Learning Path — Fourier & Harmonic Analysis
Suggested preparation for Fourier & Harmonic Analysis. Choose an entry point that fits your preparation.
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Entry / bridge route · Additional preparation for research reading · All paths
Entry / bridge route
Calculus
Use limits, derivatives, integrals and series; distinguish computation from justification.
Calculus, Volume 1 — Gilbert Strang and Edwin 'Jed' Herman. Direct open PDF
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You can solve representative differentiation, integration and convergence exercises and explain the hypotheses used.
Proofs & Mathematical Writing
Read quantified statements; write direct, contrapositive, contradiction and induction arguments.
Book of Proof — Richard Hammack. Direct open PDF
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You can explain the quantifiers in a statement and write these forms of proof without relying on a worked solution.
Real Analysis
Prove convergence, continuity, compactness and differentiation/integration results.
Basic Analysis I — Jiří Lebl. Direct open PDF
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You can work with epsilon arguments and prove standard compactness and convergence results, not just calculate limits.
Measure & Integration
Build measures, measurable functions, integrals and convergence theorems.
Measure, Integration & Real Analysis — Sheldon Axler. Direct open PDF
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You can apply monotone/dominated convergence and Tonelli/Fubini with their hypotheses checked.
Linear Algebra
Work with vector spaces, bases, linear maps, kernels, rank and eigenvalues.
Linear Algebra — Jim Hefferon. Open online text / archive
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You can prove rank–nullity, change bases and distinguish diagonalizable maps from general linear maps.
Functional Analysis
Work with Banach/Hilbert spaces and bounded linear operators.
Introduction to Functional Analysis (18.102) — Richard Melrose. Direct open PDF
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You can use completeness, duality and basic operator examples; verify the specific text’s spectral or measure requirements.
Fourier & Harmonic Analysis
Study Fourier methods, maximal operators and singular integrals.
The Fourier Transform and its Applications — Brad Osgood. Direct open PDF
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You are comfortable with Lp estimates, Fourier transforms and interpolation; advanced multiplier topics remain separate.
Research reading: additional tools
Choose according to the particular text, not as an obligation to complete every foundation again.