Deltasquared

Deltasquared, often denoted Δ^2 or spoken as "delta squared," is a concept in discrete mathematics and numerical analysis referring to the second finite difference operator applied to a real-valued function defined on a discrete domain. It captures the change in the rate of change of the function, effectively measuring discrete curvature.

Definition: For a function f defined on the integers, the first difference is Δf(n) = f(n+1) − f(n). The second difference is Δ^2 f(n) = Δ(Δ f(n)) = f(n+2) − 2 f(n+1) + f(n). With uniform spacing h, Δ_h^2 f(n) = f(n+2) − 2 f(n+1) + f(n) and Δ_h f(n) = f(n+1) − f(n). This operator relates to the second derivative via Δ_h^2 f(n) ≈ h^2 f''(nh) for smooth f.

Properties and uses: Δ^2 is linear, since Δ is linear. It is a discrete analogue of the continuous second derivative and is central to finite difference methods for approximating differential equations and for estimating curvature, convexity, and acceleration in time series. In practice, Δ^2 f(n) is used in numerical differentiation, interpolation, and in constructing quadratic splines or three-point stencils.

See also: finite difference method; discrete second derivative; binomial expansion of Δ; note that Δ^2 is not the same as the square of a Dirac delta distribution, which is not a well-defined object in distribution theory.