Fourier series and orthogonality, completeness

[kidsasd987]

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

 

[DrClaude]

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]

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?

 

[DrClaude]

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).

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.

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.

 

[CellCoree]

I would say that the key here is to understand the assumptions and limitations of the Fourier series and the concept of orthogonality and completeness. The example given in the document may seem contradictory at first, but it is important to note that the assumptions made in pg 9 are not applicable to all functions. In particular, the assumption that f(x) is equal to the sum of the individual terms in the Fourier series is only valid for certain types of functions.

Furthermore, the concept of orthogonality and completeness in Fourier series is based on the idea that the functions used in the series are orthogonal to each other and that the series is complete in the sense that it can represent any function in the given interval. However, this does not mean that the series will always converge to the exact function being represented. In some cases, it may only converge to an approximation of the function.

Therefore, it is important to understand the assumptions and limitations of Fourier series and to use them appropriately in different situations. In this case, the inequality arises because the function being represented may not fulfill the assumptions made in pg 9, and the series may only be an approximation of the function. As scientists, we must always be critical and cautious in our use of mathematical tools and understand their limitations in order to make accurate and meaningful conclusions.