Mathematics · Advanced university
Fourier coefficients and convergence · revision notes
On [−π,π], distinct integer sine or cosine modes are orthogonal under integration.
Key ideas and conditions
Orthogonality
On [−π,π], distinct integer sine or cosine modes are orthogonal under integration.
Convention
For f=a₀/2+sum(a_n cos nx+b_n sin nx), a_n and b_n have the factor 1/π.
Convergence
Pointwise, uniform and mean-square convergence are different claims with different hypotheses.
Jumps
Under standard piecewise-smooth hypotheses the Fourier series at a jump converges to the average of one-sided limits.
Explain without looking
- Distinguish convergence in mean square from convergence at every point.
- If f=2cos x−7sin(4x), what is b₄?
Further reading and review references
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