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## Main Question or Discussion Point

So the other day in class my teacher gave a proof for the completeness of [itex] \phi_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx} [/itex] in [itex] L^2([-\pi,\pi]) [/itex]. And I'm trying to convince my self I understand it at least a little. He defined Frejer's Kernel

[itex]

K_n(x) = \frac{1}{2\pi(n+1)} \sum_{k=-n}^{n}(n+1-|k|)e^{ikx}

[/itex]

He claimed without showing that it could be written as

[itex]

K_n(x) = \frac{1}{2\pi(n+1)} \frac{sin^2((n+1)\frac{x}{2})}{sin^2(x/2)}

[/itex]

He showed it was a summability kernel, and then claimed that any summability kernel approaches the identity. And used this to prove that the only vector orthogonal to all [itex] \phi_n [/itex] is the zero vector. I understand why the last step completes the proof, I'm just looking for some more insight into the rest of the proof.

What I'd like to be able to do, is understand where these definitions come from. What is this kernel and why is it so useful? For instance, if i was trying to prove completeness of the equivalent sines and cosines what would be different? Would the kernel look any different?

Also, in my class on Fourier series, we learned that [itex] sin(nx) [/itex] and [itex] cos(nx) [/itex] are complete on their own, in [itex] L^2([0,\pi]) [/itex]. How does this relate? How could I prove this? Would I have to use a new kernel?

Thanks in advance!!

[itex]

K_n(x) = \frac{1}{2\pi(n+1)} \sum_{k=-n}^{n}(n+1-|k|)e^{ikx}

[/itex]

He claimed without showing that it could be written as

[itex]

K_n(x) = \frac{1}{2\pi(n+1)} \frac{sin^2((n+1)\frac{x}{2})}{sin^2(x/2)}

[/itex]

He showed it was a summability kernel, and then claimed that any summability kernel approaches the identity. And used this to prove that the only vector orthogonal to all [itex] \phi_n [/itex] is the zero vector. I understand why the last step completes the proof, I'm just looking for some more insight into the rest of the proof.

What I'd like to be able to do, is understand where these definitions come from. What is this kernel and why is it so useful? For instance, if i was trying to prove completeness of the equivalent sines and cosines what would be different? Would the kernel look any different?

Also, in my class on Fourier series, we learned that [itex] sin(nx) [/itex] and [itex] cos(nx) [/itex] are complete on their own, in [itex] L^2([0,\pi]) [/itex]. How does this relate? How could I prove this? Would I have to use a new kernel?

Thanks in advance!!