TY - JOUR

T1 - Analytic approximation of rational matrix functions

AU - Peller, V. V.

AU - Vasyunin, V. I.

PY - 2007/10/29

Y1 - 2007/10/29

N2 - For a rational matrix function Φ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation AΦ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of 2 × 2 matrix functions. It turns out that "genetically" degΦ ≤ deg Φ - 2. We prove that for an arbitrary 2×2 rational function Φ, deg AΦ <, 2 degΦ - 3 whenever degΦ > 2. On the other hand, for k ≥ 2, we construct a 2 × 2 matrix function Φ, for which degΦ = k, while deg AΦ = 2k-3. Moreover, we conduct a detailed analysis of the situation when the inequality deg AΦ≤ degΦ-2 can violate and obtain best possible results.

AB - For a rational matrix function Φ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation AΦ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of 2 × 2 matrix functions. It turns out that "genetically" degΦ ≤ deg Φ - 2. We prove that for an arbitrary 2×2 rational function Φ, deg AΦ <, 2 degΦ - 3 whenever degΦ > 2. On the other hand, for k ≥ 2, we construct a 2 × 2 matrix function Φ, for which degΦ = k, while deg AΦ = 2k-3. Moreover, we conduct a detailed analysis of the situation when the inequality deg AΦ≤ degΦ-2 can violate and obtain best possible results.

KW - Hankel operator

KW - McMillan degree

KW - Rational matrix function

KW - Superoptimal approximation

UR - http://www.scopus.com/inward/record.url?scp=35448952481&partnerID=8YFLogxK

U2 - 10.1512/iumj.2007.56.3075

DO - 10.1512/iumj.2007.56.3075

M3 - Article

AN - SCOPUS:35448952481

VL - 56

SP - 1913

EP - 1938

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 4

ER -