Title: An AppletBased Presentation of the Chebyshev Equioscillation Theorem
1An Applet-Based Presentation of the Chebyshev
Equioscillation Theorem
- Robert Mayans
- Fairleigh Dickinson University
- January 5, 2007.
2Statement of the Theorem
- Let f be a continuous function on a,b.
- Let pn be a polynomial of degree n that best
approximates f, using the supremum norm on the
interval a,b. - Let dn f pn inf f p
deg(p) n
3Statement of the Theorem
- There exists a polynomial pn of best
approximation to f, and it is unique. - It alternately overestimates and underestimates
the function f by exactly dn, at least n2
times. - It is the unique polynomial to do so.
4An Example
- Function y f(x) ex, on -1,1
- Best linear approximation
- y p1(x) 1.1752x1.2643
- Error f - p1 0.2788
5An Example
6An Example
E(-1) 0.2788 E(0.1614)
-0.2788 E(1) 0.2788
7Another Example
- Function y f(x) sin2(x), on -p/2,p/2
- Best quadratic approximation
- y p2(x) 0.4053x20.1528
- Error f p2 0.1528
8Another Example
9Another Example
Alternating set -p/2, -1.0566, 0, 1.0566, p/2
f p2 0.1528
10A Difficult Theorem?
The proof is quite technical, amounting to a
complicated and manipulatory proof by
contradiction. -- Kendall D. Atkinson, An
Introduction to Numerical Analysis
11Weierstrass Approximation
- Let f be a continuous function on a,b.
- Then there is a sequence of polynomials q1, q2,
that converge uniformly to f on a,b. - In other words, f - qn ? 0 as n ? 8.
12 Bernstein Polynomials
- Suppose f is continuous on 0,1.
- Define the Bernstein polynomials for f to be the
polynomials - The Bernstein polynomials Bn(f,x) converge
uniformly to f.
13 Bernstein Polynomials
- An applet to display the Bernstein polynomial
approximation to sketched functions. - We use of de Casteljaus algorithm to calculate
high-degree Bernstein polynomials in a
numerically stable way.
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.
16Alternating 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(xi) - p(xi) alternate sign for
i0,1, , n-1 and - and for each i, f(xi)-p(xi) f - p .
17An Example
18Alternating 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.
19Variable Alternating Set
- Definition
- A variable alternating set for f, p on a,b is
like an - alternating set x0, , xn-1, in that f(xi) -
p(xi) alternate - in sign, except that the distances di f(xi) -
p(xi) - need not be the same.
20Example Variable Alternating Set
Variable Alternating Set -1.1, -0.4, 0.3, 1.2
21Variable Alternating Set
- If x0,, xn-1 is a variable alternating set for
f, p with distances d0, , dn-1, and g is a
continuous function close to f, then x0,,xn-1 is
a variable alternating set for g, p. - Close means that f g lt min di .
- Proof is obvious.
22Variable Alternating Set
Variable Alternating Set -1.1, -0.4, 0.3, 1.2
23Variable 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 dn f pn , where pn is a
polynomial of best approximation. - If f,q has a variable alternating set of length
n2 with distances d0, , dn1, then dn min di
24Variable Alternating Set
- Proof
- If not, f and q are close, so x0, xn1 is a
variable alternating set for q, pn - Thus q - pn has at least n2 changes in sign,
hence at least n1 zeroes, hence q pn, which
is impossible.
25Variable Alternating Set
- Corollary
- Let p be a polynomial approximation to f of
degree n. - If f,p has an alternating set of length n2, then
p is a polynomial of best approximation.
26Sectioned Alternating Sets
- A sectioned alternating set is an alternating set
x0,...,xn-1 - together with nontrivial closed intervals
I0,...,In-1, called - sections, with the following properties
- The intervals partition a,b.
- For every i, xi is in Ii
- .
- If xi is an upper point, then Ii contains no
lower points. If xi is an lower point, then Ii
contains no upper points.
27Sectioned Alternating Sets
- An example on -1,1
- f(x) cos( pex), p(x)0
28An Example
Function y f(x) cos( pex) on
-1,1 Alternating set (0, ln 2) Sections
-1,0.3, 0.3, 1
29Sectioned Alternating Sets
Theorem Any alternating set can be extended
into a (possibly larger) alternating set with
sections. Applet
30Improving the Approximation
- Suppose f, p has a sectioned alternating set of
length n1. - Then there is a polynomial q of degree n such
that f (pq) lt f - p - Applet
31Proof 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 n2 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.
32Proof of the Theorem
- We can extend that alternating set to one of
length n2 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 (pq)/2. We can show
that p-q has n2 changes in sign, hence n1
zeros, hence pq. - We conclude that the polynomial of best
approximation is unique, and the theorem is
proved.
33Finding the Best Approximation
- Cannot solve in complete generality
- Use the Remez algorithm to find polynomials.
- Applet
34The Remez (Remes) Algorithm
- Start with an approximation p to f on a,b and a
variable alternating set x0, , xn1 of length
n2. - Start Loop Solve the system of equations
- This is a linear system of n2 equations (i0,
, n1) in n2 unknowns (c0, , cn, E). - Using the new polynomial p, find a new variable
alternating set, by moving each point xi to a
local max/min, and including the point with the
largest error. - Loop back.
35Hypertext for Mathematics
- This proof with applets is part of a larger
hypertext in mathematics. - Also published in the Journal of Online
Mathematics and its Applications - Link to the Mathematics Hypertext Project
36Open Architecture, Math on the Web
- Java applets
- JavaScript functionality
- MathML
- Scalable Vector Graphics (SVG)
- TEX family
- Mathematical fonts
- Unicode
37Sad 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.
38Sad State of Tools
- Tool development by practitioners will be crucial
to make this architecture work.