Homework 7 Math 8052, Spring 2009
1. Show that Q∞
n=1 cos(z/n) converges uniformly on compact subsets of Cto an
entire function f. Find the zeros of f.
2. Do problems 9, 12, and 13, page 174, of the textbook.
—B./C—
Problem 2 of Homework 4 (assertions 1-4 below hold with far weaker assumptions):
1. Let χ:R→Rbe a Ckfunction with support contained in the interval (−ε, ε)
(here and below ε > 0), let Ibe an open interval, and let f:I→Rbe continuous.
The convolution (f∗χ)(x) can be defined for xin the set
Iε={x∈I: dist(x, R\I)> ε}.
Show that f∗χis of class Ckon Iε, and
d`
dx`(f∗χ) = f∗d`χ
dx`for `≤k.
2. Let χ:R→Rbe continuous with support in the interval (−1,1). Let
c=Zχ(x)dx.
Define χε(x) = ε−1χ(x/ε). Let Ibe an open interval, f:I→Rcontinuous. Show
that for any compact K⊂I, (f∗χε)|K, which is defined for all sufficiently small
ε > 0, converges uniformly to cf.
3. Show that if φand χare continuous, nonnegative, and compactly supported,
then φ∗ψis nonnegative and compactly supported.
4. Let Ibe an open interval, f:I→Rcontinuous, and χ:R→R,χεas in
Problem 2. If ϕ:I→Ris a continuos function with compact support and εis
small enough, then
ZI
(f∗χε)(x)ϕ(x)dx =ZI
f(y)(ˇχε∗ϕ)(y)dy
where ˇχε(x) = χε(−x).
5. Do Problem 2 of Homework 4.
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