| Fourier series | ||
|---|---|---|
| Hilbert spaces and L2(T) | Assignment #1 | |
| Fourier series in L2(T) | Assignment #2 | |
| Fejér's method | ||
| Fourier series in L1(T) | Assignment #3 | (1) in q-n 2: Pr(y) = Σ-∞<n<∞ r|n|en(y); (2) in q-n 3-a the assumption should be cn-cn-1 = o(1/n) rather than cn = o(1/n). |
| Applications of Fourier series | Assignment #4 | q-n 5 becomes consistent with diffusivity ½, i.e. the heat equation should read: ∂u/∂t = ½ ∂2u/∂x2. |
| Fourier series of measures | ||
| Multidimensional Fourier series | Assignment #5 | |
| Fourier transform | ||
| Fourier transform of Schwartz functions | ||
| Fourier transform in L2(R) | Assignment #6 | q-n 6-b: compact in C(R). |
| Fourier transform in L1(R) | ||
| Poisson summation formula | Assignment #7 | (1) q-n 3: the term exp(2πi x ξ) in the conclusion is redundant; (2) 5-a: the solution that I know requires the Riesz–Thorin interpolation theorem which we have not studied. |
| Heisenberg uncertainty principle | ||
| Fourier transform of measures | Assignment #8 | solution to q-n 3 |
| Multidimensional Fourier transform | ||
| Fourier transform in the complex domain | ||
| Analytic continuation of periodic functions | Assignment #9 | q-n 3-a: jd''+(d-1)ξ-1jd'=—jd; c: limd→∞ jd(ξ√d) = exp(-ξ2/2) |
| Some applications | ||
| Paley–Wiener theorems | ||
| Quasianalytic classes | Assignment #10 | q-n 2: the assumption shoud read: kN / N3/4 → 0; q-n 6: Carleman's inequality ∑n(a1...an)1/n ≤ e ∑nan may be of help |
| Discrete Fourier transform | ||
| Fourier transform on Z/qZ | Assignment #11 | |
| Fast Fourier transform | ||