Kolmapäev, veebruar 26, 2025

Aja viide. Aja lugu.

Mart Helme

In mathematics, the secondary polynomials $\{q_{n}(x)\}$ associated with a sequence $\{p_{n}(x)\}$ of polynomials orthogonal with respect to a density $\rho (x)$ are defined by^[1]

q_{n}(x)=\int _{\mathbb {R} }\!{\frac {p_{n}(t)-p_{n}(x)}{t-x}}\rho (t)\,dt.

^[2]

To see that the functions $q_{n}(x)$ are indeed polynomials, consider the simple example of $p_{0}(x)=x^{3}.$ Then,

{\begin{aligned}q_{0}(x)&{}=\int _{\mathbb {R} }\!{\frac {t^{3}-x^{3}}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!{\frac {(t-x)(t^{2}+tx+x^{2})}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!(t^{2}+tx+x^{2})\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!t^{2}\rho (t)\,dt+x\int _{\mathbb {R} }\!t\rho (t)\,dt+x^{2}\int _{\mathbb {R} }\!\rho (t)\,dt\end{aligned}}

which is a polynomial $x$ provided that the three integrals in $t$ (the moments of the density $\rho$ ) are convergent.

See also

Secondary measure

References

^ https://lexique.netmath.ca/en/second-degree-polynomial-function/#:~:text=Polynomial%20function%20whose%20general%20form,is%20called%20a%20quadratic%20function. {{cite web}}: Missing or empty |title= (help)
^ Groux, Roland (2007-09-12). "Sur une mesure rendant orthogonaux les polynômes secondaires [About a measure making secondary polynomials orthogonal]" (PDF). Comptes Rendus Mathematique (in French). 345 (7): 1 – via Comptes Rendus Mathematique.

Allikas: https://en.wikipedia.org/wiki/Secondary_polynomials

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