The Adjoint of Differentiation

Abstract

Let n be any nonnegative integer. Let V = P_n be the vector space of polynomials of degree at most n, equipped with the inner product ⟨f, g⟩ = ∫¹₀f(x)g(x) dx. Let D: V → V be the differentiation operator, D(f) = f′. Then D has an adjoint D*. We have closed-form expressions for D*, which were conjectured by computing D* for small values of n and finding a pattern. (If f(x) is a polynomial of degree k ≤ n, then while the value of D(f(x)) is independent of n, the value of D*(f(x)) depends on n.) We also find formulas for D* in terms of classical Legendre polynomials, shifted to the interval [0, 1]. Using these formulas, it is easy to prove that our closed-form expressions are correct. An alternative approach yields combinatorial identities involving the entries of the inverses of Hilbert matrices.

Keywords::

• differentiation
• adjoint
• Legendre polynomials
• Hilbert matrix
• binomial coefficients
• Primary 47A05
• Secondary 33C45
• 15A04
• 15B99
• 05A19