Fourier Analysis

模块目标

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

课程大纲

1. 01 Orthogonal expansions 20h

   学习结果 Derive coefficients for an orthogonal system in an inner-product space.

   形成性检查 书面推导

2. 02 Fourier series: pointwise and L² 30h

   学习结果 State a convergence theorem and a function that fails a stronger naive claim.

   形成性检查 反例

3. 03 Convolution and approximate identities 20h

   学习结果 Use convolution with an approximate identity at synopsis grain.

   形成性检查 书面推导

4. 04 Fourier transform 30h

   学习结果 Compute a transform of a standard function and state inversion hypotheses.

   形成性检查 书面推导

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

   学习结果 Apply a Parseval identity for Fourier coefficients on the circle, with completeness named as a pending extra.

   形成性检查 四选一检查

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

   学习结果 State what the L1 Fourier transform does and does not licence for L2 on the line.

   形成性检查 四选一检查

7. 07 Synthesis 30h

   学习结果 Combine a series or transform computation with a named convergence theorem.

   形成性检查 书面推导

学习结果

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

学科技能

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

可迁移技能

Keep the function class next to the inversion formula.