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 $.