14.2. Statement of Parseval’s Theorem

Try these first

1. 1.

   What does Parseval’s Theorem relate in the context of a periodic function?

   Hint

   Think about how energy is defined mathematically.

2. 2.

   Define what a square integrable function is.

   Hint

   Focus on the integral of the function squared.

3. 3.

   What does Parseval’s Theorem equate in terms of a periodic function?

   • A) The function's value at a specific point.
   • B) The total energy in time domain and the sum of squares of Fourier coefficients.
   • C) The rate of change of the function.

   Hint

   Consider energy in both domains.

4. 4.

   True or False: A function must be square integrable for Parseval’s Theorem to hold.

   • True
   • False

   Hint

   Think about what it means for an integral to be finite.

5. 5.

   A periodic function has Fourier coefficients a_0 = 2, a_1 = 3, and b_1 = 4. Using Parseval’s theorem, calculate the total energy over one period.

   Hint

   Be careful with the coefficients you sum.

6. 6.

   Show that for a square wave function with coefficients derived, Parseval's theorem holds true by calculating energy both ways (directly from f(x) and using coefficients).

   Hint

   Refer to the properties of the square wave and its Fourier coefficients.