27.16. Bessel’s Inequality and Parseval’s Identity

Try these first

1. 1.

   What does Bessel's Inequality express?

   Hint

   Think about how vectors relate to their components.

2. 2.

   In which contexts might Parseval's Identity be applied?

   Hint

   Consider where decomposing functions is common.

3. 3.

   What does Bessel's Inequality state regarding an orthonormal set and a vector?

   • The total length can exceed the vector's norm.
   • The sum of inner product squares is less than or equal to the vector's norm.
   • The sum must equal the vector's length.

   Hint

   Think about the relationship of projections onto the basis.

4. 4.

   True or False: Parseval's Identity applies only to finite orthonormal sets.

   • True
   • False

   Hint

   Consider completeness in the context of series.

5. 5.

   Let v = (3, 4, 0) in R^3. Verify Bessel's Inequality with the orthonormal set { (1, 0, 0), (0, 1, 0), (0, 0, 1) }.

   Hint

   Calculate inner products step by step before summing.

6. 6.

   In the context of Fourier Series, express a square wave function using Parseval's Identity. How does it relate to energy conservation?

   Hint

   Remember the relationship between harmonic components and energy.