LECTURE NOTES ON DYADIC HARMONIC ANALYSIS

13

operators can be decomposed as a a piece which is close to a translation invariant

(or convolution) operator plus some paraproducts:

T

=

S

+

7rb

1

+

1r;2

;

where

1r;

is the adjoint of

1fb.

We will state this more precisely in the fourth lecture.

1.5. Dyadic analogues. In this section we introduce dyadic analogues of each

of the operators discussed above (not necessarily in the same order).

Intervals of the form [k2-j, (k + 1)2-1) for integers j, k are called dyadic in-

tervals. The collection of all dyadic intervals is denoted by V, and VJ denotes all

dyadic intervals

I,

such that

III

= 2-1, also called the j-th generation. It is clear

that each V

1

provides a partition of the real line, and that V = UjE'llVJ.

EXERCISE 1.22. Show that given

I,

J E

V,

then either they are disjoint or one

is contained in the other.

This "martingale" property is what makes the dyadic intervals so useful.

Each dyadic interval

I

is in a unique generation

V

1

,

and there are exactly two

subintervals of

I

in the next generation VJ+

1

,

the children of

I,

which we will denote

Ir, Iz,

the right and left child respectively. Clearly,

I=

h

U

Ir·

Associated to any interval I there is a Haar function defined by:

1

hi(x)

=

III

112

[xdx)- XIJx)],

where XI(x) = 1 if x

E

I,

XI(x) = 0 otherwise. It is not hard to see that {hi}IED

is an orthonormal basis in L

2

(R),

that is the content of the next exercise.

EXERCISE 1.23. Show that

j

hi = 0, lihiii2 = 1 and that (hi, hJ) = JI,J

6

for

all

I,

J

E

V.

Furthermore show that if(!, hi)= 0 for all

IE

V,

then

f

= 0 in L

2

.

We will introduce here two operators that will play the role of Qt and Pt

in the continuous case. Denote the average of a function

f

on the interval I by

mi f =

1

}

1

JI f(t) dt. Then the expectation operators are defined by

Enf(x)

=

mif, x

E IE

Vn;

and the difference operators by

l:!.nf(x)

=

En+d(x)- Enf(x).

EXERCISE 1.24. Show that l:!.nf(x) = LIEDn (!, hi)hi(x). Show that for all

f

E

L

2

(R)

EN+d(x)

=

L

l:!.nf(x).

n-~N

This provides another proof of the completeness of the Haar system, after observing

that limn---oo Enf =fin L

2

;

i.e.: f = LnE'll l:!.nf·

The Haar functions were introduced by A. Haar in 1909, see

[Ha].

They provide

the oldest example of a wavelet basis.

EXERCISE 1.25. Let

I=

[k2-j, (k+1)2-1), show that hi(x) =

2JI2h(2Jx-k)

=

h1,k(x),

where h = hro,l]·

6As

usual lh,J = 0 if If J, 81,J = 1 if I= J.