INDEX

Bessel’s Inequality

closure, 191 compression

with Daubechies wavelets

with FFT

with general wavelets

with Haar wavelets

computational complexity (number of operations)

convergence

comparison between uniform and L²,

in L²

pointwise

uniform

convergence in the mean

convergent sequence

convolution

discrete periodic signals

functions

operator

periodic function

sequences

data compression

via Fourier Series

via linear predictive coding

with general wavelets

with Haar wavelets

Daubechies scaling function

Daubechies wavelet

decomposition

algorithm

summary

via Daubechies wavelets

via Haar

via multiresolution analysis

downsampling

Fast Fourier Transform

filter

causal

low-pass

time invariant

filtering

via Fourier Series

via wavelets

Fourier coefficients

Fourier inversion formula

Fourier Series

complex form of,

computation of,

over other intervals, AA-Ad

convergence at a point of continuity

convergence at a point of discontinuity

convergence in the mean

cosine and sine expansions,

data compression

filtering with

history of,

partial differential equations,

uniform convergence of,

Fourier Transform

adjoint of

basic properties,

convolution

definition of

discrete,

discrete

definition of

properties of,

inverse of

Fredholm Alternative

frequency band-limited

Gibbs phenomenon

Gram Schmidt orthogonalization

Haar scaling function

definition

properties

Haar wavelet

Haar wavelet function

heat equation

high pass filter

impulse response function

inner product

on C

on Rⁿ,

general definition

inverse Fourier Transform

definition

rigorous proof of,

isometry between L² and I²

L²

convergence

definition

inner product

l²

definition

inner product

Laplace Transform

leastsquares

method of,

left differentiable

length

linear operator

adjoint

bounded

definition

linear predictive coding,

low-pass filter

multiresolution analysis

definition

properties,

norm

orthogonal

complement

projection

subspaces

vectors

orthonormal

basis

projection onto

Fourier Transform criteria

general definition

system

Parseval’s equation

Fourier series

Fourier transform

orthonormal basis

piecewise continuous

piecewise smooth

Plancherel formula

Poisson’s formula

quadrature mirror filter

reconstruction

algorithm

summary

via Daubechies

via Haar

via multiresolution analysis,

relative L² error

computation of

relative error

Riemann Lebesgue Lemma

right differentiable

sampling theorem

scaling equation

Fourier Transform criteria

scaling function

dyadic values,

iterative procedure for,

scaling relation

Schwarz inequality

Si(z)

system function

transfer function

quadrature mirror interpretation,

triangle inequality

two-scale relation

uncertainty principle

upsampling

wavelet construction

wavelet packets

wavelet transform

definition

inversion formula

Weierstrass M-Test

Z-transform

definition

properties