universal functions a construction using fourier approximations

UNIVERSAL FUNCTIONS

A Construction Using Fourier Approximations

Presentation Overview

1. An intuitive concept of Universal Functions

2. Function Approximation and - Bans about Functions

3. Fourier Interpolation on the Interval [0, 2]

4. Fourier Approximation on an interval [a,b]

5. The Construction of a Universal Function

6. Other Universal Functions

7. The Category of Universal Functions

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.

2 4 6 12 16 20 48 64 80

-6

-4

-2

2

4

6

Graph of Ux... ...

6,2inFor

04

42

2

x

xxU

xxU

20,12inFor

0)2sin(416

2sin416

x

xxU

xxU

80,48inFor

0arctan64

arctan64

x

xxU

xxU

- 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].

xfxpxf

or

xfxp

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]. a b

f(x)+

f(x)-

f(x)

Fourier Interpolation

The Fourier Method of Interpolation is a way constructing a function to exactly agree with a set of data points that uses combinations of sin(qx) and cos(qx) where q is an integer.

Trigonometric Polynomials

A trigonometric polynomial of degree m (am0 or bm0) is a polynomial of the form:

m

kk

m

kk

a

m

ma

tkbtka

mtbtbtbtb

mtatatatatp

112

321

3212

)sin()cos(

)sin()3sin()2sin()sin(

)cos()3cos()2cos()cos()(

0

0

Degree of a Trigonometric Interpolating Polynomial

For a data set with n data points the degree of the polynomial depends on if n is even or odd. If m is the degree of the trigonometric polynomial the relation between n and m is that:

)(2

1oddisnif

nm

)(

2evenisnif

nm

Interpolating Polynomials

Because of the distinction between even and odd the interpolating polynomials take on two different forms. One if the data set has an even number of data items another if it is odd.

This is the form if n (the number of data points) is odd:



321

32120

mtbtbtbtb

mtatatatatp

m

ma

This is the form if n (the number of data points) is even (notice the am term):



321

232120

mtbtbtbtb

mttatatatp

m

aa m

In either case the aj and bj terms are given by:

1

0

21

0

2 sin2

cos2 n

knk

kj

n

knk

kj jxn

bandjxn

a

Example: Let’s use a Fourier Interpolation of the data {1,3,-5,2}

The number of data points is n=4 so m=4/2=2 as before. We begin by computing a0, a1, a2, b1, b2. (Notice from the formula we can assume b0 is always 0.)

2

12531

2

10cos20cos50cos30cos1

2

1


2

0

432

422

412

402

0

a

a

305012

1cos2cos5cos30cos1

2

1


2

23

21

432

422

412

402

1

a

a

2

92531

2


2

1


2

2

432

422

412

402

2

a

a

000002

10sin20sin50sin30sin1

2

1


2

0

432

422

412

402

0

b

b

2

12030

2

1sin2sin5sin30sin1

2

1


2

23

21

432

422

412

402

1

b

b

000002


2

1


2

2

432

422

412

402

2

b

b

The interpolating polynomial is:

)sin(2

1)2cos(

4

9)cos(3

4

1

)2sin()sin()2cos()cos()( 2121220

ttt

tbtbttatp aa

2

3 2

2

-5

-4

-3

-2

-1

1

2

3

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.

numbersrealandsincos2 00

0kk

m

kkk

m

kkk dcxdbxca

axp

Below we show how (x-4)2 can be interpolated on the interval [2,6].

2 4 6

1

2

3

4

Interpolation on2,6with 20 data points

2 4 6

1

2

3

4

Interpolation on2,6with 40 data points

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

kkkk

m

kkkk fxdbexca

axp

00

0 sincos2

2 4 6

-1

1

2

3

4

5

An epsilon ban around the function

2 4 6

-1

1

2

3

4

5

An epsilon ban around the function

2 4 6 12 16 20

-1

1

2

3

4

5

Two Different Functions on Disjoint Intervals

2 4 6 12 16 20

-1

1

2

3

4

5

Two Different Functions on Disjoint Intervals

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].

The Construction of a Universal Function

Seidel and Walsh (W. Seidel and J.L. Walsh, "On Approximation by Euclidean and Non-Euclidean 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

kkkk

m

kkkk fxdbexca

axp

00

0 sincos2

This implies that this set of functions can be enumerated call them {pm(x)}.

