Homework #10: Filters. Due on Monday, November 19.
Each question is four points.
2
1. Consider the smoothing operator Su(x) := 21 0 K (x s) u(s) ds. Prove that
(Su)N = S (uN ), where wN denotes the L2 -projection of w L2 (0, 2 ) into the span
of cfw_eikx
Homework #7: Quadrature rules. Due on Monday, October 22.
Each question is four points. The last one is optional.
b
1. We approximate the integral a u(s) ds by the mid-point rule and by the
trapezoidal rule. Compute the Peano kernel for each of these rule
Homework #8: Interpolation with trigonometric functions. Due on
Monday, October 29.
Each question is four points.
1. Here, we compare the truncated Fourier series of the function u dened on
(0, 2 ), uN , with its trigonometric interpolate I2N u. Here we t
Homework #9: Interpolation with polynomials. Due on Monday, November 5.
Each question is four points. The last two problems are optional.
In what follows, PN u is the polynomial interpolant of the function u : (1, 1) R
at the points xj , j = 0, . . . , N
Homework #6: Greens functions of Sturm-Liouville problems. Due on
Monday, October 15.
Each question is four points. The last one is optional.
1. Consider the following regular Sturm-Liouville problem:
d2
u = u
dx2
u=0
in (0, ),
on cfw_0, .
We know that it
Homework #5: The Sturm-Liouville problem for the Chebyshev polynomials. Due on Monday, October 8.
Each question is four points. The last one is optional.
1. Show that the function dened by Ti (cos ) = cos(i), satises the following
Sturm-Liouville problem:
Homework #1: Norms. Due on Monday, September 10. Each question is
four points. The last one is optional.
1. Prove that
1/p
b
b
q
|g (x)| dx
|f (x)| dx
|f (x) g (x)| dx
,
a
a
a
1/q
b
p
for any p 1 and 1/p + 1/q = 1. This is the so-called Hlder inequality.
Homework #2: Convergence. Due on Monday, September 17.
Each question is four points.
1. Find a sequence cfw_n nN C 0 (a, b) such that n L2 (a,b) = 1 and such that
limn n L1 (a,b) = 0. Argue that for any such a sequence we must have that
limn n L (a,b) = .
Homework #3: Convergence of Fourier series. Solutions.
1. We have
:=
(x) u(x) dx
(x) uN (x) dx
(x) (u(x) uN (x) dx
(x) N (x) (u(x) uN (x) dx ,
by denition of uN . Applying the Cauchy-Schwarz inequality we now get that
N
u uN
d
u uN
,
dx
(N + 1)
since i
Homework #4: Approximation by Chebyshev and Legendre polynomials. Due on Monday, October 1.
Each question is four points. The last one is optional. In what follows, all the
functions we consider are dened on the interval (1, 1). The L2 (1, 1)-norm with
we