translation
invariant sine and cosine solutions of the multidimensional Helmholtz equations in
handling periodic problems. In 2D case, we have

16

Nx Ny M

P ( x, y ) ≅ ∑∑∑ cij ,k sin(2πi ( x − x k ))cos(2πj ( y − y k )) .

(49)

i = 0 j = 0 k =1

For the diffusion and convection-diffusion problems, the similar DFW expansion can be
made via exp(-τ(x-xk)) and exp(-τ(y-yk)) and their combination with the velocity term as
shown in (43). In some particular application, we can construct a special translation
invariant distance functions which reflects the systematic characteristics (PDE and outer
forcing terms).
It is very desirable to have the orthogonal DFW expansion of the form:
Nx Ny M

u ( x, y , t ) ≅ ∑∑∑ bij ,k φix ( x − x k )φ jy ( y − y k ) ,

(50)

i =0 j = 0 k =1

where φ ix and φ i y are orthogonal eigenfunctions. It is very interesting to note that unlike
the fundamental solution and general solution DFWs in [1] and the preceding sections,
the polynomial and trigonometric DFW series developed in this section could have their
basis functions from the PDE solution (the separation of the variables) under the
rectangular or hypercube domains instead of the circle or hypersphere domains. As such,
(50) can nevertheless be simply used to handle arbitrary domain problems due to its
translation invariant property. For plenty of PDEs, orthogonal eigenfunctions under
rectangular and cube domains can be found in literature. Thus, it is not a difficult task to
construct such translate DFW series to solve a broad variety of problems. However, we
do not know if these eigenfunctions are still orthogonal under arbitrary domains.
The essential distinction of the present DFW expansions from the other coordinate
variable expansion approaches lies in that we use the translation invariant distance
variable solutions of the high order PDE to get the meshfree and multiscale interpolation
schemes. It is stressed that as the spherically symmetric solutions are closely related to
the fundamental and general solutions, most of the coordinate variable kernel solutions of

17

PDEs under regular domains are translate invariant solutions for arbitrary domain
problems.
The above polynomial DFW series in space can be extended to the time-space DFW. It is
known that the wave and heat problems have the time-harmonic solutions, which are in
some sense similar to the spatial harmonic functions of the Laplacian. In particular, the
wave equation has the time-reversal invariance solution. We may have the time-space
polynomial DFW series
Nt N x N y M

u ( x, y , t ) ≅ ∑∑∑∑ a sij ,k φ st (t − t k )φ ix ( x − x k )φ jy ( y − y k ) ,

(51)

s = 0 i = 0 j = 0 k =1

where t-tk≥0 for the diffusion problem and c 2 (t − t k ) − rk2 ≥ 0 for wave problems are
2

required to ensure the principle of casualty.
To illustrate the difference between the translation and rotational polynomial DFW series
of harmonic functions, we briefly list the rotational invariant Laplacian DFW series [2]

∞

∞

Q (x ) = Q0 ( x ) + ∑∑α mk u L*m ( x − x k ) ,
n

m =1 k =1

n≥2,

(52)

where u L*m is the fundamental solution of the m-th order Laplacian ∇2(m+1)u. When the
n

order m≥1, the high-order Laplacian fundamental solutions of 2D and 3D cases [17] are
no longer singular at the origin. Q0(x) is evaluated via the Green second identity

(

∂u L*0 x − x j
 ∂Q (x j ) *
Q 0 ( x ) = ∫ n −1 
u L0 x − x j −Q (x j ) n
n
S
n
∂
∂n


(

)

18

)

dx j ,


(53)

where Sn-1 are the surface of finite domains. (53) can be easily evaluated by the boundary
element method (BEM) with boundary data, and then Q0(x) at any inner locations can be
calculated via (52). The boundary knot method [10] is also an alternative to the BEM for
this task. For more details on the rotational invariant DFW series see the section 5 of [2].
(52) has simpler form and fewer expansion coefficients for high dimension problems than
the preceding monomial Laplacian DFW series, and u L*m of different orders m are
n

orthogonal. The drawback is that the calculation of the Laplacian fundamental solution is
more costly than that of monomials.

7. Supplementary results

The DFW is easily extended to hand the problems with anisotropic parameters via the
modified definition of the distance variable [18]. For instance, consider the Helmholtz
equation

− ∆ ,
∇(η∇u ) + λ2 u =  i
 0,

in Ω,

(54)

where η={ηj} are the anisotropic parameters in different coordinate directions. If all ηj
are constants, we have the common fundamental and general solutions via the modified
Euclidean distance variable

rˆk =

n

∑η
j =1

j

( x j − x kj ) 2 .

