Alphabeta MathFourier Analysis
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
5 pagesThis page builds the character group of an abelian topological group with the compact-open topology and proves the elementary properties of the Pontryagin dual that do not…
1 definition, 10 lemmas, 2 theoremsExamples & counterexamples →This page keeps the algebraic Dirichlet-kernel identities separate from the oscillatory arguments that make pointwise convergence local.
2 definitions, 6 lemmas, 4 theorems, 2 corollariesExamples & counterexamples →This page develops the discrete Fourier transform on the finite cyclic group ℤ/Nℤ from finite sums alone, with no convergence, regularity or topological hypothesis anywhere.
5 definitions, 4 lemmas, 4 theoremsExamples & counterexamples →This page develops translation-invariant Fourier multipliers and the Fourier characterisations of the Sobolev scale.
4 definitions, 3 lemmas, 4 theorems, 1 corollaryExamples & counterexamples →This page develops the unit-normalized Riesz potential on complex Euclidean Lebesgue spaces and proves the strict-range Hardy–Littlewood–Sobolev fractional integration theorem.
1 definition, 2 lemmas, 1 theoremExamples & counterexamples →
Level 1
5 pagesThis page develops the representation, inversion and Plancherel theory of the Fourier transform on a locally compact Hausdorff abelian group without assuming Pontryagin biduality.
2 definitions, 13 lemmas, 5 theorems, 1 corollaryExamples & counterexamples →This page treats the two positive summation methods for one-period Fourier series.
1 definition, 2 lemmas, 7 theorems, 1 remarkExamples & counterexamples →This page develops L²-density Fourier restriction theory for a compact curved hypersurface.
2 definitions, 16 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →This page carries the Euclidean lattice refinement of the published Schwartz Poisson theorem: it fixes the objects a lattice supplies (a full-rank lattice Λ=Aℤⁿ, its covolume…
2 definitions, 9 lemmas, 3 theorems, 1 corollary, 1 remarkExamples & counterexamples →This page isolates three inequivalent notions of localisation of a function on ℝⁿ and states precisely how much can be said about each.
1 definition, 7 lemmas, 4 theorems, 1 corollary, 3 remarksExamples & counterexamples →
Level 2
5 pagesAbsolute summability makes Fourier synthesis uniformly convergent and turns coefficient convolution into a Banach-algebra product.
2 definitions, 5 lemmas, 3 theorems, 2 corollariesExamples & counterexamples →This construction produces an integrable function on the period-one circle whose symmetric Fourier partial sums are unbounded almost everywhere.
1 definition, 6 lemmas, 1 theoremExamples & counterexamples →Hadamard gaps make finite Fourier sums behave like orthogonal random sums in every finite Lᵖ scale.
2 definitions, 3 lemmas, 2 theorems, 2 corollariesExamples & counterexamples →This page completes the duality theory of locally compact Hausdorff abelian groups whose topology interface was built on the prerequisite page on character groups and elementary duals.
1 definition, 12 lemmas, 6 theorems, 2 corollariesExamples & counterexamples →- The Carleson–Hunt Time–Frequency Theorem14 results
The argument establishes the Fourier partial-sum maximal inequality on the circle for every exponent strictly between one and infinity.
3 definitions, 9 lemmas, 2 theoremsExamples & counterexamples →
Level 3
1 pageThe symmetric Fourier partial sums on T=ℝ/ℤ can diverge at a prescribed point even for a continuous real function.
1 definition, 2 lemmas, 1 corollary, 1 counterexample, 2 remarksExamples & counterexamples →
Level 4
1 page- Hilbert and Riesz Transforms16 results
This page proves the strict-range mapping theory of the periodic conjugate function and the L² theory of the Hilbert and Riesz transforms on ℝⁿ.
3 definitions, 8 lemmas, 2 theorems, 2 corollaries, 1 remarkExamples & counterexamples →
Level 5
1 pageThis page develops the Calderón–Zygmund decomposition and the mapping theory of singular integrals, and then applies it to the Hilbert and Riesz transforms and to Mihlin multipliers.
4 definitions, 12 lemmas, 4 theorems, 3 corollaries, 2 remarksExamples & counterexamples →
Level 6
2 pagesThis page develops the Muckenhoupt weighted theory that upgrades the unweighted maximal and singular-integral estimates of the Fourier analysis track to measures w dλ.
5 definitions, 19 lemmas, 4 theorems, 2 corollaries, 1 remarkExamples & counterexamples →This page develops the real Hardy spaces Hᵖ(ℝⁿ) for 0<p<∞. The space is defined by the radial maximal function of a fixed admissible Schwartz kernel φ with ∫φ≠0, and the page…
4 definitions, 16 lemmas, 4 theorems, 2 corollaries, 1 remarkExamples & counterexamples →
Level 7
1 page- BMO, John-Nirenberg, and H1 Duality20 results
This page develops the space BMO(ℝⁿ) of bounded mean oscillation modulo constants, the John-Nirenberg exponential inequality, and the duality (H¹(ℝⁿ))^ congBMO(ℝⁿ)/ℂ of the real Hardy space H¹ of the companion FR-9 page.
1 definition, 12 lemmas, 4 theorems, 3 corollariesExamples & counterexamples →
Level 8
1 pageThis page develops the inhomogeneous Littlewood–Paley theory on Euclidean space: a fixed smooth dyadic frequency partition with a retained low-frequency block, the resulting…
4 definitions, 8 lemmas, 3 theorems, 1 corollary, 1 remarkExamples & counterexamples →