السنة الأولى من الدراسات العلياAnalysisالدرس متاح
Fourier and harmonic analysis
Decomposing functions into frequencies: Fourier series and the Fourier transform, L² theory and Plancherel, convolution and approximate identities, pointwise convergence and its subtleties, the Poisson summation formula, the uncertainty principle, and a glimpse of harmonic analysis on groups.
math.analysis.harmonic-analysis
يعتمد على
عدد المتطلبات السابقة الإلزامية (حتى الأساس): 43
ستتعلّم
- Prove Plancherel's theorem and use convolution with approximate identities to show Fourier series of continuous functions are Cesàro summable.
- Compute Fourier transforms, including of Gaussians, and apply Poisson summation.
- State and prove the Heisenberg uncertainty inequality, and describe the honest state of pointwise convergence (Carleson; Kolmogorov).
تعريف
- The n-th Fourier coefficient of an integrable 2π-periodic f is f̂(n) = (1/2π) ∫_{−π}^{π} f(x) e^{−inx} dx.
- The Fourier transform of f ∈ L¹(ℝ) is f̂(ξ) = ∫_ℝ f(x) e^{−2πixξ} dx.
- The convolution of f and g is (f ∗ g)(x) = ∫ f(x − y) g(y) dy.
مبرهنة
- Plancherel: the Fourier transform extends to a unitary map of L²(ℝ).
- Fejér: the Cesàro means of the Fourier series of a continuous periodic function converge uniformly to it.
- Carleson: the Fourier series of a function in L²(𝕋) converges almost everywhere; Kolmogorov constructed an L¹ function whose Fourier series diverges almost everywhere.
- Poisson summation: Σ_n f(n) = Σ_k f̂(k) for suitable f, such as Schwartz functions.
مفاهيم خاطئة شائعة
- Thinking the Fourier series of a continuous function always converges pointwise; du Bois-Reymond found a continuous function whose Fourier series diverges at a point.
- Believing the Gibbs overshoot disappears as more terms are added; its height near a jump tends to about 9% of the jump.
اكتشافات اختيارية
- Analytic number theory Poisson summation proves the functional equation of the theta function, from which Riemann derived the functional equation of the zeta function.
- Mathematical foundations of quantum mechanics Position and momentum wave functions are Fourier transforms of each other; the Heisenberg uncertainty principle is a theorem of Fourier analysis.
- ← Random structures: walks, percolation and random geometry
- ← Extremal and additive combinatorics
مسارات التعلّم
قراءات إضافية
- Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction (2003)
- Yitzhak Katznelson, An Introduction to Harmonic Analysis (1968)
- Loukas Grafakos, Classical Fourier Analysis (2008)