Transcript UNIVERSAL FUNCTIONS - Muskingum University

UNIVERSAL FUNCTIONS
A Construction Using Fourier
Approximations
UNIVERSALITY
To find one (or just a few) mathematical relationships (functions
or equations) to describe a certain connection between ideas.
Examples of this are common in science
Ideal Gas Law
Newton’s Law of Falling Bodies
𝑃𝑉 = 𝑛𝑅𝑇
ℎ = −𝑔𝑡 2 + 𝑣0 𝑡 + ℎ0
Maxwell’s Equations:
𝛻 ∙ 𝐃 = 𝑝𝑣
𝛻∙𝐁=0
𝜕𝐁
𝛻×𝐄=−
𝜕𝑡
𝜕𝐃
𝛻×𝐇=
+𝐉
𝜕𝑡
Einstein’s General Theory of
Relativity
𝐸 = 𝑀𝐶 2
In Calculus we learned how to reduce area formulas to a single equation!
𝐴 = 12 𝑏1 + 𝑏2 ℎ
𝐴 = 12𝑏ℎ
Beginning Calculus
Ω
𝛾
Multivariate Calculus
g(x)
a
𝑏
𝐴𝑟𝑒𝑎 =
f(x)
b
𝐴𝑟𝑒𝑎 =
𝐴 = 𝜋𝑟 2
𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥
𝑎
1 𝐷𝐴 =
Ω
−1
∳𝛾 2 𝑦𝑑𝑥
+
1
𝑥𝑑𝑦
2
Universal Functions
(An Intuitive Concept)
A universal function is a function whose behavior on an interval (or part of its
graph) is "like any" continuous function you might select. Think of it as a single
function that can be used to describe all other functions.
The Universal Function we will construct in this presentation will be a function whose
translations (shift in their graph) will approximate any continuous function we can think of
on a given closed bounded interval (i.e. U(x+t)f(x)).
Think of the graph of such a function call it U(x) has the property that if you look along the
x-axis the graph of U(x) will be "close" to being the graph of any continuous function f(x)
(such as x2, 4+sin(2x) or arctan(x) etc.) you might select.
Graph of U x
6
4
...
2
-2
2
4
6
...
12
16
20
48
64
80
-4
-6
U x  4  x 2
U x  4  x 2  0
For x in 2,6
U  x  16  4  sin 2 x 
U  x  16  4  sin(2 x)   0
For x in 12,20
U x  64  arctanx 
U x  64  arctanx   0
For x in 48,80
The Construction of a Universal Function
Seidel and Walsh (W. Seidel and J.L. Walsh, "On Approximation by Euclidean and NonEuclidean Translations of Analytic Functions", Bulletin of the American Mathematical
Society, Vol. 47, 1941, pp. 916-920) were the first to use a similar method of construction
using ordinary polynomials instead of trigonometric polynomials.
The set of finite linear combinations of trigonometric functions with rational parameters
ak, bk, ck, dk, ek, fk of the form given below is countable.
m
a0 m
px     ak cosck x  ek    bk sin d k x  f k 
2 k 0
k 0
This implies that this set of functions can be enumerated call them {pm(x)}.
pm x  p1 x, p2 x, p3 x,
Any rational translation of one of these functions is another function of this form.
p j x  r pm x for any rationalnumber r
 - Bans of Functions on an Interval
The concept of a translation of U(x) coming "close" to being a function f(x) on a closed
bounded interval [a,b] has a more formal mathematical characterization.
We say a function p(x) approximates a
function f(x) within  (think of  as a
small positive number) on an interval
[a,b] if the following condition is satisfied
for all x in [a,b].
px   f x   
or
f x     px   f x   
Intuitively we can think of this as the graph
of p(x) must lie below the graph of f(x)+
and above the graph of f(x)-. In other
words, the graph of p(x) must remain in
the shaded area between the two graphs
for all of the points x in the interval [a,b].
f(x)+
f(x)
f(x)-
a
b
Approximation on a Closed Interval [a,b]
We can interpolate any data set on an interval [a,b] by translating the interval [a,b] to the
interval [0,2] then translating back. The trade off we make is that the function p(x) that is
used to do this takes a slightly different form.
m
a0 m
px     ak cosck x    bk sin d k x 
2 k 0
k 0
ck and d k real numbers
Below we show how (x-4)2 can be interpolated on the interval [2,6].
Interpolation on
2,6
with 20 data points
Interpolation on
2,6
with 40 data points
4
4
3
3
2
2
1
1
2
4
6
2
4
6
An interpolating function for a set of data will exactly match that set of data. We can find
and approximating function that will remain in an -Ban around the function p(x) no
matter how small of a number  we take. This function p(x) can be found so that all of the
numbers ak, bk, ck, dk, ek and fk are rational numbers:
m
a0 m
px     ak cosck x  ek    bk sin d k x  f k 
2 k 0
k 0
An epsilon ban around the function
5
4
3
2
1
2
-1
4
6
In the construction of a universal function we will need to be able to find a trigonometric
polynomial with rational parameters that can behave like two different functions on two
different intervals. Below is an example of how we can have a function that behaves like
(x-4)2 on the interval [2,6] and the function 4+sin(2(x-16)) on the interval [12,20].
We define two sequences of closed bounded intervals Cn that are intervals centered at
powers of 4 and In intervals centered at the origin as given below.

