27.12. Inner Product and Orthogonality in Function Spaces

Try these first

1. 1.

   Define the inner product of two functions.

   Hint

   Think about the integral of the product of the functions over the given interval.

2. 2.

   What does orthogonality mean for two functions?

   Hint

   Recall the definition of inner product in relation to zero.

3. 3.

   What is the definition of the inner product of two functions?

   Hint

   Recap the definition we talked about.

4. 4.

   Are sin(x) and cos(x) orthogonal over the interval [0, 2π]?

   • True
   • False

   Hint

   Think about their integral product.

5. 5.

   Prove that the functions f(x) = x and g(x) = x^2 are not orthogonal over the interval [-1, 1].

   Hint

   Remember to use the odd property of the function x^3 over symmetric limits.

6. 6.

   Construct the Fourier series for a square wave using orthogonal sine functions over the interval [0, T].

   Hint

   Think about how coefficients are calculated using the integral of the square wave function multiplied by the sine functions.