UNIVERSAL FUNCTIONS

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 Ideal Gas Law Newton’s Law of Falling Bodies Maxwell’s Equations: Einstein’s General Theory of Relativity

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

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 x2, 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 ak, bk, ck, dk, ek, fk of the form given below is countable. This implies that this set of functions can be enumerated call them {pm(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 ak, bk, ck, dk, ek and fk 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 Cn that are intervals centered at powers of 4 and In 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 Cn are disjoint: 2. The In are nested: 3. In and Cn+1 are disjoint: 4. In contains C1, C2,…, Cn: … Cn+1 C1 C2 Cn ] [ ] [ ] [ ] [ ] 4n 4n+1 0 4 16 In

11. 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. 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 In the sequence {n(x)} will be a Cauchy Sequence. 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). 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).