1. UNIVERSAL FUNCTIONS A Construction Using Fourier Approximations

2. 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

3. In Calculus we learned how to reduce area formulas to a single equation! ab f(x)f(x) g(x)g(x)

4. 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.

5. 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 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.

6.  - 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)

7. 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].

8. 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:

9. 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].

10. 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+1 0 4 16 4n4n 4 n +1 InIn …

11. 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.

12. 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.

13. 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). 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).