Symmetric second derivative ($D_s^{2}F(t)={\lim }_{h\to 0}[F(t+h)-2F(t)+F(t-h)]/h^2$)

The symmetric second derivative is the displayed limit when it exists. For a twice continuously differentiable function it equals the ordinary second derivative by Taylor expansion; the definition itself does not require ordinary differentiability. A continuous function with vanishing symmetric second derivative throughout an interval must be affine, by vanishing symmetric second derivative forces an affine function. For a twice-integrated Fourier series, sinc-squared weights express this quotient, connecting the operation to Riemann summation of a convergent trigonometric series.