1
vote
0answers
24 views

Fourier Transform of Fractional Laplacian

I'm trying to solve a PDE with a spectral method. The PDE has a fractional Laplacian... $\Delta^s$. In regards to a numerical implementation, will the "s" term simply become the exponent of the ...
0
votes
0answers
41 views

Nomenclature for a truncation approximation

The function $f(x)$ can be written exactly in terms of a set of linearly independent basis functions $\phi_n(x)$: $$ f(x) = \sum_{n=0}^\infty k_n \phi_n(x). $$ For a large enough $N$, however, $f(x)$ ...
2
votes
0answers
35 views

Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
1
vote
1answer
71 views

Convergence of Fourier series - strange graph in proof

I am reading a text that states the following related to convergence of Fourier series: $$g_K(x) = > ...
2
votes
0answers
51 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
1
vote
0answers
43 views

Variation on a completeness relation

The completeness relation for the spherical harmonics is: $$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
2
votes
1answer
235 views

Fourier transform physical meaning [closed]

What is the physical meaning of the Fourier transform expressed at the spectral density? Also, what is the relationship between the Fourier transform and the total energy of an oscillating system? ...