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Mathematics · Advanced university

Fourier coefficients and convergence

On [−π,π], distinct integer sine or cosine modes are orthogonal under integration.

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Prerequisites

Integration, trigonometry and function spaces.

Study notes

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.

Worked examples

For f(x)=3sin(2x)+5cos x on [−π,π], find b₂ in the stated real Fourier convention.

Orthogonality removes the cosine term and all other sine modes.

3

For the constant function f(x)=4, find a₀ in the convention f=a₀/2+... .

The constant term a₀/2 must equal 4.

8

Check your understanding

  1. At a jump from left limit 2 to right limit 6, what is the usual Fourier-series limit under piecewise-smooth hypotheses?

    1. 6
    2. 8
    3. 2
    4. 4

  2. What is the integral of sin(2x)sin(3x) from −π to π?

  3. Explain why a finite Fourier sum cannot equal a discontinuous function pointwise everywhere.

  4. For a real odd function on [−π,π], which coefficients vanish?

    1. Every coefficient necessarily
    2. Only b₁
    3. a₀ and all cosine coefficients
    4. All sine coefficients

Answers and marking guidance

Answers and marking notes
  1. 4

    The limit is the midpoint of the one-sided limits.

  2. 0

    Distinct integer sine modes are orthogonal.

  3. Every finite sum of sines and cosines is continuous. A discontinuous target cannot be identical to a continuous function at every point.

    Credit continuity of finite sums and the contradiction.

  4. a₀ and all cosine coefficients

    Odd integrands have zero symmetric integrals for the constant and cosine products.

Further reading and review references

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