Question

Let \( T : L^2[-1, 1] \to L^2[-1, 1] \) be defined by \( T f = \tilde{f} \), where \( \tilde{f}(x) = f(-x) \) almost everywhere. If \( M \) is the kernel of \( I - T \), then the distance between the function \( \varphi(t) = e^t \) and \( M \) is

• A. \( \frac{1}{2} \sqrt{e^2 - e^{-2} + 4} \)
• B. \( \frac{1}{2} \sqrt{e^2 - e^{-2} - 2} \)
• C. \( \frac{1}{2} \sqrt{e^2 - 4} \)
• D. \( \frac{1}{2} \sqrt{e^2 - e^{-2} - 4} \)

Hint:

When calculating the distance between a function and a subspace in \( L^2 \), focus on the orthogonal projection of the function onto the subspace. In this case, we are projecting onto the space of even functions.

Answer 1

The Correct Option is D

The problem involves determining the distance between the function \( \varphi(t) = e^t \) and the kernel of the operator \( T \). The kernel of \( T \), \( M \), consists of functions that are unchanged by the operation of \( T \), meaning that these are even functions. The distance is determined by calculating the \( L^2 \)-norm of the difference between \( \varphi(t) \) and the closest even function. Using the given properties of \( T \) and the function \( \varphi(t) \), we can compute the distance as follows: \[ \frac{1}{2} \sqrt{e^2 - e^{-2} - 4}. \] Thus, the correct answer is (D).