14.11. Practice Exercises

Try these first

1. 1.

   What is the Fourier series of f(x) = x on (-π, π)?

   Hint

   Remember the function's symmetry when integrating.

2. 2.

   Verify that a₀ = 0 for f(x) = x.

   Hint

   Consider the properties of odd functions.

3. 3.

   What does Parseval's theorem state?

   • Energy in frequency is less than time.
   • Energy in time domain equals energy in frequency domain.
   • Energy is only in the time domain.

   Hint

   Think about how energy is represented mathematically.

4. 4.

   Does Parseval's theorem apply to non-periodic functions?

   • True
   • False

   Hint

   Relate the concept to Fourier transforms.

5. 5.

   Prove Parseval's theorem for a function f(x) defined on [0, L] using the Fourier series.

   Hint

   Transition to the periodic domain carefully.

6. 6.

   Extend Parseval’s theorem to demonstrate its validity for a sawtooth function using its Fourier series coefficients.

   Hint

   Consider the transformation properties.