Cn  x  4n  2n  4n  2n ,4n  2n




I n  x  4 n  2 n  1   4 n  2 n  1 ,4 n  2 n  1
The particular lengths of the intervals have been chosen so that the intervals have the
following properties.
1. The Cn are disjoint:
C j  Ck  
2. The In are nested:
I1  I 2  I 3  
3. In and Cn+1 are disjoint:
I n  Cn 1  
4. In contains C1, C2,…, Cn:
C j  In
C1
[ ]
0
4
…
C2
[
]
16
In
[
Cn
4n
for j  n
Cn+1
] ]
[
4n+1
]
A sequence of trigonometric polynomials {m(x)} can be chosen from the set {pm(x)} using
a recursive definition. This can be done using the previous result.
1
Choose  1 x  so that : p1 x  4   1 x  
x  C1
2
1

for x  I1
  1  x    2  x   4
Choose 2 x  so that: 
 p x  4 2    x   1 for x  C
2
2
 2
4


In general if n-1(x) has been defined the function n(x) can be chosen as follows.
1

  n 1 x    n x   2 n
Choose n x  so that: 
 p x  42   x   1
n
 n
2n


for x  I n 1
for x  Cn
For any x in the interval In the sequence {n(x)} will be a Cauchy Sequence.
 m  k x    m x    m  k x    m  k 1 x    m  k 1 x    m  k  2 x      m 1 x    m x 

1
2m k
1
 m 1
2

1
2 m  k 1

1
2 m 1
This implies the sequence {n(x)} will converge on the
interval In. This means that the limit will exist for all x
in this interval. The intervals In can be as large as you
wish so for any x in (-,) We can define the function
U(x) as a limit of n(x).
U x   lim  n x 
n 
Because the sequence {n(x)} is Cauchy, the function U(x) will can be written as a
convergent telescopic series.
U x   n x   n1 x   n x   n2 x   n1 x   n3 x   n2 x  
It turns out that a similar method can also be used to construct Universal Functions on
different domains (even sets in the complex plane) that will have a different operation in
which the function will be universal.
For the domain that is the real line with 0 deleted (i.e. (-,)U(0,)) a universal function U(x)
can be constructed so that a dilation or contraction of U(x) approximates a function f(x).
This was done by Heins (1955).
U cx  f x 
c0
For the domain that is the open interval |x|<1 (i.e. (-1,1)) a universal function U(x) can
be constructed so that a rational transformation of U(x) approximates a function f(x).
This was done by Zappa (1988).
 xc 
U
  f x 
 1  cx 
c 1