Inner product spaces: general Ideas
such that
Energy:Use inner product to measure correlation :
well-correlated:
uncorrelated:
, Length:
Recall definition: inner product spacewhen
, spaces:
Index set : finite or infinite
discrete case
all sequences:
with finite energy:
Inner product:
Examples:
,
finite means
, spaces:
Continuous case
Suppose integrals defined on
all functions
with finite energy:
Inner product:
Examples:
Inner product spaces: general properties
Cauchy-Schwartz Inequality:
Triangle Inequality:
Polarization:
Inner product spaces: general Ideas
Remember: can now be infinite-dimensional.
So we have to take more care! Family in
orthonormal when
Bessel’s Inequality:
Inner product spaces: general Ideas
Complete orthonormal family:
When complete:
in sense
Parseval:
Examples orthonormal families:
1.
2.
3.
4.
standard basis, Fourier basis complete in
Haar type
complete in translates
of box function in . Complete? NO
translates and dilates
of box function in . Complete? NO
More Examples:
5.translates
of Haar function in . Complete? NO
Prototypical wavelet idea:
6.
splitting
translates
of box and Haar function in
Complete? NO
.
But complete in
Operators on inner product spaces
In infinite dimensions need to worry about
controlling energy: bounded when
Energy-preserving when
Basic example: Synthesizing
Basic example
Adjoint: Analyzing
Synthesizing
orthonormal in
Check energy:
Bessel’s inequality
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