Theory of the approximation/Demonstrations

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A polynomial of $n$ degree which provides a function of error having $n+2$ extrema of the same size in absolute value, reaching them with a change of sign, is optimal.

Demonstration

Let us start by showing it on a graph. Let us pose $n=4$. Let us suppose that $P$ is a polynomial of $n$ degree having the properties above, in the sense that $P-f$ oscillates between $n+2$ extrema of alternated signs, of $+\; \backslash \; varepsilon$ to $-\; \backslash \; varepsilon$.

The function $P-f$ error could resemble the red graph:

$P-f$ reached $n+2$ extrema (of which two are at the ends), which has above the same size in absolute value located in 6 intervals on the graph.

Let us suppose now that $Q$, another polynomial of $n$ degree, is a strictly better approximation than $P$. That means that the extrema its function error must all have in absolute value a value strictly smaller than $\backslash \; varepsilon$, so that they are strictly localized inside the graph of error for $P$. The function error for $Q$ could have above a chart resembling the blue graph. That means that $(PF)\; -\; (Q-f)$ must oscillate between strictly positive and strictly negative nonnull numbers, a full number of $n+2$ time. But $(PF)\; -\; (Q-f)$ is equal to $P-Q$ which is a polynomial of $n$ degree. It must have at least the $n+1$ roots located between various points in which the function polynomial takes nonnull values. According to a consequence of the theorem of Alembert, it is impossible.

Category: Polynomial Category: Theory

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