Abstract
This paper presents a criterion for uniqueness of a critical point inH 2,R rational approximation of type (m, n), withm≥n-1. This criterion is differential-topological in nature, and turns out to be connected with corona equations and classical interpolation theory. We illustrate its use with three examples, namely best approximation of fixed type on small circles, a de Montessus de Ballore type theorem, and diagonal, approximation to the exponential function of large degree.
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Abbreviations
- T,U,V :
-
unit circle, open unit disk, complement in\(\bar C\) of the closed unit disk
- T r,U r,V r :
-
circle of radiusr, open disk of radiusr, complement in\(\bar C\) of the closed disk of radiusr (with centers at the origin)
- P n :
-
space of real polynomials of degree at mostn; regarding the coefficients as coordinates, we endowP n with the Euclidean topology ofR n+1
- M n :
-
monic real polynomials of degreen
- \(\bar {\mathcal{M}}_n \) :
-
real polynomials of degree at mostn with constant coefficient equal to 1
- \(\mathcal{M}_n^r \) :
-
monic real polynomials of degreen having all their roots inU r
- \(\bar {\mathcal{M}}_n^r \) :
-
real polynomials of degree at mostn with constant coefficient equal to 1 having all their roots inV r
- Δ n :
-
real monic polynomials of degreen having all their roots in\(\bar U\) alternatively closure ofM 1 n with respect to the Euclidean topology ofP n
- \(\tilde \Delta _n \) :
-
real polynomials of degree at mostn with constant coefficient equal to 1 having all their roots in\(\bar V\); alternatively, closure of\(\widetilde{\mathcal{M}}_n^1 \) with respect to the Euclidean topology ofP n
- ‖·‖∞,‖·‖2:
-
norms inL ∞(T) and inL 2(T), respectively
- <·,·>:
-
scalar product inL 2(T)
- L 2,R (T):
-
real subspace ofL 2(T) consisting of functions with real Fourier coefficients
- H 2,R (U):
-
real Hardy space of exponent 2 of the unit disk consisting of functions inL 2,R(T) whose Fourier coefficients with negative index vanish
- H 02,R (V):
-
real Hardy space of exponent 2 of the complement of the closed unit disk restricted to those functions vanishing at infinity; alternatively, orthogonal complement ofH 2,R (U) inL 2,R (T)
- P +,P − :
-
orthogonal projectionsL 2,R (T)→H 2,R (U) andL 2,R (T)→H 02,R (V), respectively
- H ∞,R (U):
-
real subspace ofH 2,R (U) consisting of essentially bounded functions
- \(\mathcal{R}_{m,n}^0 \left( V \right)\) :
-
subset ofH 02,R (V) consisting of rational functionsp/z m-n+1 q withp ∈P m andq ∈M 1 n
- \(\widetilde{\mathcal{R}}_{m,n}^0 \left( U \right)\) :
-
subset ofH 2,R (U) consisting of rational functions\(\tilde p/\tilde q\) with\(\tilde p \in \mathcal{P}_m \) and\(\tilde q \in \widetilde{\mathcal{M}}_n^1 \)
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The research of this author was supported, in part, by NSF-INRIA cooperative research grant INT-9417234 as well as NSF grant DMS-9501130.
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Baratchart, L., Saff, E.B. & Wielonsky, F. A criterion for uniqueness of a critical point inH 2 rational approximation. J. Anal. Math. 70, 225–266 (1996). https://doi.org/10.1007/BF02820445
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DOI: https://doi.org/10.1007/BF02820445