Finite Difference - Calculus of Finite Differences

Calculus of Finite Differences

The forward difference can be considered as a difference operator, which maps the function f to Δ_h. This operator amounts to

where T_h is the shift operator with step h, defined by, and I is the identity operator.

The finite difference of higher orders can be defined in recursive manner as or, in operator notation, Another equivalent definition is

The difference operator Δ_h is a linear operator and it satisfies a special Leibniz rule indicated above, Δ_h(f(x)g(x)) = (Δ_hf(x)) g(x+h) + f(x) (Δ_hg(x)). Similar statements hold for the backward and central differences.

Formally applying the Taylor series with respect to h, yields the formula

where D denotes the continuum derivative operator, mapping f to its derivative f'. The expansion is valid when both sides act on analytic functions, for sufficiently small h. Thus, T_h=eD, and formally inverting the exponential yields

This formula holds in the sense that both operators give the same result when applied to a polynomial.

Even for analytic functions, the series on the right is not guaranteed to converge; it may be an asymptotic series. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to f’(x) mentioned at the end of the section Higher-order differences.

The analogous formulas for the backward and central difference operators are

The calculus of finite differences is related to the umbral calculus of combinatorics. This remarkably systematic correspondence is due to the identity of the commutators of the umbral quantities to their continuum analogs (h→0 limits),

A large number of formal differential relations of standard calculus involving functions f(x) thus map systematically to umbral finite-difference analogs involving f(xT_h−1).

For instance, the umbral analog of a monomial xn is a generalization of the above falling factorial (Pochhammer k-symbol), so that

hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function f(x) in such symbols), and so on.

For example, the umbral sine is

As in the continuum limit, the eigenfunction of also happens to be an exponential,

and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials. This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols.

Thus, for instance, the Dirac delta function maps to its umbral correspondent, the cardinal sine function,

and so forth. Difference equations can often be solved with techniques very similar to those for solving differential equations.

The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator.