MATH 5667-001 : Introduction to Approximation Theory,
Fall 2002, University of Colorado Denver

MIDTERM

1. Consider the space R^2 equipped with the x-y coordinate system and
the standard 2-norm. Let A be a unit circle (not a disk!), i.e.,
A = (x,y): x^2 + y^2 = 1. Attempt to apply theorems of Ch. 1-2
to determine
(a) the existence and
(b) the uniqueness
of the best approximation of an arbitraty element f in R^2 by
elements of the set A.
(c) Draw a diagram of where the best
approximations will be for different cases.

2. Consider the n+1 Chebyshev interpolation points on the interval
[-5,5] for some degree n+1 > 1. Denote the largest and the smallest
point by x_0 and x_n. Is it true that x_0 = -x_n for any n?

3. Explain what the Lagrange and Newton's interpolations have in common
and what makes them different.

4. Consider the Bernstein interpolation operator B_3 that produces
polynomials of the degree not larger than 3. Show that it is not a
projection.