# 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 $.

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