Bessel interpolation formula

The Bessel interpolation formula is one of the interpolation formulas with equidistant support points. With their help, functions can be represented as polynomials of the nth degree. n is determined from the ( n + 1 ) support points. It was named after Friedrich Wilhelm Bessel , its author.

Difference table

First you create a so-called difference table in which the interpolation points follow one another at equal intervals. This distance h is calculated according to . lies in the middle of the bases. The differences are now calculated as follows:, all others in the same way . ${\displaystyle x_{i}}$ ${\displaystyle h=x_{i+1}-x_{i}}$ ${\displaystyle x_{0}}$ ${\displaystyle \Delta f_{i}=f_{1+i}-f_{i}}$ ${\displaystyle \Delta ^{k}f_{i}=\Delta ^{k-1}f_{i+1}-\Delta ^{k-1}f_{i}}$

The formula

The polynomial φ is then calculated using the formula:

${\displaystyle \varphi =f_{0}+u\Delta f_{0}+{\frac {u(u-1)}{2}}\cdot {\frac {\Delta ^{2}f_{-1}+\Delta ^{2}f_{0}}{2}}+{\frac {u(u-1)(u-0,5)}{3!}}\cdot \Delta ^{3}f_{-1}}$ ${\displaystyle +{\frac {u(u^{2}-1)(u-2)}{4!}}\cdot {\frac {\Delta ^{4}f_{-2}+\Delta ^{4}f_{-1}}{2}}+...}$ ${\displaystyle ...+{\frac {(u-0,5)u(u^{2}-1)...(u^{2}-(n-1)^{2})(u-n)}{(2n+1)!}}\cdot \Delta ^{2n+1}f_{-1}}$

with . ${\displaystyle u={\frac {x-x_{0}}{h}}}$