Lecture 1 -- Polynomials of best approximation

Roots of polynomials

Chebyshev polynomials

Definition, recurrence relation, identities

Best approximation, alternation properties

Markov's inequality for the coefficients

Uniqueness of extremal polynomials

Lagrange interpolation formula

Application to best approximation

Chebyshev alternation theorem

Best approximation on a finite set of points

Generalization to Chebyshev systems

Lecture 2 -- Derivatives of Polynomials on the real line

Complex numbers

Fundamental theorem of algebra

De Moivre's Theorem

Complex form and generating functions for Chebyshev polynomials

Complex derivatives and the maximum principle

Markov's inequality for the derivatives

Lemma of I. Schur

Three inequalities of Bernstein - trig, real, complex polys

Proof of Markov inequality for first derivative

Extension of Schaeffer and Duffin

Lecture 3 -- Applications of Complex Analysis to Polynomials

Gauss-Lucas theorem

Physical interpretations

Bernstein's comparison of polynomial derivatives

Proof using maximum principle and conformal mapping

Ilieff-Sendov conjecture

Lecture 4 -- Coefficients of Polynomials Satisfying Inequalities

Polynomials with given bounds at n points

Rogosinski's inequalities for the derivatives

Proof of Lemma of I. Schur

Polynomial growth outside an interval where bounded

Applications when points are symmetric

Optimal choices for the n points

Estimating the top three coefficients

Divided differences

Construction of extremal polynomials

Chebyshev distribution on [-1,1]

Equivalent distribution on R

Deduction of inequalities of Bernstein and Szego

Lecture 5 -- Polynomials in abstract variables

Generalization of Bernstein's inequalities

Abstract definition of polynomial and its Fréchet derivative

Hörmander's theorem on zeros of polynomial derivatives

Bernstein's theorem for Cⁿ

Applications

Norms of multilinear mappings

Szegö's extension of Bernstein's trigonometric inequality

Polynomial roots

1)

Show that if p(x) is a polynomial of degree n and if r is a root of the polynomial, then there exists a polynomial q(x) of degree at most n-1 with p(x) = (x-r)q(x). Use this to show that p cannot have more than n distinct roots unless .

2)

Show that if a polynomial p has a local maximum or minimum value v at some point r, then is a factor of p(x)-v.

3)

Use the intermediate value theorem to show that a polynomial of odd degree has a real root.

1)

Show that .

2)

Show that the solutions of are exactly the numbers , where .

3)

Show that for .

4)

Find the minimum value of

over all numbers with . Find the points where this minimum is attained.

5)

Show that the polynomial of degree at most n-1 which is the best approximation to on an interval [a,b] is given by

Chebyshev alternation theorem

1)

Let f be a continuous function on [a,b] and suppose p is a polynomial of degree at most n-1 such that f(x) - p(x) alternates in sign at n+1 consecutive points in [a,b]. Put

Show that there is no polynomial q of degree at most n-1 which satisfies for all x with .

2)

Let f be a continuous function on [a,b] with f''(x) > 0 for a < x < b.

a)

Show that there is a unique number c with

b)

Show that y = f'(c) x + B is the best linear approximation to f on [a,b] when

3)

Find the quadratic polynomial which is the best approximation to the function f(x) = 1/x at the points x=1,2,3,4.

1)

Show that for all . (Hint: Use the addition formula for the sine and induction.)

2)

Show that . (Hint: Use the chain rule.)

3)

Show that is a linear combination of .

4)

Let p be a polynomial of degree n with distinct roots . Show that if , then

5)

Use Markov's theorem to show that if p(x) is a polynomial of degree at most n that satisfies whenever , then .

6)

Show that if a non-constant polynomial of degree at most n assumes its maximum and minimum values in an interval [a,b] at the points and , respectively, then .

7)

Show that if p is a polynomial of degree at most n and if for , then

for and . (Hint: If also , then q(t)=p(t/2+ x -1/2) satisfies the same hypothesis as p so by the Schur inequality.)

8)

Show that is a solution of the second order linear differential equation

9)

Use mathematical induction to prove that

a)

b)

.

Complex numbers

1)

Let be n complex numbers and let

Show that K is convex and is contained in every convex set containing . (Hint: For the second part, use induction and the identity

when .)

2)

Let z be any complex number. Find the maximum of over all complex numbers with .

3)

Let z and w be complex numbers. Show that exactly when for all complex numbers satisfying .

4)

Show that if p(z) is a polynomial of degree at most n, then so is .

