Question

Let \( \{ \varphi_0, \varphi_1, \varphi_2, \dots \} \) be an orthonormal set in \( L^2[-1, 1] \) such that \( \varphi_n = C_n P_n \), where \( C_n \) is a constant and \( P_n \) is the Legendre polynomial of degree \( n \), for each \( n \in \mathbb{N} \setminus \{0\} \). Then, which of the following statements are TRUE?

• A. \( \varphi_6(1) = 1 \)
• B. \( \varphi_7(-1) = 1 \)
• C. \( \varphi_7(1) = \sqrt{\frac{15}{2}} \)
• D. \( \varphi_6(-1) = \sqrt{\frac{13}{2}} \)

Hint:

For Legendre polynomials, \( P_n(1) = 1 \) and \( P_n(-1) = (-1)^n \), which can help in evaluating the values of \( \varphi_n(1) \) and \( \varphi_n(-1) \).

Answer 1

The Correct Option is C, D

The functions \( \varphi_n \) are given as the product of a constant \( C_n \) and the Legendre polynomial \( P_n \). The values of \( \varphi_n(1) \) and \( \varphi_n(-1) \) are related to the values of \( P_n(1) \) and \( P_n(-1) \), which are known properties of Legendre polynomials. Specifically, \[ P_n(1) = 1 \quad \text{and} \quad P_n(-1) = (-1)^n. \] For \( n = 6 \) and \( n = 7 \), we have the following calculations: - For \( n = 7 \), \( \varphi_7(1) = C_7 \times P_7(1) = C_7 \times 1 \). From standard Legendre polynomial values, we know that \( P_7(1) = 1 \) and the constant \( C_7 \) is \( \sqrt{\frac{15}{2}} \), hence \( \varphi_7(1) = \sqrt{\frac{15}{2}} \). - For \( n = 6 \), \( \varphi_6(-1) = C_6 \times P_6(-1) = C_6 \times (-1)^6 = C_6 \). From standard Legendre polynomial values, we know that \( P_6(-1) = 1 \) and the constant \( C_6 \) is \( \sqrt{\frac{13}{2}} \), hence \( \varphi_6(-1) = \sqrt{\frac{13}{2}} \). Thus, the correct answers are (C) and (D).