Applied Mathematics, 2010, 1, 439-445
doi:10.4236/am.2010.16058 Published Online December 2010 (http://www.SciRP.org/journal/am)
On Approximating Two Distributions from a Single
Complex-Valued Function
William Dana Flanders, George Japaridze
Department of Epidemiology, Rollins School of Public Health, Atlanta, USA
Department of Physic s, Clark At l ant a Uni ver si t y, Atlanta, USA
Received May 1, 2010; revised July 10, 2010; accepted July 15, 2010
Abstract
We consider the problem of approximating two, possibly unrelated probability distributions from a single
complex-valued function

and its Fourier transform. We show that this problem always has a solution
within a specified degree of accuracy, provided the distributions satisfy the necessary regularity conditions.
We describe the algorithm and construction of

and provide examples of approximating several pairs of
distributions using the algorithm.
Keywords: Probability Distribution, Sinc Approximation, Cardinal Function, Fourier Transform, Wavelet
1. Introduction
In this paper we consider the problem of reconstructing
two different probability distributions from a single com-
plex-valued function with a specified degree of accuracy.
This problem was suggested by the interpretation of the
wave function  which is a solution of the Schrödinger
equation. As is well known
 
x
x
★ is the proba-
bility density in a position space and



ˆˆ
pp
★ is
the corresponding probability density in a momentum
space, where

x
★ is the complex conjugate of

x
 and

ˆp is the Fourier transform of


x

[1]. We raise the related question as to whether any two
distributions can be approximated within a specified error
from a single function by calculating its modulus or that
of its Fourier transform, calculations like those used in
Quantum Mechanics.
Our answer to this question is affirmative.
We prove this assertion by constructing a complex-
valued function

which approximates two given distri-
butions within a given error, provided the two distribu-
tions satisfy some regularity requirements specified
below. In particular, we show that the cumulative distri-
butions can be approximated pointwise using the indefi-
nite integral of t he m odul us o f

or of ˆ

.
The paper is organized as follows. In the next section
we list assumptions, conv entio ns and notation . In Section
3 we first introduce an expression for evaluating the error
in the approximation of the two given distributions from
the modulus of

or its Fourier transform. We then
show how to construct this function. In the next section
we list 4 pairs of distributions approximated using cons-
truction described in Section 3. In Section 5 we discuss
the method, limitations and possible generalizations. In
the appendix we sketch a proof that the function

approximates any two given distributions on subintervals
as claimed and a proof that the cumulative distributions
can be approximated pointwise (uniformly on closed
intervals) using the indefinite integral of the modulus of

or of ˆ

.
2. Definitions, Conventions and Assumptions
2.1. Assumptions
1)
X
and P are any two variables;


X
f
x is the
probability density function for
X
with corresponding
cumulative distribution


X
f
x;


P
f
p is the probabi-
lity density function for P with corresponding cumula-
tive distribution


P
F
p
2)


X
f
x and


P
f
p have finite variances and
they (and their square roots) are continuously differen-
tiable with Fourier transforms in 2 L and in 1 L (for
definitions of 1
L and 2 L see [2]; throughout, we
denote the norm in 2 L as 2)
3) for every >0

, there exist b
X
and b P such
that: a) for all


,
bb
x
XX and for all


,
bb
pPP ,


>0
X
fx and


>0;
P
fp and, b)

</16
Xb
FX  ,

W. D. FLANDERS ET AL.
440


</16
Pb
FP  ,

>1 /16
Xb
FX  , and


>1 /16
Pb
FP  .
2.2. Definitions and Notations
1) Let n be any integer, set =2
n
J, =/
b
x
XJ, and
=/,
b
pPJwhere b
X
and b P are given in assumption
3; if necessary, increasen( and/orb P) so that

/2 >1/16;
Pb
FP p   2
J
is the number of intervals
of width
x
and pin


,
bb
X
Xand


/2,/2 ;
bb
PpPp
2)
s
is a number defined in the Construction below;
it indicates the number of subintervals into which each
interval
x
 is divided. It is used to control the size of
the error in the approximations ;
3)

=/ts xp ;
4) j and k are integers, between
s
J, and 1sJ

;
l, m and m are integers between
J
, and 1
J

or
J
, unless noted othe rwis e;
5)

s
x
 is a function of the form used in the “sinc
approximation” [3,4]:
 
