Fourier series | ||
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Hilbert spaces and L_{2}(T) |
Assignment #1 | |

Fourier series in L _{2}(T) |
Assignment #2 | |

Fejér's method | ||

Fourier series in L _{1}(T) |
Assignment #3 | (1) in q-n 2: P; _{r}(y) = Σ_{-∞<n<∞} r^{|n|}e_{n}(y)(2) in q-n 3-a the assumption should be c rather than _{n}-c_{n-1} = o(1/n)c._{n} = 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 ∂.^{2}u/∂x^{2} |

Fourier series of measures | ||

Multidimensional Fourier series | Assignment #5 | |

Fourier transform | ||

Fourier transform of Schwartz functions | ||

Fourier transform in L _{2}(R) |
Assignment #6 | q-n 6-b: compact in C(.R) |

Fourier transform in L _{1}(R) |
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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: j; c: lim_{d}''+(d-1)ξ^{-1}j_{d}'=—j_{d}_{d→∞} jexp(_{d}(ξ√d) = -ξ)^{2}/2 |

Some applications | ||

Paley–Wiener theorems | ||

Quasianalytic classes | Assignment #10 | q-n 2: the assumption shoud read: k; q-n 6: Carleman's inequality _{N} / N^{3/4} → 0∑ may be of help_{n}(a_{1}...a_{n})^{1/n} ≤ e ∑_{n}a_{n} |

Discrete Fourier transform | ||

Fourier transform on Z/qZ |
Assignment #11 | |

Fast Fourier transform |

Exam #1

Exam #2

- H. Dym and H. McKean, Fourier series and integrals
- Y. Katznelson, An introduction to harmonic analysis
- H. L. Montgomery, Early Fourier analysis

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