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Title: An AppletBased Presentation of the Chebyshev Equioscillation Theorem


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 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

3
Statement 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.

4
An Example
  • Function y f(x) ex, on -1,1
  • Best linear approximation
  • y p1(x) 1.1752x1.2643
  • Error f - p1 0.2788

5
An Example
6
An Example
E(-1) 0.2788 E(0.1614)
-0.2788 E(1) 0.2788
7
Another 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

8
Another Example
9
Another Example
Alternating set -p/2, -1.0566, 0, 1.0566, p/2
f p2 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 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.

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(xi) - p(xi) alternate sign for
    i0,1, , n-1 and
  • and for each i, f(xi)-p(xi) 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 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.

20
Example Variable Alternating Set
Variable Alternating Set -1.1, -0.4, 0.3, 1.2
21
Variable 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.

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 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

24
Variable 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.

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 n2, then
    p is a polynomial of best approximation.

26
Sectioned 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.

27
Sectioned Alternating Sets
  • An example on -1,1
  • f(x) cos( pex), p(x)0

28
An Example
Function y f(x) cos( pex) 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. Applet
30
Improving 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

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 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.

32
Proof 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.

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 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.

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 Applications
  • Link to the Mathematics 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.
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