(55)

Then, we can replace the standard Euclidean variable in the HFT and HLT with the above
definition of distance variable. These modified HFT and HLT will be more efficient to
handle anisotropic problems.

19

For nonlinear problems, the distance variable may be defined by

rˆk =

n

∑η (u(x ))( x
j

j =1

j

− x kj ) 2 .

(56)

Another useful technique handling nonlinear problems is to use the Kirchhoff transform
as in the boundary element method. For detailed discussions of different definitions of the
Euclidean distance variable see [18]. For data processing, we need to keep in mind the
physical mechanism (PDE model) behind the data and consider the PDE-specific DFW
technique.
The distance function wavelets have some potential important applications. For example,
the Helmholtz-Fourier transform may be a competitive alternative to the Fourier
transform in various data processing techniques such as filtering, modulation and
correlation.
The DFW transform and series are established on the kernel distance variable solutions of
various linear time-space invariant PDE systems. In more general sense, the strategy can
be extended to the gauge invariant PDE model (gauge group) such as Yang-Mills
equation. Further, the inverse-scattering transform reveals that invariance (the
superposition principle) also holds in some important nonlinear PDE models. We are now
under way to research the DFW transform and series based on such nonlinear invariance
mechanism. [13] puts it “the invariance has powerful consequences which lie at the heart
of the physical idea”. We add that “distance” (various relative differences) is an
underlying fundamental devise depicting the invariant relativity. Such philosophy implies
the distance function wavelets, which underlie invariance and relativity, may be a
powerful tool to display, analyze and exploit universe symmetry, simplicity and beauty
out of mundane irregularity, clutter and difficulty.
It is worth pointing out again that most results in this report are intuitively attained
without any rigorous mathematical analysis. The readers need to note their conjecture

20

status. Further research is now under way and the author would like to get any comments
and opinions

References

1. W. Chen, Distance function wavelets – Part I: Helmholtz and convection-diffusion
transforms and series, CoRR preprint, http://xxx.lanl.gov/abs//0205019,

Research report of Informatics Dept. of Univ. of Oslo, May, 2002.
2. W. Chen, Distance function wavelets - Part III: “Exotic” transform and series, CoRR

preprint, Research report of Simula Research Laboratory, May, 2002.
3. T. Blu and M.Unser, Wavelets, fractals and radial basis functions, IEEE Trans. Signal

Processing, (to appear), 2002.
4. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives:

theory and applications, Gordon and Breach Science Publishers, 1993.
5. L.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with

Applications in Optics and Signal Processing, John Wiley & Sons, Ltd, 2001.
6. M. ZÄHLE: Fractional differentiation in the self-affine case. V - The local degree of
differentiability. Math. Nachr. 185, 279-306, 1997.
7. M. ZÄHLE: Harmonic calculus on fractals -- A measure--geometric approach II.
(Preprint).
8. N. Makris and M.C. Constantinou, Models viscoelasticity with complex-order
derivative, J. Engng. Mech., 119(7), 1453-1464, 1993.
9. B. Davies, Integral Transforms and Their Applications, Springer-Verlag, 1978.
10. W. Chen, Boundary knot method for Laplace and biharmonic problems. In the Proc.
of the 14th Nordic Seminar on Computational Mechanics, Lund, Sweden, Oct. 2001.
11. P.K. Kythe, Fundamental Solutions for Differential Operators and Applications,
Birkhauser, Boston, 1996.
12. J. Sladek and V. Sladek, The BIE analysis of the Berger equation, Ingenieur-Archiv,
53, 385-397, 1983.

21

13. W.A. Strauss, Partial Differential Equations, an Introduction, John Wiely & Sons,
1992.
14. W. Chen, Kernel distance function – Part III: Data processing and numerical PDEs

CoRR preprint, Research report of Simula Research Laboratory, May, 2002.
15. Li BQ, Evans JW. Boundary element solution of heat convection-diffusion problems.

J. Comp. Phy., 1991;93:225-272.
16. W. Chen, High-order fundamental and general solutions of convection-diffusion
equation and their applications with boundary particle method, Engng. Anal. with

BEM, (in press), 2002.
17. A.J. Nowak and A.C. Neves (ed.). The Multiple Reciprocity Boundary Element
Method. Southampton: Comput. Mech. Publ., 1994.
18. W. Chen, Definitions of distance function in radial basis function approach, CoRR

preprint, http://mpej.unige.ch/mp_arc/a/00-304, July, 2000.

22

