Fourier Analysis

Module aims

Prove or apply a convergence theorem with hypotheses, and compute a transform only when the integral or series is justified.

Course outline

1. 01 Orthogonal expansions 20h

   Learning result Derive coefficients for an orthogonal system in an inner-product space.

   Formative check Written derivation

2. 02 Fourier series: pointwise and L² 30h

   Learning result State a convergence theorem and a function that fails a stronger naive claim.

   Formative check Counterexample

3. 03 Convolution and approximate identities 20h

   Learning result Use convolution with an approximate identity at synopsis grain.

   Formative check Written derivation

4. 04 Fourier transform 30h

   Learning result Compute a transform of a standard function and state inversion hypotheses.

   Formative check Written derivation

5. 05 Parseval on the circle (taught grain) 10h

   Learning result Apply a Parseval identity for Fourier coefficients on the circle, with completeness named as a pending extra.

   Formative check Four-option check

6. 06 Plancherel on the line (taught grain) 10h

   Learning result State what the L1 Fourier transform does and does not licence for L2 on the line.

   Formative check Four-option check

7. 07 Synthesis 30h

   Learning result Combine a series or transform computation with a named convergence theorem.

   Formative check Written derivation

Learning results

State a Fourier inversion or convergence result with function-class hypotheses, and give a counterexample outside that class.

Subject skills

Fourier series, orthogonal expansions, Fourier transform at taught grain, convolution where licensed.

Transferable skills

Keep the function class next to the inversion formula.