# Fourier series and orthogonality, completeness

• kidsasd987
In summary, the conversation discusses the concept of completeness and orthogonality in Fourier series. The example on page 10 highlights the importance of completeness in obtaining a non-zero solution for the right-hand side of the equation. The conversation also touches on the fact that complete sets can still have zero coefficients, as seen in the example on page 12. The discussion concludes with the idea that signals can be approximated if they are not complete or if only part of the signal is known.

#### kidsasd987

http://ms.mcmaster.ca/courses/20102011/term4/math2zz3/Lecture1.pdf

On pg 10, the example says f(x)=/=0 while R.H.S is zero. It is an equations started from the assumption in pg 9; f(x)=c0f(x)0+c1f(x)1…, then how do we get inequality?

if the system is complete and orthogonal, then (f(x),ϕ_n(x))=0, which makes sense only when f(x)=0.
but we know for Fourier series, we get values for Rhs and Lhs.

kidsasd987 said:
if the system is complete and orthogonal, then (f(x),ϕ_n(x))=0, which makes sense only when f(x)=0.
That's the key word. The comment on p. 10 is for an arbitrary orthogonal system. That is why completeness is necessary, as explained on p. 11.

• kidsasd987
DrClaude said:
That's the key word. The comment on p. 10 is for an arbitrary orthogonal system. That is why completeness is necessary, as explained on p. 11.

Could you explain the details?

Also, if (f(x),ϕ_n(x))=0 holds in general for complete series, then Fourier series must be also zero since they are complete.

however we know that for signals we do get values within the interval of pi and -pi. is this because what we usually solve for signal input f(t) is not complete?

so we are approximating the signal?

kidsasd987 said:
Could you explain the details?
Look at p. 12, where an example is given of an orthogonal system that is not complete. Using only cosines, you could never write an expression for e.g. sin(x).

kidsasd987 said:
Also, if (f(x),ϕ_n(x))=0 holds in general for complete series, then Fourier series must be also zero since they are complete.
But it's the other way around. If ##(f, \phi_n) = 0## for a given ##f(x)## and for all ##\phi_n##, then ##\{\phi_n\}## is not a complete set.

kidsasd987 said:
however we know that for signals we do get values within the interval of pi and -pi. is this because what we usually solve for signal input f(t) is not complete?

so we are approximating the signal?
I'm not sure what you are asking here. If the signal is periodic but not on the interval ##[-\pi,\pi]##, then it is trivial to scale/shift it to that interval. If the signal is finite in time, then it is delt with the same way. If the signal is not finite, or only part of it is known, then indeed thre are approximations being made.