On linear operators preserving the set of positive polynomials. (English) Zbl 1160.12002
Let \( \mathbb R[x]\) be the real space of univariate polynomials and denote by \(\mathbb R_n[x]\) the subspace consisting of polynomials of degree\(\leq n.\) A polynomial in \(\mathbb R[x]\) is called by the authors (only?) hyperbolic/elliptic, respectively, if all/none of its roots are real. It is positive/nonnegative, respectively, if for all \(x\in \mathbb R,\) \(p(x)>0/p(x)\geq 0.\)
G. Pólya and I. Schur [J. Reine Angew. Math 144, 89–113 (1914; JFM 45.0176.01)] studied which w.r.t. the monomial basis diagonal linear operators on \(\mathbb R[x]\) are hyperbolicity preservers. Related questions for diagonal operators were pursued by T. Craven and G. Csordas [Methods Appl. Anal. 2, No. 4, 420–441 (1995; Zbl 0853.30018)] and M. D. Kostova [C. R. Acad. Bulg. Sci. 36, 23–25 (1983; Zbl 0551.30022)].
In the present paper it is shown that a linear differential operator of finite order \(k\geq 1,\) \(U_Q=\sum_{i=0}^k q_i(x) \frac{d^i}{dx^i}\) with \(q_i \in \mathbb R[x]\), \(q_k\not\equiv 0,\) does not preserve any of the sets of nonnegative/positive/elliptic polynomials of degree \(2k;\) this contrasts to the 1913 Hurwitz-Remak result that an infinite order linear differential operators with constant coefficients \(\sum_{i=0}^\infty \alpha_i \frac{d^i}{dx^i}\) preserves positivity iff the infinite Hankel matrix \(((i+j)!\alpha_{i+j})_{i,j\geq 0}\) represents a positive semidefinite quadratic form iff there exists a politive measure \(\mu_\alpha\) whose moment sequence \(\int_{-\infty}^{\infty} t^k d\mu_\alpha(t) =k!\alpha_k,\) \(k=0,1,2,\dots\)
Another result is that a linear operator \(\Phi: \mathbb R[x] \rightarrow \mathbb R[x]\) preserves the set of elliptic polynomials iff either \(\Phi\) or \(-\Phi\) preserves the set of positive polynomials. If one requires \(\Phi(1)>0,\) then an analogue for \(\mathbb R_n[x]\) holds. Section 3 is devoted to diagonal operators preserving positivity, i.e. \(\lambda\)-sequences; in particular some not quite correct claims found in L. Iliev’s book [Laguerre entire functions. Sofia: Publishing House of the Bulgarian Academy of Sciences (1987; Zbl 0691.30001)] are corrected and illustrated with counterexamples. Sections 4,5 are devoted to proofs, refinements, and strengthenings of the mentioned non-preservation- and the Hurwitz-Remak- results.
G. Pólya and I. Schur [J. Reine Angew. Math 144, 89–113 (1914; JFM 45.0176.01)] studied which w.r.t. the monomial basis diagonal linear operators on \(\mathbb R[x]\) are hyperbolicity preservers. Related questions for diagonal operators were pursued by T. Craven and G. Csordas [Methods Appl. Anal. 2, No. 4, 420–441 (1995; Zbl 0853.30018)] and M. D. Kostova [C. R. Acad. Bulg. Sci. 36, 23–25 (1983; Zbl 0551.30022)].
In the present paper it is shown that a linear differential operator of finite order \(k\geq 1,\) \(U_Q=\sum_{i=0}^k q_i(x) \frac{d^i}{dx^i}\) with \(q_i \in \mathbb R[x]\), \(q_k\not\equiv 0,\) does not preserve any of the sets of nonnegative/positive/elliptic polynomials of degree \(2k;\) this contrasts to the 1913 Hurwitz-Remak result that an infinite order linear differential operators with constant coefficients \(\sum_{i=0}^\infty \alpha_i \frac{d^i}{dx^i}\) preserves positivity iff the infinite Hankel matrix \(((i+j)!\alpha_{i+j})_{i,j\geq 0}\) represents a positive semidefinite quadratic form iff there exists a politive measure \(\mu_\alpha\) whose moment sequence \(\int_{-\infty}^{\infty} t^k d\mu_\alpha(t) =k!\alpha_k,\) \(k=0,1,2,\dots\)
Another result is that a linear operator \(\Phi: \mathbb R[x] \rightarrow \mathbb R[x]\) preserves the set of elliptic polynomials iff either \(\Phi\) or \(-\Phi\) preserves the set of positive polynomials. If one requires \(\Phi(1)>0,\) then an analogue for \(\mathbb R_n[x]\) holds. Section 3 is devoted to diagonal operators preserving positivity, i.e. \(\lambda\)-sequences; in particular some not quite correct claims found in L. Iliev’s book [Laguerre entire functions. Sofia: Publishing House of the Bulgarian Academy of Sciences (1987; Zbl 0691.30001)] are corrected and illustrated with counterexamples. Sections 4,5 are devoted to proofs, refinements, and strengthenings of the mentioned non-preservation- and the Hurwitz-Remak- results.
Reviewer: Alexander Kovačec (Coimbra)
MSC:
| 12D99 | Real and complex fields |
| 12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
| 44A60 | Moment problems |
| 15A04 | Linear transformations, semilinear transformations |
