List of top Mathematics Questions on Orthogonal Polynomials

Consider the inner product space of all real-valued continuous functions defined on \( [-1, 1] \) with the inner product \[ \langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \, dx. \] If \( p(x) = \alpha + \beta x^2 - 30x^4 \), where \( \alpha, \beta \in \mathbb{R} \), is orthogonal to all the polynomials having degree less than or equal to 3, with respect to this inner product, then \( \alpha + 5\beta \) is equal to (in integer).

Let \( \{x_k\}_{k=1}^\infty \) be an orthonormal set of vectors in a real Hilbert space \( X \) with inner product \( \langle \cdot, \cdot \rangle \). Let \( n \in \mathbb{N} \), and let \( Y \) be the linear span of \( \{ x_k \}_{k=1}^n \) over \( \mathbb{R} \). For \( x \in X \), let \[ S_n(x) = \sum_{k=1}^n \langle x, x_k \rangle x_k. \] Then, which of the following is/are TRUE?

Consider the inner product space of all real-valued continuous functions defined on \( [-1, 1] \) with the inner product \[ \langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \, dx. \] If \( p(x) = \alpha + \beta x^2 - 30x^4 \), where \( \alpha, \beta \in \mathbb{R} \), is orthogonal to all the polynomials having degree less than or equal to 3, with respect to this inner product, then \( \alpha + 5\beta \) is equal to (in integer).