Question

Consider the function $f(x)$ for an interval $(-\pi,\pi)$ then which one of the following is NOT correct Dirichlet's condition for a Fourier series?

• A. The function $f(x)$ is single-valued.
• B. The function $f(x)$ is not bounded.
• C. The function $f(x)$ has at most a finite number of maxima and minima.
• D. The function $f(x)$ has finite number of discontinuities.

Hint:

Dirichlet conditions are "sufficient" but not "necessary." Most physical signals satisfy these because they don't have infinite energy or infinite oscillations.

Answer 1

The Correct Option is B

Concept: Dirichlet conditions are sufficient conditions for a real-valued, periodic function $f(x)$ to be represented by a convergent Fourier series. If a function satisfies these criteria, the Fourier series converges to the function at points of continuity.

Step 1: Review the standard Dirichlet conditions.
For a function to have a valid Fourier series expansion, it must satisfy: 1. $f(x)$ must be single-valued and periodic. 2. $f(x)$ must be absolutely integrable over a period, which implies the function must be bounded. 3. $f(x)$ must have a finite number of maxima and minima within a single period. 4. $f(x)$ must have a finite number of discontinuities within a single period, and these discontinuities must be finite.

Step 2: Identify the incorrect statement.
- Option (1), (3), and (4) are all standard requirements for Dirichlet conditions. - Option (2) states the function is "not bounded." This contradicts the requirement that the function must be absolutely integrable and have finite values. If a function is not bounded (e.g., it goes to infinity at a point), the Fourier coefficients may not exist or converge.

Step 3: Conclusion.
Since $f(x)$ must be bounded to satisfy Dirichlet's conditions, the statement that it is "not bounded" is the one that is NOT correct.