10
Banach Alegebras, Wieners Theorem
There is a theorem due to Wiener that asserts the following.
Theorem 10.1. Suppose f (x) on the d-torus has an absoutely convergent
Fourier Series and f (x) is nonzero on the d-torus. Then the function g (x) =
1
also h
2
Singular Integrals
We start with a very useful covering lemma.
Lemma 1. Suppose K S is a compact subset and I is a covering of K .
There is a nite subcollection cfw_Ij such that
1. cfw_Ij are disjoint.
2. The intervals cfw_3Ij that have the same midp
Final Take Home Examination.
Due before the end of term.
Q1. Let f (t) 0 be a weight on [0, 2 ]. Let
2
2 = inf
0
aj eijt |2 f (t)dt
|1
j =0
be the error in approximating 1 by functions involving only non zero frequencies in L2 (f dt).
When is 2 > 0 and
4
Riesz Kernels.
A natural generalization of the Hilbert transform to higher dimension is
mutiplication of the Fourier Transform by homogeneous functions of degree
0, the simplest ones being
Ri f ( ) =
i
f ( )
| |
(4.1)
Since the functions ki ( ) = |i| a
12
Representations SO(3)
We will consider the irreducible representations of the group G of rotations
in R3 . These are orthogonal transformations of determinant 1, i.e. that
preserve orientation. An element g G is represented as the matrix
t1,1 (g ) t1,2
3
Multidimensional Versions
The problem of convergence of Fourier Series in several dimensions is more
complicated because there is no natural truncation. If n = cfw_n1 , . . . , nd is a
multi-index, then the sum
an ein.x
n
is natuarally computed by summ
Lecture 1.(Jan 19, 2000)
1. Construct explicitly a continuous function f on S , such that the Fourier
Series of f does not converge uniformly.
2. Show that the Fourier Series of any f , which is Hlder continuous with
o
some exponent > 0, converges uniform
7
Hardy Spaces.
For 0 < p < , the Hardy Space Hp in the unit disc D with boundary
S = D consists of functions u(z ) that are analytic in the disc cfw_z : |z | < 1,
that satisfy
1
0r<1 2
2
sup
0
|u(rei )|p d <
(7.1)
From the Poisson representation formula
Chapter 1
Fourier Series
We will consider complex valued periodic functions with period 2 . We can
view them as functions dened on the circumference S of the unit circle
in the complex plane or equivalently as function f dened on [, ] with
f ( ) = f ( ).
9
Elliptic PDEs
We will apply the results of singular integrals particularly the estimate that
the Riesz transforms are bounded on evry Lp (Rd ) for 1 < p < to prove
existence of solutions u W2,p (Rd ) for the equation
u(x)
ai,j (x)
i,j
2u
xi xj
bj (x)
j
11
Compact Groups. Haar Measure.
A group is a set G with a binary operation G G G called multiplication
written as gh G for g, h G. It is associative in the sense that (gh)k =
g (hk ) for all g, h, k G. A group also has a special element e called the
iden
8
BMO
The space of functions of Bounded Mean Oscillation (BMO) plays an important role in Harmonic Analysis.
A function f , in L1 (loc) in Rd is said to be a BMO function if
sup inf
x,r
a
1
|Bx,r |
y Bx,r
|f (y ) a|dy = u
BM O
<
(8.1)
where Bx,r is the ba