1
=
/
=/ ,
/
sJ
sX
jsJ
x
jxs
xfjxssinc
xs









where
 
sin
x
sinc x
x


;
6)

max ,
xX
Qfx for


,;
bb
xXX

min 0,
xX
Pfx
for


,;
bb
x
XX
min() >0,
pP
Pfp for


/2, ;
bb
pPpP 

max/,
xX
Qdfxdx
 for


,;
bb
xXX
7)

1/2
/
jX
x
afjxs
s





for
=, 1,,1jsJsJ sJ
8) 2
;, =,
jz
x
jz m
mj
pr a

 for 0;zsJj


9) ;
p
l
pr =


1lp
P
lp
f
zdz


 for 1;JlJ 
10) ˆ(.)
f
denotes the Fourier transform of a function
(.),
f
*(.)fdenotes the complex conjugate of a function
(.);
f
11)
j
M
is a function, defined recursively in the constr-
uction, mapping the integers

,1,,1jsJsJ sJ


to the numbers
  
/,1/,,1/,/ ;
s
JxsJx sJxsJx 
12) =/
jj
mMxs ;
j
m is an integer;
13)


:,1,,1
l
SjjsJsJ sJ and


=/
j
M
ls x for =, 1,,lJJ J. Equivalently,


1/,
lM
Sflsx


 where 1(.)
M
f is the inverse
mapping.
3. Construction
Given




,
XP
f
xf p,

and n, and b
X
and b P
with the properties described in Assumptions we cons-
truct a complex-valued function

which satisfies the
following two inequalities
 

1
12*
1
=2
ˆˆ <
lp
J
P
lJ lp
tqt qtfqdq










 
 (1)
and



11*
=<
Jlx
X
lx
lJ
xxfxdx






 (2)
where t is a scale parameter defined as
=
s
t
x
p

 (3)
and
s
is defined to be the smallest integer so that
Inequalities (4)-(8) are satisfied:
<4
x
xQ
s
J

 (4)
 

1/2 3
4
4322<
42 1
x
xQlogsJ
sJ
 



  (5)
 
4
2</82 1
sX
xfx J 

 (6)
 


22
4
||22 242 1
x
xx
xQ
JQlogsJ
sP J

 
(7)
 

 
1
=
11
1/
/
==
/=
/8
sJ
XbXX
XbjsJ
sJ sJ
jxs
XX
jxs
jsJ jsJ
x
fxdx fjxs
s
x
fxdxfjxs
s
















(8)
Many of the functions already defined, such as ()
s
x

,
or to be defined, such as ()
x

and
j
M
, depend on
s
and n. For brevity we suppress explicit notation of that
dependency.
The next step in the construction of

is to define
subintervals. First we split the interval


,
bb
X
X into
2
J
intervals of length :
x


,1 ,lxl x 

=, 1,,1lJJ J

 . Next we further divide each
interval


,1lxlx





 into
s
subintervals and

W. D. FLANDERS ET AL.
441
number them from =,1,,jsJsJ to 1sJ

.
Define

=, 1,,1jsJsJ sJ 
1/2
=
jX
xjx
af
ss
 

 


(9)
We continue the construction by introducing the
function
j
M
which maps the integers

,1,,1jsJsJ sJ  to the numbers


 
/,1/,,1/,/ ,
s
JxsJ xsJ xsJx  using
the following recursive algorithm: initialize j to
s
J

and l to
J
. Then proceed with the following steps:
1) Set ;, =0
xjz
pr for =1z and set
2
;, =
=jz
x
jz m
mj
pr a

 for each integer

0,1,,,zsJj
and set

1
2
;1
2
lp
pl P
lp
prfz dz








 for 1JlJ
2) If an integer z exists between 1 and
s
Jj

such that the inequalities


;;,
0/221,
plx jz
pr prJ   
and ;,1; >0
xjzpl
pr pr
 hold, then set =kz
; otherwise,
set =ksJj
3) If 0k then set all ,,
j
jk
MM

 equal to
/ls x
4) Set =1jjk and =1ll; if =,jsJ stop,
otherwise go to step 5)
5) If 1=lJ, set 1 ,1,,
j
jsJ
MM M

 all to
/
s
Jx an d stop; otherwise return to step 1).
Step 2) always returns a value of k. The iterative
algorithm continues until stopping, either at step 4) or 5).
Properties of
j
M
that will be used subsequently
include: for each ,, 1jsJ sJ , the mapping
j
M
is a non-decreasing function of j; each /
j
M
sx

 is
an integer (i.e., /=
j
j
M
xs m for some integer
,,
j
mJJ ).
Using this definition of
j
M
, we can now define

x

appearing in Equations (1)-(2) as:
12
=
12
=
/
()= /
/
=/
sJ iMx
j
j
jsJ
sJ iMx
j
X
jsJ
sxjxs
xasinc e
xxs
jxx jxs
fsince
sxs













