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 x 2, 4 + sin ( 2x ) or arctan ( x ) etc.) you might select.

 - 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 ]. 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)

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 ( a m  0 or b m  0) is a polynomial of the form: 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:

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: This is the form if n (the number of data points) is even (notice the a m term): In either case the a j and b j terms are given by:

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 a 0, a 1, a 2, b 1, b 2. (Notice from the formula we can assume b 0 is always 0.) The interpolating polynomial is:

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

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 a k, b k, c k, d k, e k and f k are rational numbers:

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 ) 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 a k, b k, c k, d k, e k, f k of the form given below is countable. This implies that this set of functions can be enumerated call them { p m ( x )}. Any rational translation of one of these functions is another function of this form.

We define two sequences of closed bounded intervals C n that are intervals centered at powers of 4 and I n intervals centered at the origin as given below. The particular lengths of the intervals have been chosen so that the intervals have the following properties. 1. The C n are disjoint: 2. The I n are nested: 3. I n and C n+1 are disjoint: 4. I n contains C 1, C 2,…, C n : [ ] ] C1C1 C2C2 CnCn C n n4n 4 n +1 InIn …

A sequence of trigonometric polynomials {  m ( x )} can be chosen from the set { p m ( x )} using a recursive definition. This can be done using the previous result. In general if  n -1 ( x ) has been defined the function  n ( x ) can be chosen as follows.

For any x in the interval I n the sequence {  n ( x )} will be a Cauchy Sequence. This implies the sequence {  n ( x )} will converge on the interval I n. This means that the limit will exist for all x in this interval. The intervals I n 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 ). Because the sequence {  n ( x )} is Cauchy, the function U ( x ) will can be written as a convergent telescopic series.

Consider a point x in the set C n. Look at how U ( x ) and p n ( x -4 n ) will differ. If we replace x by x +4 n in the inequality above we get the following relation for any x in the closed bounded interval [-2 n,2 n ]. 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 [-2 n,2 n ], then find a lager value for n so that | f(x)-p n (x) |<  /2 and an even larger value for n so that the inequality above is less than  /2.

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. Add a polynomial to it. Add a trig functions to it. Translations and dialations. 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 ). 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 ).

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 f n ( x ) in S that has the following property: A set of functions is "open" if its complement is closed. This is called the topology of almost uniform convergence. Using the trigonometric functions { p i ( 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. 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.