SYSTEMATIC MATHEMATICS

Harmonic Analysis & Wavelets

How can one theorem control averages at every radius, how can a singular kernel define a bounded operator, and how can a signal be separated simultaneously by location, frequency, and scale? This course answers those questions in forty visible bilingual chapters across eight content-sized units of lengths 5, 5, 6, 5, 5, 5, 5, and 4. It begins with translations, dilations, modulation, convolution, approximate identities, and dense test functions; then develops Fourier summability, operator interpolation, maximal estimates, differentiation, singular integrals, dyadic frequency localization, Sobolev control, Haar and multiresolution wavelets, sampling, inverse stability, and a complete multiscale dossier. Every norm, exponent, kernel cancellation, convergence mode, constant dependence, and numerical truncation is named. The course does not present a coefficient heatmap as proof, turn weak type into strong type, or claim to replace abstract locally compact group harmonic analysis, nonlinear time-frequency analysis, research-level pseudodifferential theory, or application-specific sensing and noise validation.

Before this course: Completed Measure & Lebesgue Integration, Functional Analysis, and Integral Transforms & Special Functions, with their Real Analysis, Linear Algebra, Differential Equations, and Complex Analysis prerequisites. Students should already know Lp spaces, almost-everywhere convergence, product integration, weak convergence, bounded and compact operators, Hilbert projection, Fourier conventions, convolution, tempered distributions, and basic PDE kernels. No wavelet software, FFT package, probability noise model, or locally compact group theory is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Harmonic analysis asks how a function behaves across location, frequency, and scale

Objective: Which dilation would preserve the L² norm instead?

A single global frequency coefficient can detect oscillation but forget where it occurs. A local average detects location but smooths fine detail. Harmonic analysis organizes the trade between these descriptions and proves bounds that stay valid across all scales. Begin by naming the space, norm, scale, kernel, normalization, and equality or convergence sense. The chapter’s exact result is: Translation and modulation preserve every L^p norm, while ||D_af||_p=a^{-d(1−1/p)}||f||_p for 1≤p≤∞.

The precise object is: For f on R^d, translation is τ_yf(x)=f(x−y), L¹-normalized dilation is D_af(x)=a^{-d}f(x/a) for a>0, and modulation is M_ξf(x)=e^{iξ·x}f(x). Harmonic analysis compares location and frequency across scales; a picture of oscillations is evidence only after the operator and error notion are stated.

Use the exact problem “In dimension d=1, let f=1 on [0,1] and zero elsewhere. For a=2, find D_2f and its L¹ and L² norms.” to expose every constant and hypothesis. Rebuild the three steps, verify the result in a second representation, and test this failure boundary: Different dilation conventions preserve different quantities. Using an L¹-normalized formula while claiming L² energy preservation loses a factor a^{-d/2}.

Translation and modulation preserve every L^p norm, while ||D_af||_p=a^{-d(1−1/p)}||f||_p for 1≤p≤∞.

Translation uses u=x−y and Lebesgue-measure invariance; modulation has magnitude one.

For dilation, compute ∫|a^{-d}f(x/a)|^pdx and set u=x/a, so dx=a^ddu.

The p-th power norm gains a^{-dp+d}; taking the p-th root gives a^{-d(1−1/p)}, with the supremum case checked directly.

In dimension d=1, let f=1 on [0,1] and zero elsewhere. For a=2, find D_2f and its L¹ and L² norms.

  1. D_2f(x)=(1/2)f(x/2), so it equals 1/2 on [0,2] and zero elsewhere.
  2. Its L¹ norm is length 2 times height 1/2, equal to 1.
  3. Its squared L² norm is 2·(1/2)²=1/2, so ||D_2f||₂=1/√2.

Result: D_2f=(1/2)1_[0,2], with L¹ norm 1 and L² norm 1/√2.