Harmonic analysis (spring 2016)

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(2/2)
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 

Exam #1
Exam #2


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

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