StudyBrixFree worksheets, study guides and courses

Mathematics · lesson 3 of 5

Fourier coefficients and convergence

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

Advanced university8 practice questionsNotes + worked examples
Take the quizStudy guideRevision sheet

Learn the ideas

Prerequisites: Integration, trigonometry and function spaces.

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

Example 1

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

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

3

Orthogonality removes the cosine term and all other sine modes.

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

Show answer and explanation

8

The constant term a₀/2 must equal 4.

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

Show answer and explanation

4

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

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

Show answer and explanation

0

Distinct integer sine modes are orthogonal.

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

Sample answer and marking guidance

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.

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

Show answer and explanation

a₀ and all cosine coefficients

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

7 Distinguish convergence in mean square from convergence at every point.

Sample answer and marking guidance

Mean-square convergence controls the integral of squared error; isolated points or small sets can behave differently without violating that integral bound.

Credit an integral norm versus pointwise control.

8 If f=2cos x−7sin(4x), what is b₄?

Show answer and explanation

-7

Read the coefficient using the specified convention.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.