 
 

 


(10)
Apart from

exp 2j iMx  and the
s
Jth term in the
series (10), the function

x
 represents the
Whittaker-Shannon interpolation formula [5,6] for

X
f
x.
In addition to approximations given in (1)-(2) we also
prove uniform approximations to the cumulative distri-
butions: let us define
 
/2
=p
PP
Pp
b
Fp fqdq


 
*
/2 ˆˆ
=p
PPp
b
F
ptqtqt dq 


 (11)
and
 
=x
XX
Xb
F
xfzdz


 
*
=x
XXb
F
xzzdz 


 (12)
Then, it can be shown that with



,
XP
f
xf p, and
*

given and with b
X
and b P having the properties
described in Assumptions, one can choose n large
enough that the complex-valued function

given by
the construction can be used to uniformly approximate
the cumulative distributions

X
F
x and


P
F
p for


,
bb
x
XX , and for


/2, /2
bb
pPpPp :


*
XX
Fx Fx



 (13)
and


*
PP
Fp Fp



 (14)
For derivation of (1) and (13) see Appendix.
Approximations (2) and (14) are derived in a similar
fashion.
4. Examples
To illustrate the approximation, we compare the integral
of the probability density function over a series of
intervals with the correpsonding approximation
generated by

from (1), (2) and (10). Specifically, in
P space we plot

/2
/2
pp
P
pp
f
qdq


 and the correspon-
ding approximation
 
/2 *
/2 ˆˆ
pp
pp
tqt qtdq 

 
. Similarly,
in
X
space we plot

/2
/2
xx
X
xx
f
qdq


 and the corres-
ponding approximation
 
/2 *
/2
xx
xx qqdq


 
; we app-
roximate the last integral by
 
*
x
xx  , as
x

is
small.
We applied the method to four pairs of distributions;
for each pair, we use a single

(and its Fourier tran-
sform) to approximate them with n = 7,
s
= 64 and
n
==5/20.039xp :
1) Figure 1 illustrates the approximation using
Equation (10), for the distributions

22
11
88
22
4
= 0.750.25
2
xx
X
fxe e

 
 
 
 










W. D. FLANDERS ET AL.
442
Figure 1. Approximation to the integrals

2
2
pp p
pp
f
qdq



(top) and

2
2
xx X
xx
f
qdq


 (bottom) based on Equation (10)
for the pair of distributions



2
exp 2pfpp

 , and

22
11
88
22
40.75 0.25
2
xx
X fx e e

 
 
 
 






.

2/2
1
and=2
p
P
fp e


2) Figure 2 illustrates the approximation using
Equation (10), for the distributions
 
2/2
=,0 and=12
xp
XP
fx exfpe 


3) Figure 3 illustrates the approximation using
Equation (10), for the distributions
 
2/2
1
=and=,0
2
xp
XP
fxefp ep



4) Figure 4 illustrates the approximation using
Figure 2. Approximation to the integrals

2
2
pp p
pp
f
qdq



(top) and

2
2
xx X
xx
f
qdq


(bottom) based on Equation (10)
for the pair of distributions

22
1
2
p
p
fp e


 and


x
X
f
xe

 for 0x and


0Xfx for <0x.
Equation (10), for the distributions
 
22
/2 /2
11
=and =
22
xp
XP
fx efpe


5. Discussion
Based on the algorithm described in Section 3 we have
shown how to approximate two given distributions from
a single complex-valued function with a specified
accuracy. The main result of our paper is given in
expressions (1-2), (10) and (13-14). Expression (10) can
be viewed as an extension of well-known numerical
methods based on the sinc function [3], providing a

