1. An Applet-Based Presentation of the Chebyshev Equioscillation Theorem Robert Mayans Fairleigh Dickinson University January 5, 2007.

2. Statement of the Theorem Let f be a continuous function on [a,b]. Let p n * be a polynomial of degree ≤ n that best approximates f, using the supremum norm on the interval [a,b]. Let d n * = || f – p n * || = inf { || f – p || : deg(p) ≤ n }

3. Statement of the Theorem There exists a polynomial p n * of best approximation to f, and it is unique. It alternately overestimates and underestimates the function f by exactly d n *, at least n+2 times. It is the unique polynomial to do so.

4. An Example Function: y = f(x) = e x, on [-1,1] Best linear approximation: y = p 1 (x) = 1.1752x+1.2643 Error: || f - p 1 || = 0.2788

5. An Example

6. E(-1) = 0.2788 E(0.1614) = -0.2788 E(1) = 0.2788

7. Another Example Function: y = f(x) = sin 2 (x), on [-π/2,π/2] Best quadratic approximation: y = p 2 (x) = 0.4053x 2 +0.1528 Error: || f – p 2 || = 0.1528

8. Another Example

9. Alternating set: -π/2, -1.0566, 0, 1.0566, π/2 || f – p 2 || = 0.1528

10. A Difficult Theorem? “The proof is quite technical, amounting to a complicated and manipulatory proof by contradiction.” -- Kendall D. Atkinson, An Introduction to Numerical Analysis

11. Weierstrass Approximation Let f be a continuous function on [a,b]. Then there is a sequence of polynomials q 1, q 2, … that converge uniformly to f on [a,b]. In other words, || f - q n ||  0 as n  ∞.

12. Bernstein Polynomials Suppose f is continuous on [0,1]. Define the Bernstein polynomials for f to be the polynomials: The Bernstein polynomials B n (f,x) converge uniformly to f.

13. Bernstein Polynomials An applet to display the Bernstein polynomial approximation to sketched functions.applet We use of de Casteljau’s algorithm to calculate high-degree Bernstein polynomials in a numerically stable way.de Casteljau’s algorithm

14. Existence of the Polynomial Sketch of a proof that there exists a polynomial of best approximation. Define the height of a polynomial by Mapping from polynomials of degree n: p → || f - p || is continuous.

15. Existence of the Polynomial As ht(p) tends to infinity, then || f – p || tends to infinity. By compactness, minimum of || f – p || assumed in some closed ball { p : ht(p) ≤ M } This minimum is the polynomial of best approximation.

16. Alternating Sets Let f be a continuous function on [a,b]. Let p(x) be a polynomial approximation to f on [a,b]. An alternating set for f,p is a sequence of n points such that f(x i ) - p(x i ) alternate sign for i=0,1,…, n-1 and and for each i, | f(x i )-p(x i ) | = || f - p ||.

17. An Example

18. Alternating Sets Let f be a continuous function on [a,b]. Let p(x) be a polynomial of best approximation to f on [a,b]. We claim that f,p has an alternating set of length 2. This alternating set has the two points that maximize and minimize f(x) – p(x) If they are not equal and of opposite sign, we could add a constant to p and make a better approximation.

19. Variable Alternating Set Definition: A variable alternating set for f, p on [a,b] is like an alternating set x 0, …, x n-1, in that f(x i ) - p(x i ) alternate in sign, except that the distances d i = | f(x i ) - p(x i ) | need not be the same.

20. Example: Variable Alternating Set Variable Alternating Set: -1.1, -0.4, 0.3, 1.2

21. Variable Alternating Set If x 0,…, x n-1 is a variable alternating set for f, p with distances d 0, …, d n-1, and g is a continuous function close to f, then x 0,…,x n-1 is a variable alternating set for g, p. “Close” means that || f – g || < min d i. Proof is obvious.

22. Variable Alternating Set Variable Alternating Set: -1.1, -0.4, 0.3, 1.2

23. Variable Alternating Set Theorem: (de la Vallee Poussin) Let f be continuous on [a,b] and let q be a polynomial approximation of degree n to f. Let d * n = || f – p * n ||, where p * n is a polynomial of best approximation. If f,q has a variable alternating set of length n+2 with distances d 0, …, d n+1, then d * n ≥ min d i

24. Variable Alternating Set Proof: If not, f and q are close, so x 0, … x n+1 is a variable alternating set for q, p * n Thus q - p * n has at least n+2 changes in sign, hence at least n+1 zeroes, hence q = p * n, which is impossible.

25. Variable Alternating Set Corollary: Let p be a polynomial approximation to f of degree n. If f,p has an alternating set of length n+2, then p is a polynomial of best approximation.

26. Sectioned Alternating Sets A sectioned alternating set is an alternating set x 0,...,x n-1 together with nontrivial closed intervals I 0,...,I n-1, called sections, with the following properties: The intervals partition [a,b]. For every i, x i is in I i. If x i is an upper point, then I i contains no lower points. If x i is an lower point, then I i contains no upper points.

27. Sectioned Alternating Sets An example on [-1,1]: f(x) = cos( πe x ), p(x)=0

28. An Example Function: y = f(x) = cos( πe x ) on [-1,1] Alternating set: (0, ln 2) Sections: [-1,0.3], [0.3, 1]

29. Sectioned Alternating Sets Theorem: Any alternating set can be extended into a (possibly larger) alternating set with sections. AppletApplet:

30. Improving the Approximation Suppose f, p has a sectioned alternating set of length ≤ n+1. Then there is a polynomial q of degree n such that || f – (p+q) || < || f - p || Applet:Applet

31. Proof of the Theorem Let f be a continuous function on [a,b]. Let p(x) be a polynomial of best approximation to f on [a,b]. If f, p has a alternating set of length n+2 or longer, then p is a polynomial of best approximation. On the other hand, if p is a polynomial of best approximation, then it must have an alternating set of length 2.

32. Proof of the Theorem We can extend that alternating set to one of length n+2 or longer. Otherwise we can change p by a polynomial of degree n and get a better approximation. Finally, if p, q are both polynomials of best approximation, then so is (p+q)/2. We can show that p-q has n+2 changes in sign, hence n+1 zeros, hence p=q. We conclude that the polynomial of best approximation is unique, and the theorem is proved.

33. Finding the Best Approximation Cannot solve in complete generality Use the Remez algorithm to find polynomials. Applet:

34. The Remez (Remes) Algorithm Start with an approximation p to f on [a,b] and a variable alternating set x 0, …, x n+1 of length n+2. Start Loop: Solve the system of equations: This is a linear system of n+2 equations (i=0, …, n+1) in n+2 unknowns (c 0, …, cn, E). Using the new polynomial p, find a new variable alternating set, by moving each point x i to a local max/min, and including the point with the largest error. Loop back.

35. Hypertext for Mathematics This proof with applets is part of a larger hypertext in mathematics. Also published in the Journal of Online Mathematics and its ApplicationsJournal of Online Mathematics and its Applications Link to the Mathematics Hypertext ProjectMathematics Hypertext Project

36. Open Architecture, Math on the Web Java applets JavaScript functionality MathML Scalable Vector Graphics (SVG) TEX family Mathematical fonts Unicode

37. Sad State of Tools Java applets cannot display mathematical text No practical conversion to HTML Deep incompatibilities between TEX and MathML Poor or nonexistent capabilities for formula search and formula syntax-check Wide variety of setups for the browser.

38. Sad State of Tools Tool development by practitioners will be crucial to make this architecture work.