,,, 321 xpxpxpxpm Any rational translation of one of these functions is another function of this form.

rxprxp mj numberrationalanyfor

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.

124,124124

24,2424

nnnnnn

n

nnnnnnn

xI

xC

The particular lengths of the intervals have been chosen so that the intervals have the following properties.

1. The Cn are disjoint:

2. The In are nested:

3. In and Cn+1 are disjoint:

4. In contains C1, C2,…, Cn:njIC

CI

III

CC

nj

nn

kj

for1

321

[ ] [ ] [ ]

[ ]]C1 C2 Cn

Cn+1

0 4 16 4n 4n+1

In

…

222

2

121

2

4

14

4

1

:thatsoChoose

Cxforxxp

Ixforxxx

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.

1111 2

14:thatsoChoose Cxxxpx

In general if n-1(x) has been defined the function n(x) can be chosen as follows.

nnnn

nnnn

n

Cxforxxp

Ixforxxx

2

14

2

1

:thatsoChoose2

11

For any x in the interval In the sequence {n(x)} will be a Cauchy Sequence.

1

11

1211

2

12

1

2

1

2

1

m

mkmkm

mmkmkmkmkmmkm xxxxxxxx

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).

xxU nn

lim

Because the sequence {n(x)} is Cauchy, the function U(x) will can be written as a convergent telescopic series.

xxxxxxxxU nnnnnnn 23121

Consider a point x in the set Cn. Look at how U(x) and pn(x-4n) will differ.

1

21

121

2

12

1

2

1

2

1

44

n

nnn

nnnnnn

nn

n xxxxxxpxpxU

If we replace x by x+4n in the inequality above we get the following relation for any x in the closed bounded interval [-2n,2n].

12

14

nnn xpxU

If we begin with an arbitrary function f(x) on an interval [a,b] and a value for >0. First find a a large enough n so that [a,b] is in [-2n,2n], then find a lager value for n so that |f(x)-pn(x)|</2 and an even larger value for n so that the inequality above is less than /2.

2244 xfxpxpxUxfxU nnnn

Other Universal Functions

You can add or subtract certain types of function to a universal function and get another universal function. For instance adding or subtracting one of the combinations of finite trigonometric polynomials with rational parameters. If we start with a Universal Function U(x) each of the following will also be universal.

723 24 xxxU Add a polynomial to it.

xxxU sin54cos3 Add a trig functions to it.

xexU 7973 Translations and dialations.

xxexU x sin94 7 Combinations of them.

Examples of other Universal functions

Once you have found one there seem to be an infinite number of possibilities. A strange fact is that if you add two universal functions together you might not get a universal function.

It turns out that this 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).

0 cxfcxU

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).

11

cxfcx

cxU

The Category of Universal Functions

By knowing one Universal Function it is easy to create an infinite number of other Universal Functions a typical question mathematicians like to ask is:

How "common" are Universal Functions among continuous functions?

The word common to a mathematician needs to be more precise. It is too vague to be able to get any type of results from.

One way to think of meaning of the word common would be in what mathematicians call the measure of the functions. We could rephrase this question as:

If I picked a function at random what is the chance it is Universal?

Another way to think of the meaning of the word common would be in what mathematicians call the topology of functions. This question can be rephrased as:

Can the complement of the set of Universal Functions be broken into a countable union of sets of functions so that each set of functions is "closed"

and each set contains no "open" set of functions?

Once the concept of the meaning of and "open" and "closed" set of functions is known, this is what topologists call a Residual Set. A set being residual means its elements are "commonly" found.

The concept of a set of functions being "closed" is a complicated concept when you are first introduce to it. If S is a set of continuous functions we say that S is closed if we take a sequence of functions fn(x) in S that has the following property:

SxfKxfxfnKxn

thensetboundedclosedanyfor0maxlimIf

A set of functions is "open" if its complement is closed. This is called the topology of almost uniform convergence. Using the trigonometric functions {pi(x)} we used in the construction of U(x), define the following sets of functions indexed by three positive integer parameters i, j and k.

kijxt

xptxgxgkjiS 1

,maxinf,,

kjiSxhxhNkji

,,functionuniversalanotis,,

It can be proven each S(i,j,k) is closed and contains no open subsets. We can also show the following:

This means that Universal Functions are common in the topological sense.

universal functions a construction using fourier approximations

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