# What does aequidistant mean in mathematics

## Equidistant vs. Chebyshev polynomial interpolation

In the classical polynomial interpolation, the task is to find a function \$ f: [a, b] \ to \ mathbb {R} \$, from which function values ​​\$ f (t_i) \$ an \$ n + 1 \ in \ mathbb {N} \$ different in pairs Interpolation node \$ t_0, \ ldots, t_n \ in [a, b] \$ knows to approximate by a polynomial \$ P \$. So it should

\ [P (t_i) = f (t_i), \]

for all \$ i \ in {0, \ ldots, n} \$ hold (Interpolation property) and \$ \ Vert P - f \ Vert \$ “small” with respect to a suitable norm \$ \ Vert \ cdot \ Vert \$. Since a \$ n \$ -th degree polynomial is clearly defined by \$ n + 1 \$ support points, it makes sense to use precisely this polynomial for the interpolation of \$ f \$. In purely formal terms, \$ P \$ can be represented as follows

\ [P (t): = \ sum_ {i = 0} ^ nf (t_i) L_ {i, n} (t), \ quad \ text {where} \ quad L_ {i, n} (t): = \ prod _ {\ substack {j = 0 \ j \ neq i}} ^ n \ frac {t - t_j} {t_i - t_j}, \]

that \$ i \$ -th denotes Lagrange polynomial of degree \$ n \$. According to the definition, \$ L_ {i, n} (t_j) = \ delta_ {i, j} \$, which explains the interpolation property of \$ P \$.

As with any numerical problem, we must / should ask ourselves about the condition of polynomial interpolation. It turns out that the absolute condition number is equal to the so-called Lebesgue constants

\ [\ kappa _ {\ text {abs}} = \ Gamma_n: = \ max_ {t \ in [a, b]} \ sum_ {i = 0} ^ n | L_ {i, n} (t) |, \ ]

is. It is interesting here that \$ \ Gamma_n \$ increases exponentially in \$ n \$ for an equidistant selection of the support points \$ t_i \$. The degree of the interpolation polynomial \$ P \$ should not be too large for equidistant support point selection!

### Applet

In the applet below you can clearly see how the drastic deterioration in conditions with an equidistant selection of interpolation points affects the interpolation quality of \$ P \$ with increasing \$ n \$.