W. D. FLANDERS ET AL.
443
Figure 3. Approximation to the integrals

2
2
pp p
pp
f
qdq



(bottom) and

2
2
xx X
xx
f
qdq


 (top) based on Equation (10)
for the pair of distributions

22
X1
2
x
fx e


, and




expP
f
pp for 0p and


0Pfp for <0p.
Figure 4. Approximation to the integrals

2
2
xx X
xx
f
qdq



(bottom) and

2
2
xx X
xx
f
qdq


 (top) based on Equation (10)
for the pair of distributions

22
1
2
p
p
fp e


, and

22
1
2
x
X fx e


.
method for approximating two distributions simulta-
neously.
The construction and arguments given here show that
there always exists a function of the form given by (10)
which approximates two given distributions with aspe-
cified degree of accuracy provided our assumptions (1-3)
are satisfied. For successively higher degrees of accuracy,
the method allows construction of a sequence of appro-
ximating functions but this sequence may not necessarily
converge, say in 2
L, to a limiting function.
A modification of our approach or a similar method
based not on the sinc function but on some other basis or
wavelet series [7] might be used to approximate the two
distributions and converge to a limiting function. For
example, one might attempt to use the Hermite poly-
nomials multiplied by the appropriate Gaussian function
as a basis.
It remains speculative, however, whether our approach
can be modified so as to approximate arbitrary pairs of
distributions with a basis other than the sinc functions.
Further speculation about this potential generalization is
beyond the scope of this paper. We would like to stress
that the main goal of the present work is not to find the
best numeric approximation (though we can always meet
an increasing demand in accuracy) but to establish the
existence of an algorithm allowing uniform approxi-
mation of the cumulative distributions.
6. References
[1] J. von Neumann, “Mathematical Foundations of Quantum

W. D. FLANDERS ET AL.
444
Mechanics,” Princeton University Press, Princeton, 1955.
[2] W. Rudin, “Functional Analysis,” McGraw-Hill, New
York, 1991.
[3] F. Stenger, “Summary of Sinc Numerical Methods,”
Journal of Computational and Applied Mathematics, Vol.
121, No. 1-2, September 2000, pp. 379-420.
[4] P. L. Butzer, J. R. Higgins and R. L. Stens, “Classical and
Approximate Sampling Theorems; Studies in the


2RL
and the Uniform Norm,” Journal of Approximation
Theory, Vol. 137, No. 2, December 2005, pp. 250-263.
[5] J. M. Whittaker, “Interpolation Function Theory,” Cam-
bridge Tracts in Mathematics and Mathematical Physics,
No. 33, Cambridge University Press, Cambridge, 1935.
[6] C. E. Shannon, “Communication in the Presence of
Noise,” Proceedings of Institute of Radio Engineers, Vol.
37, No. 1, January 1949, pp. 10-21.
[7] I. Doubechis, “Ten Lectures on Wavelets,” CBMS-NSF
Regional Conference Series for Applied Mathematics,
1992.
Appendix
We sketch the derivation of (1) and (13). First we write
down the Fourier transform of

x
 :
 


2
122
=
12
=
ˆ=
/
=/
=
ixq
sJ iMx ixq
j
j
jsJ
x
sJ ijqM
j
s
jj
jsJ
qxedx
sxjxs
aesince dx
xxs
xx
aRqMe
ss




 
  


 















(1)
In (1)
R
is a rectangular function:

11
1when<q<,
22
11
whenq= ,
=22
0otherwise
Rq









(2)
We consider
  
1
*
2
1
2
ˆˆ ,
lp
lp
I
ltqtqt dq 









 (3)
the integral of
 
*
ˆˆ
tqt qt
  over the interval
11
,
22
lplp


 




,for each ,1,,1lJJJ .
Using expression (1), rescaling the integration variable,
and using definition of integer :=/
jj j
mmxM s, for
this integral we obtain:



 
11
2
1,=
2
2
()= sJ
l''
j
kj
ljk sJ
'
ijkqjmkm
jk
'k
I
ldqaaRqm
Rqme

 





 






where

1/2
=//
jX
axfjxss 

.
Property (2) of the rectangular function R implies
that each ,jk term in the integrand is either 0 for the
full range of integration or vanishes for all but a
subinterval of length 1. The latter holds only if =
j
k
mm
and we have:

11
2
j
lml

  if and only if =
j
lm,
for some integer
j
m. After interchanging the order of
integration and summation, each integral in the sum is 0
due to the factor

2'
ijkq
e


 , unless =jk. On the other
hand, if =jk the integral is just 2
j
a. Thus, we can
write


=, 1,,1lJJ J

 :
  
1
*2 2
1
2
ˆˆ
==,
lp
j
lp jS
l
I
ltqtqt dqa 






 

