Theorem of Chudnovsky

The theorem of Chudnovsky is a theorem which shows under certain conditions that a continuous function is limiting uniform of functions polynomials to whole coefficients . It is a refinement of the Théorème of Stone-Weierstrass.

Statement

That is to say $f: I \ to \ mathbb {R}$ a function continues definite on a segment $I =$ not containing entireties. Then there exists a continuation $(P_n) _ {N \ in \ mathbb {NR}}$ of polynomials to whole coefficients uniformly converging towards $f$ on $I$ .

Idea of the proof

We bring back if $\ subseteq] 0; 1$ . The first stage of the proof consists in showing modestly that the constant function $X \ in [has; B \ mapsto \ frac {1} {2}$ is limiting uniform of polynomials to whole coefficients. One can even clarify this continuation $(P_n) _ {N \ in \ mathbb {NR}}$ of polynomials by:

P_0 = X, P_ {n+1} = 2 (1 - P_n) P_n

In the second time, one widens this result with all the constant functions: indeed, the continuous applications of

\ mathbb {R}

provided with the uniform standard form an algebra on

\ mathbb {R}

which one notes

\ mathcal {C}

. The whole of the uniform limits of polynomials to whole coefficients

\ overline {\ mathbb {Z}}

one is closed containing all the constant functions towards a dyadic number.

But the dyadic numbers are dense in $\ mathbb {R}$ , therefore $\ overline {\ mathbb {Z}}$ contains all the constant functions. But it is also an algebra which contains $X$ and $\ mathbb {R}$ , it thus contains $\ mathbb {R}$ , and a fortiori $\ overline {\ mathbb {R}}$ . However the theorem of Stone-Weierstrass ensures us that $\ overline {\ mathbb {R}} = \ mathcal {C}$ .

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