L2 norm of complex functions?


by divB
Tags: complex, functions, norm
divB
divB is offline
#1
Jan15-13, 12:19 AM
P: 76
Hi,

I want to show:

[tex]
\|f-jg\|^2 = \|f\|^2 - 2 \Im\{<f,g>\} + \|g\|^2
[/tex]

However, as far as I understand, for complex functions [itex]<f,g> = \int f g^* dt[/itex], right? Therefore:

[tex]
\|f-jg\|^2 = <f-jg, f-jg> = \int (f-jg)(f-jg)^* dt = \int (f-jg)(f+jg) dt = \int f^2 + jfg - jfg + g^2 dt = \|f\|^2 + \|g\|^2
[/tex]

Where is my wrong assumption?
Thanks.
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
CompuChip
CompuChip is offline
#2
Jan15-13, 03:57 AM
Sci Advisor
HW Helper
P: 4,301
Are f and g complex functions? Then you should write f = Re(f) + j Im(f), g = Re(g) + j Im(g).

Or are they just the real parts of a single function? Because then you've just shown that ||f|| = ||Re f|| + ||Im f||, which makes sense, right?
Fredrik
Fredrik is offline
#3
Jan15-13, 08:40 AM
Emeritus
Sci Advisor
PF Gold
Fredrik's Avatar
P: 9,010
Quote Quote by divB View Post
[tex]
\int (f-jg)(f-jg)^* dt = \int (f-jg)(f+jg) dt
[/tex]
This one should be $$\int (f-jg)(f-jg)^* dt = \int (f-jg)(f^*+jg^*) dt.$$ But why use the definition of <,> at all? I assume that you have already proved that it's an inner product. So why not just use that?

divB
divB is offline
#4
Jan15-13, 03:01 PM
P: 76

L2 norm of complex functions?


Hi, thank you. Ok, no integrals, but only use <,>

I am again confused :(

[tex]\|v\|^2 = <v,v>[/tex], as far as I understand also for complex functions. But then, with using only the inner product, I have no chance to obtain an imaginary part only:

[tex]
\|f-jg\|^2 = <f-jg,f-jg> = <f,f-jg>-<jg,f-jg> \\
= <f,f> - <f,jg> - (<jg,f>-<jg,jg>) \\
= <f,f> - <f,jg> - <jg,f> + <jg,jg> \\
= <f,f> - j<f,g> - j<g,f> - <g,g>
= \|f\|^2 - 2j<f,g> - \|g\|^2
[/tex]

But [itex]2j<f,g>[/itex] is not [itex]2\Im<f,g>[/itex]...
CompuChip
CompuChip is offline
#5
Jan15-13, 03:04 PM
Sci Advisor
HW Helper
P: 4,301
What is <cf, g> and what is <f, cg> if c is a complex number?
Fredrik
Fredrik is offline
#6
Jan15-13, 06:31 PM
Emeritus
Sci Advisor
PF Gold
Fredrik's Avatar
P: 9,010
Quote Quote by divB View Post
[tex]<f,f> - <f,jg> - <jg,f> + <jg,jg> \\
=<f,f> - j<f,g> - j<g,f> - <g,g>\\
= \|f\|^2 - 2j<f,g> - \|g\|^2
[/tex]
These steps are both wrong. What are the properties of an inner product on a complex vector space?


Register to reply

Related Discussions
Complex Convergence with Usual Norm Calculus & Beyond Homework 4
The limit of p-norms is the max norm in the space of continuous functions in [a,b] Calculus & Beyond Homework 0
Convergence of a sequence of functions to zero in the L1 norm? Calculus 7
How do i determine if a rock is qtz-norm, ol-norm, hy-norm or ne-norm? Earth 1
Convergence with L2 norm functions Calculus & Beyond Homework 2