Neumannegyenlet

Neumann’s inequality, also known as the Neumann–Titchmarsh theorem or Neumann’s constant ratio law, is a fundamental result in the theory of orthogonal polynomials and special functions. It was first formulated by the German mathematician Carl Neumann in the late 19th century and later expanded upon by other mathematicians, including Edward Titchmarsh. The theorem provides a relationship between the coefficients of orthogonal polynomials and their asymptotic behavior, particularly in the context of Jacobi polynomials and related systems.

The inequality is closely tied to the study of orthogonal polynomials on the real line, which satisfy a three-term recurrence relation. Neumann’s work demonstrated that under certain conditions, the ratio of consecutive coefficients of these polynomials tends to a constant as the degree increases. This constant is now referred to as Neumann’s constant, denoted by *ρ*. The theorem states that if a sequence of orthogonal polynomials {Pₙ(x)} satisfies specific growth and orthogonality conditions, then the limit

lim (aₙ₊₁ / aₙ) = ρ

exists, where *aₙ* represents the leading coefficient of the nth polynomial. This result is particularly useful in analyzing the asymptotic properties of orthogonal polynomials and their applications in approximation theory, numerical analysis, and mathematical physics.

Neumann’s inequality has broader implications in the study of orthogonal expansions and integral transforms. It ensures the stability and convergence of certain series expansions, which are essential in solving differential equations and other problems in mathematical analysis. The theorem also connects to the theory of moment problems and the distribution of zeros of orthogonal polynomials, further solidifying its importance in classical and modern analysis.

While Neumann’s original work focused on specific cases, later generalizations extended the theorem to more abstract settings, including orthogonal polynomials on the unit circle and in higher dimensions. These developments have contributed to the deeper understanding of functional analysis and harmonic analysis, reinforcing Neumann’s inequality as a cornerstone of mathematical research.