theorem4.5.9

View full entry at my blogspot.com blog: fourier analysis: orthonormal bases

One of the reasons Fourier analysis works so well is that $\left\{ e_n := e^{inx}\right\}_{n=-\infty}^\infty$ forms an orthogonormal subspace of $L^2[-\pi,\pi]$ functions. Of course the inner product is the usual one for complex functions on the circle, $<f,g> := \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) \overline{g(x)} dx$

Hence it is not surprising that Hilbert space geometry plays a crucial...

View full entry at my blogspot.com blog: road to reality: chapters 6-7

Levels of smoothness:

I really enjoyed how Penrose spent so much time talking about what a mathematician means by smooth. The hierarchy he has is

one continuous derivative $\rightarrow$ derivatives of all order $\rightarrow$ analytic (has a power series expansion)

I like mentioning this because before I had not considered analyticy as the next level of "smoothness",...

View full entry at my blogspot.com blog: fourier analysis: infinite series

There are some clever tricks for getting information from divergent series that I have not seen before. I will probably forget this in the future, so I would like to take some time and write some notes on Cesaro and Abel summability techniques (which just so happen to be useful in Fourier analysis!) Note: this post covers chapter 2, section 5 of the text.

I am going to make up some terminology...

View full entry at my blogspot.com blog: fourier analysis: nonzero solution of the laplacian

In the last remark of chapter 2, the authors make mention that if $u$ is harmonic on the disc (i.e. $\triangle u = 0$) and converges uniformly to $0$ as $r \rightarrow 1$, then $u$ must be identically $0$. However, if we relax the boundary condition to pointwise convergence, the conclusion no longer holds. Here is some matlab code I wrote up to investigate this problem (the solution is provided...

View full entry at my blogspot.com blog: fourier analysis: convolutions

I did not understand convolutions well so I tried to find an analogous operation for finite dimensional vector spaces. This turned out unsatisfactory but rewarding. What I focused on was the result
$$S_N(f)(x) := \sum_{n=-N}^{N}\hat{f}(n)e^{inx} = (f *D_N) $$

where $D_N = \sum_{n=-N}^{N}e^{inx}$, the so called Dirichlet kernel. Basically, convolution replaces the pointwise product in many...