Su un nuovo tipo di sviluppo di una funzione in serie di polinomi

Ghizzetti, Aldo ; Ossicini, Alessandro

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 43 (1967) no. 1-2, pp. 21-29

Résumé

We consider the sequence of polynomials $P_{s,n} (x)$ having the following property: in the interval [-1, 1] the polynomial $P_{s,n}^{2s+1}$ is orthogonal to polynomials $P_{s,k}$ with k n. Ossicini has already demonstrated the existence of such polynomials. Other properties of the $P_{s,n} (x)$ are given here. Then a series expansion of the type $$\sum_{n=0}^{\infty} c_{n} \, P_{s,n}^{2s+1}(x)$$ is associated to every $f(x)$, with certain values of the coefficients $c_{n}$ that are determined by the property of orthogonality mentioned above. The Bessel inequality and the Fischer-Riesz theorem are extended to such expansions.