 
 (4)
where we recall the definition of l
S: l S is the set of all
,1,,jsJsJ sJ

 for which

/=
j
M
xs l .
There are 2
J
terms


I
l of the form given in (4);
the approximation given b y Inequality (1) will be proven
if we can show that for =, 1,,1jJJ J  the
following relation holds:

1
22 2
;1
2
=2
lp
jpl jP
lp
jS jS
ll
apr afzdz
J










 

 (5)
Either Inequality (5) holds for=, 1,,1lJJ J ,
and we are done, or there is a smallest l violating (5),
say 1 l, such that
 
1
22
1
12
||>/2.
lp
jp
jS
llp
afzdxJ 










This implies that 11jsJ , where


1=max :l
jjjS. Indeed, if this were not the case
(i.e., 1<1jsJ

), than 1
1
j
M would have been defined
by the algorithm described in section 3 as 1 l since


1
1/21
j
aJ 


 and 11j

would also have been in
1
l
S. This would be a contradiction since 1 j was
supposed to be the largest element of 1
l
S. Therefore,
11jsJ.

W. D. FLANDERS ET AL.
445
By choice of s in section 3, requirement (8):


1/
12
=/
jxs
J
jX
lJ jSjxs
lafxdx


 

  is less or equal
/8,

and because

X
f
x integrates over the interval


,
bb
X
X to at least 1/16

 (Assumption 3), we
have:
12
=1/16/8>1/4
sJ
j
jsJ
a




 
 (6)
Since ()
P
f
p is a probability density function and by
Assumption 3 and the algorithm described in section 3:


1
12
12
11
11
22 2
1
===
2
1=
=1/4.
lp
P
Pp
b
ll
sJ
lp
Pjj
lp
lJlJjS jsJ
l
fzdz
fzdza a 





 













(7)
The last inequality above is simply Inequality (6). By
the Construction,

1
2 2
1
2
lp
P
j
jS
l
lp
f
zdza




 




 for every
l. This fact and (7) imply:


112
21=
2
1
12
2
1
=2
=
1
4
l
lp
Pj
Pp
blJjS
l
llp
Pj
lp
lJ jS
l
fzdza
fzdza





 















(8)
Further, the upper bound of 1 in expression (6) implies
that:
 
1
112
2
11
1=1
22 1
3
=4
Jlp
Pp
b
PP
lp lp
ll
fzdz fzdz



 

 
 
 

 



(9)
Now l S is empty for 1 >ll since 11 =
sJ
M
l
 and
j
M
is non-decreasing. In other words 2=0
j
jS
la

 for
1
>ll. This fact and the inequality in Expression (9)
yield:

 
1
11
122
1
=1=1 211
3
=0
4
JJ
lp
lp
Pj P
lp lp
lljS ll
l
fzdz afzdz




  








(10)
Finally, Inequalities (8) and (10) lead to the desired
result, proving Inequality (1).
Proof of Inequality (2) requires no new qualitative
features but straightforward tedious calculations; we do
not cite it here.
To prove the uniform approximation for the
cumulative distributions, Inequality (13), choose n
large enough that *
=/2</2
n
xxb
QxQX   , constr-
uct ()
x

so that Inequalities (1) and (2) hold with
*/2
 and let


=XX
x
Fx Fx  . We now show
that


*
x



for all


,( 1)
x
jx jx  and for
=, 1,,1jJJ J

 . Suppose the maximum of




XX
F
xFxoccurs at 0 =,
X
xfor


0,1
x
jx jx 

.
Let
F
j
M
denote
 


max,,1.
X
F
xx jxjx 

If




00
>
XX
F
xFx
, then


FX
j
x
MFjx 


 (11)
since ()
X
F
x
 is non-decreasing. Further, the mean
value theorem implies

FX x
j
M
FjxQ x since
x
Q is th e maximum of

X
f
x. Using this result in (11)
gives




** *
/2 /2=
XxX
xFjxQxFjx

 

(12)
since




*/2
XX
FjxFjx   
 follows from (1)
and * ||/2
x
Qx    by choice of n.
On the other hand , if


00
<
XX
F
xFx
, then










 

*
** *
11
/2
/2 /2=
XXX
XXx
X
x
Fj xFjxFj x
FjxFjxQx
Fjx


 

 
 
 

(13)
The chain of estimates above follows from the fact
that


X
F
x
 is non-decreasing,
x
Qn is the maximum
of


X
f
x, and by choice of n.
The proof for


PP
F
pFp is similar.