REAL ANALYSIS I HOMEWORK 6
CIHAN
BAHRAN
Stein and Shakarchi, Chapter 3
11. If a, b > 0, let
xa sin(xb )
f (x) =
0
for 0 < x 1 ,
if x = 0 .
Prove that f is of bounded variation in [0, 1] if and only if a > b. Then, by
taking a = b, construct (for each 0 <
REAL ANALYSIS II HOMEWORK 3
CIHAN
BAHRAN
Conway, Page 49
3. Let K and k be as in Proposition 4.7 and suppose that k(x, y) = k(y, x).
Show that K is self-adjoint and if cfw_n are the eigenvalues of K, each repeated
P
2
dim ker(K n ) times, then
1 |n | <