Solvability of the Hankel determinant problem for real sequences
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Solvability of the Hankel determinant problem for real sequences. / Bakan, Andrew; Berg, Christian.
Frontiers In Orthogonal Polynomials and Q-series. ed. / M Zuhair Nashed; Xin Li. World Scientific, 2018. p. 85-117 (Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes, Vol. 1).Research output: Chapter in Book/Report/Conference proceeding › Book chapter › Research › peer-review
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TY - CHAP
T1 - Solvability of the Hankel determinant problem for real sequences
AU - Bakan, Andrew
AU - Berg, Christian
PY - 2018
Y1 - 2018
N2 - To each nonzero sequence s := {sn}n≥0 of real numbers, we associate the Hankel determinants Dn = detHn of the Hankel matrices Hn := (si+j)n i,j=0, n ≥ 0, and the nonempty set Ns := {n ≥ 1 Dn-1 ≠ 0}. We also define the Hankel determinant polynomials P0 := 1, and Pn, n ≥ 1 as the determinant of the Hankel matrix Hn modified by replacing the last row by the monomials 1, x, xn. Clearly Pn is a polynomial of degree at most n and of degree n if and only if n ε Ns. Kronecker established in 1881 that if Ns is finite then rank Hn = r for each n ≥ r - 1, where r := max Ns. By using an approach suggested by Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence {tn}n≥0 to be of the form tn = Dn, n ≥ 0 for a real sequence {sn}n≥0. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial Pn satisfying deg Pn = n ≥ 1 is preceded by a nonzero polynomial Pn-1 whose degree can be strictly less than n - 1 and which has no common zeros with Pn. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that D0 > 0, Dr-1 > 0 and Dn = 0 for all n ≥ r.
AB - To each nonzero sequence s := {sn}n≥0 of real numbers, we associate the Hankel determinants Dn = detHn of the Hankel matrices Hn := (si+j)n i,j=0, n ≥ 0, and the nonempty set Ns := {n ≥ 1 Dn-1 ≠ 0}. We also define the Hankel determinant polynomials P0 := 1, and Pn, n ≥ 1 as the determinant of the Hankel matrix Hn modified by replacing the last row by the monomials 1, x, xn. Clearly Pn is a polynomial of degree at most n and of degree n if and only if n ε Ns. Kronecker established in 1881 that if Ns is finite then rank Hn = r for each n ≥ r - 1, where r := max Ns. By using an approach suggested by Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence {tn}n≥0 to be of the form tn = Dn, n ≥ 0 for a real sequence {sn}n≥0. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial Pn satisfying deg Pn = n ≥ 1 is preceded by a nonzero polynomial Pn-1 whose degree can be strictly less than n - 1 and which has no common zeros with Pn. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that D0 > 0, Dr-1 > 0 and Dn = 0 for all n ≥ r.
KW - Frobenius rule
KW - Hankel matrices
KW - Kronecker theorem
KW - Orthogonal polynomials
UR - http://www.scopus.com/inward/record.url?scp=85045712122&partnerID=8YFLogxK
U2 - 10.1142/9789813228887_0005
DO - 10.1142/9789813228887_0005
M3 - Book chapter
AN - SCOPUS:85045712122
SN - 9789813228870
T3 - Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes
SP - 85
EP - 117
BT - Frontiers In Orthogonal Polynomials and Q-series
A2 - Nashed, M Zuhair
A2 - Li, Xin
PB - World Scientific
ER -
ID: 203599344