Mathematics / advanced
Fourier and harmonic analysis
Study frequency decomposition through convergence, transforms, convolution, interpolation, and singular integrals.
Your route through mathematics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 advanced
Fourier and harmonic analysis
Study frequency decomposition through convergence, transforms, convolution, interpolation, and singular integrals.
- 01
Trigonometric systems and Fourier coefficients
Treat periodic frequencies as an orthonormal system, compute coefficients with a fixed normalization, and distinguish energy from pointwise recovery.
34 min - 02
Dirichlet and Fejér kernels
Express truncation and averaging as circle convolutions, and see how positivity changes stability and convergence.
35 min - 03
Pointwise convergence and summability
Separate pointwise, uniform, L² and averaged convergence, including the midpoint rule at periodic jumps.
36 min - 04
Fourier transform on Schwartz functions
Choose Euclidean Fourier normalization, justify differentiation and integration by parts, and calculate Gaussian transforms.
35 min - 05
Plancherel and Fourier inversion
Extend Fourier analysis from test functions to L² while distinguishing norm inversion from absolutely convergent pointwise integrals.
37 min - 06
Convolution, approximate identities, and Young's inequality
Control convolution in Lebesgue spaces and state exactly when shrinking kernels recover a function in norm.
37 min - 07
Distributions and Fourier multipliers
Extend Fourier operations to tempered distributions and distinguish bounded L² symbols from unbounded differentiation.
39 min - 08
Maximal functions and weak-type bounds
Measure the largest local average, derive an exact interval example, and distinguish weak L¹ control from strong norm bounds.
38 min - 09
Hilbert transform and singular integrals
Use principal-value cancellation and a signed frequency multiplier while respecting endpoint failures and periodic zero modes.
40 min - 10
Littlewood–Paley decompositions
Organize smooth frequency bands into a square function and distinguish exact L² orthogonality from Lᵖ norm equivalence.
41 min