Robert D. Bates | Searcharxiv

arXiv2015

Diagonal Differential Operators

We explore differential operators, $T$, that diagonalize on a simple basis, $\{B_n(x)\}_{n=0}^\infty$, with respect to some sequence of real numbers, $\{a_n\}_{n=0}^\infty$, and sequence of polynomials, $\{Q_k(x)\}_{k=0}^\infty$, as in $ T[B_n(x)]:=\left(\sum_{k=0}^\infty Q_k(x) D^k\right)B_n(x)=a_n B_n(x)$ for every $n\in\mathbb{N}_0$. We discover new relationships between the sequence, $\{Q_k(x)\}_{k=0}^\infty$, and the sequence, $\{a_n\}_{n=0}^\infty$. We find new relationships between polynomial interpolated eigenvalues and the sequence, $\{°(Q_k(x))\}_{k=0}^\infty$.

math.CV↗

arXiv2015

Operator Diagonalizations of Multiplier Sequences

We consider hyperbolicity preserving operators with respect to a new linear operator representation on $\mathbb{R}[x]$. In essence, we demonstrate that every Hermite and Laguerre multiplier sequence can be diagonalized into a sum of hyperbolicity preserving operators, where each of the summands forms a classical multiplier sequence. Interestingly, this does not work for other orthogonal bases; for example, this property fails for the Legendre basis. We establish many new formulas concerning the $Q_k$'s of Peetre's 1959 differential representation for linear operators in the specific case of Hermite and Laguerre diagonal differential operators. Additionally, we provide a new algebraic characterization of the Hermite multiplier sequences and also extend a recent result of T. Forgács and A. Piotrowski on hyperbolicity properties of the polynomial coefficients in hyperbolicity preserving Hermite diagonal differential operators.

math.CV↗