MA5242 Wavelets Lecture 2 Euclidean and Unitary Spaces

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MA5242 Wavelets Lecture 2 Euclidean and Unitary
Spaces

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg Tel [phone]
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Scalar Product for Euclidean Space
Definition A Euclidean space is a vector space V
over R together with a scalar product
that satisfies, for all
Linear in the first argument
Symmetric
Positive Definite
and 0 iff u 0
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Examples
Example 1.
with
Example 2.
with
where
is symmetric and positive definite
Example 3. Euclidean space V with dot product
with
Example 4.
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Gramm-Schmidt Orthogonalization
Theorem Given a linearly independent set
of a Euclidean space V, there exists an
orthogonal set
whenever i lt j ) and satisfies
(so
and
for all
Proof. Let
where
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Example of GS Orthogonalisation
Example Applying GS Orthogonalisation to
yields
where
where
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Problem Set 1
1. Orthogonalize the following basis of
2. Continue the GS Orth. on previous page to
obtain
3. Show that
contains only even powers
of x for k odd and odd powers of x for k even
4. Show that all zeros of are in -1,1
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Gramm Matrices
Definition The Gramm matrix
of a set
is defined by its entries
Example 1. The Gramm matrix for a basis of
that consists of the columns of a matrix B is
Example 2. The Gramm matrix for the monomial
basis
of
with scalar product
is the Hilbert matrix
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Orthogonal Transforms
Definition A linear transformation A V ?V
is orthogonal if
Definition The norm . V ? R on a ES is
Theorem A linear transformation A V ? V is ort.
iff
Proof. Follows from the polarization identity
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Gramm Orthonormalization
Theorem Given a linearly independent set
with Gramm matrix G
the set of vectors
defined by
where A is a matrix is orthonormal iff
Proof.
iff
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Haar Transform
Haar Transform (one stage) on
given by matrix
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Discrete Wavelet Transform
For suitable real entries this matrix is
orthogonal.
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Scalar Product for Unitary Space
Definition A Unitary space is a vector space V
over C together with a scalar product
that satisfies, for all
Linear in the first argument
Hermitian Symmetric
Positive Definite
and 0 iff u 0
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Unitary Transforms
Definition A linear transformation A V ?V
is unitary if
Definition The norm . V ? R on a US is
Theorem A linear transformation A V ? V is uni.
iff
Proof. Follows from the unitary polarization
identity
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Problem Set 2

• Prove that a set of vectors in a Euclidean or
• Unitary space is linearly independent iff its
• Gramm matrix is positive definite

2. Derive the polarization identities and theorems
3. Show that the columns of a square matrix
having complex entries are orthogonal iff its
rows are.
4. State derive the Schwarz inequality for ES
US
5. State derive the triangle inequality for ES
US
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Problem Set 3

• Derive necessary and sufficient conditions on

and
for the discrete wavelet transform to be
orthogonal
2. Show that Daubechies length 4 filters are
good
3. Show that d has zero 0-th and 1-st order
moments
4. Write a MATLAB program to make wavelet mat.
5. Use it to compute plot the Daubecheis
length-4 WT of the vector v abs(140 21.5)
analyse