An Applet-Based Presentation of the Chebyshev Equioscillation Theorem

Robert MayansFairleigh Dickinson University

January 5, 2007.

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 }

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 n+2 times.

• It is the unique polynomial to do so.

An Example

• Function: y = f(x) = ex, on [-1,1]

• Best linear approximation:

y = p1(x) = 1.1752x+1.2643

• Error: || f - p1 || = 0.2788

An Example

Linear Approximation to exp(x)

0

0.5

1

1.5

2

2.5

3

-1.5 -1 -0.5 0 0.5 1 1.5

f(x)=exp(x) p1(x)=1.2643+.1752*x

An Example

Error plot, exp(x) - p1(x)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

E(x)=f(x)-p1(x)

E(-1) = 0.2788

E(0.1614) = -0.2788

E(1) = 0.2788

Another Example

• Function: y = f(x) = sin2(x), on [-π/2,π/2]

• Best quadratic approximation:

y = p2(x) = 0.4053x2+0.1528

• Error: || f – p2 || = 0.1528

Another Example

Quadratic Approximation to sin^2(x)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

f(x) = sin^2(x) p2(x) = 0.4053x^2+0.1528

Another ExampleError with 5-point alternating set

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

E(x) = f(x)-p2(x)

Alternating set: -π/2, -1.0566, 0, 1.0566, π/2

|| f – p2 || = 0.1528

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

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 ∞.

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.

knkn

kn xx

k

nnkfxfB

1/),(

0

Bernstein Polynomials

• An applet to display the Bernstein polynomial approximation to sketched functions.

• We use of de Casteljau’s algorithm to calculate high-degree Bernstein polynomials in a numerically stable way.

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.

nnn xaaxpaaapht 0

221

20 )( if )(

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.

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 i=0,1,… , n-1 and

and for each i, | f(xi)-p(xi) | = || f - p ||.

bxxa n 10

An Example

Linear Approximation to exp(x)

0

0.5

1

1.5

2

2.5

3

-1.5 -1 -0.5 0 0.5 1 1.5

f(x)=exp(x) p1(x)=1.2643+.1752*x

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.

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.

Example: Variable Alternating Set

A Variable Alternating Set

-1

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5

p(x) f(x)

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

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 || < min di .

• Proof is obvious.

Variable Alternating SetA Variable Alternating Set

-1

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5

p(x) f(x) g(x)

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

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 d0, …, dn+1, then d*

n ≥ min di

Variable Alternating Set

Proof:

• If not, f and q are close, so x0, … xn+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.

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.

Sectioned Alternating Sets

A sectioned alternating set is an alternating set x0,...,xn-1

together with nontrivial closed intervals I0,...,In-1, calledsections, 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.

Sectioned Alternating Sets

• An example on [-1,1]:

f(x) = cos( πex), p(x)=0

An Example

Upper and Lower Sections

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Function: y = f(x) = cos( πex) on [-1,1]

Alternating set: (0, ln 2)

Sections: [-1,0.3], [0.3, 1]

Sectioned Alternating Sets

Theorem:

Any alternating set can be extended into a (possibly larger) alternating set with sections.

Applet:

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:

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.

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.

Finding the Best Approximation

• Cannot solve in complete generality

• Use the Remez algorithm to find polynomials.

• Applet:

The Remez (Remes) Algorithm

• Start with an approximation p to f on [a,b] and a variable alternating set x0, …, xn+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 (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.

Excxcxcc ininii )1(2

21

10

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

Open Architecture, Math on the Web

• Java applets• JavaScript functionality• MathML• Scalable Vector Graphics (SVG)• TEX family• Mathematical fonts• Unicode

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.

Sad State of Tools

• Tool development by practitioners will be crucial to make this architecture work.