# Fourier coefficients and convergence

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

Course: [Advanced mathematical methods](https://studybrix.com/en/courses/advanced-mathematical-methods/index.md)
Level: advanced
Prerequisites: Integration, trigonometry and function spaces.

## 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.

## Practice questions

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

Answer: 3

Explanation: 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+... .

Answer: 8

Explanation: 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?

A. 6
B. 8
C. 2
D. 4

Answer: 4

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

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

Answer: 0

Explanation: Distinct integer sine modes are orthogonal.

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

Sample answer / marking guidance: Every finite sum of sines and cosines is continuous. A discontinuous target cannot be identical to a continuous function at every point.

Explanation: Credit continuity of finite sums and the contradiction.

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

A. Every coefficient necessarily
B. Only b₁
C. a₀ and all cosine coefficients
D. All sine coefficients

Answer: a₀ and all cosine coefficients

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

Explanation: Credit an integral norm versus pointwise control.

### 8. If f=2cos x−7sin(4x), what is b₄?

Answer: -7

Explanation: Read the coefficient using the specified convention.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.