5)

This problem shows how to use the Gauss-Lucas Theorem and the maximum principle to deduce Bernstein's inequality for complex polynomials p.

a)

Suppose when |z| = 1. Use the maximum principle and the previous problem to show that when , where n is the degree of p.

b)

Let and put . Deduce that when .

c)

Apply the Gauss-Lucas theorem to get that when .

d)

Deduce that for .

e)

Apply the maximum principle to get for .

Rogosinski's Theorem

1)

Show that whenever and .

2)

Show that whenever . (Hint: Put and show that for .)

3)

In each case below, determine the largest possible values of a polynomial and its derivatives at x=a when the polynomial has degree at most 4 and satisfies for x=-2, -1, 0, 1, 2.

a)

a = 3.

b)

a = 0.

(Hint: Use Maple)

Recursive computation of the Chebyshev polynomials

> # This Maple program computes the Chebyshev polynomials T[n]
# up to degree N
N := 10;
T[0] := 1;
T[1] := x;
for n from 1 to N-1 do
T[n+1] := expand(2*x*T[n] - T[n-1])
od;

Graphs

> plot({T[1],T[2],T[3],T[4],T[5],T[6]}, x=-1.1..1.1, -3..3, title=`Chebyshev Polynomials`);

> # This Maple program expands the generating function for
# the Chebyshev polynomials
f:=(1-x*r)/(1-2*x*r+r^2);

> simplify(taylor(f,r,11));

Verify summation formula for the Chebyshev polynomials

> # This Maple program checks a summation formula for the Chebyshev polynomials.
N:=10;
for n from 1 to N do
T[n]:=simplify(sum('(-1)^k *binomial(n,2*k)*x^(n-2*k)*(1-x^2)^k',k=0..n/2))
od;

Explicit formula for coefficients

This Maple program uses the second order linear differential equation satisfied by the

Chebyshev polynomials to compute a recursion relation for the coefficients. They

are computed in descending order and a general formula is obtained.

> diffeq:=

> (1-x^2)*diff(y(x),x,x)-x*diff(y(x),x)+n^2*y(x) = 0;

> dsolve({diffeq, y(1)=1, D(y)(1) = n^2}, y(x));

Try power series method:

> terms:=sum(a[m]*x^m, m=k-2..k);

> subs(y(x) = terms, diffeq);

> simplify(%);

> low_term:= coeff(lhs(%),x^(k-2));

> expr:=solve(low_term = 0, a[k-2]);

> recursive_eq := a[k-2]=factor(expr);

> a[n]:=2^(n-1);

> recursive_eq2:= subs(k=n-m,recursive_eq);

> for m from 0 by 2 to 8 do

> a[n-m-2]:= normal(rhs(recursive_eq2))

> od;

The general formula appears to be:

A similar argument shows that a[n-1] = 0 and thus a[n-2m+1] = 0 for the same m.

Classical theory of polynomials

1. E. W. Cheney, Introduction to Approximation Theory, Reprint of 1982 McGraw-Hill Edition, AMS Chelsea Publishing, Providence, 1998.
2. Philip J. Davis, Interpolation and Approximation, Reprint of 1963 Blaisdell Edition, Dover, New York, 1975.
3. L. A. Harris, Coefficients of polynomials of restricted growth on the real line, J. Approx. Theory 93(1998), 293-312.
4. I. P. Natanson, Constructive Function Theory, Vol. I, Frederick Ungar, New York, 1964.
5. G. V. Milovanovic, D. S. Mitrinovic, Th. M. Rassias, Topics In Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore 1994.
6. G. Pólya and G. Szegó, Problems and Theorems in Analysis Vol. II, Springer-Verlag, New York, 1976.
7. T. J. Rivlin, Chebyshev Polynomials, Second ed., John Wiley & Sons, New York, 1990.
8. W. W. Rogosinski, Some elementary inequalities for polynomials, Math. Gaz. 39(1955), 7-12.
9. V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable, M.I.T. Press, Cambridge, Mass., 1968.

Abstract polynomials

1. L. A. Harris, Commentary on problems 73 and 74, The Scottish Book, R. D. Mauldin, Ed., Birkhäuser 1981, 143-150.
2.                , Bernstein's polynomial inequalities and functional analysis, Irish Math. Soc. Bull. 36(1996), 19-33.
3. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Chapter XXVI, Amer. Math. Soc. Colloq. Publ., Vol. 31, AMS, Providence, 1957.
4. T-W. MA, Classical Analysis on Normed Spaces, World Scientific, Singapore, 1995.
5. L. Nachbin, Topology on Spaces of Holomorphic Mappings, Section 3, Ergebnisse 47, Springer-Verlag, New York, 1969.

Biographical Notes