Analytic continuation, singular-value expansions, and Kramers-Kronig analysis

A. Dienstfrey, Leslie Greengard

Research output: Contribution to journalArticle

Abstract

We describe a systematic approach to the recovery of a function analytic in the upper half-plane, C+, from measurements over a finite interval on the real axis, D ⊂ R. Analytic continuation problems of this type are well known to be ill-posed. Thus, the best one can hope for is a simple, linear procedure which exposes this underlying difficulty and solves the problem in a least-squares sense. To accomplish this, we first construct an explicit analytic approximation of the desired function and recast the continuation problem in terms of a 'residual function' defined on the measurement window D itself. The resulting procedure is robust in the presence of noise, and we demonstrate its performance with some numerical experiments.

Original languageEnglish (US)
Pages (from-to)1307-1320
Number of pages14
JournalInverse Problems
Volume17
Issue number5
DOIs
StatePublished - Oct 2001

Fingerprint

Analytic Continuation
Singular Values
half planes
analytic functions
expansion
recovery
intervals
Half-plane
Research and Development
Continuation
Least Squares
Analytic function
Recovery
approximation
Numerical Experiment
Interval
Approximation
Demonstrate
Experiments

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

Analytic continuation, singular-value expansions, and Kramers-Kronig analysis. / Dienstfrey, A.; Greengard, Leslie.

In: Inverse Problems, Vol. 17, No. 5, 10.2001, p. 1307-1320.

Research output: Contribution to journalArticle

@article{c9a6d2afdb254a9e9dd3fd840f704592,
title = "Analytic continuation, singular-value expansions, and Kramers-Kronig analysis",
abstract = "We describe a systematic approach to the recovery of a function analytic in the upper half-plane, C+, from measurements over a finite interval on the real axis, D ⊂ R. Analytic continuation problems of this type are well known to be ill-posed. Thus, the best one can hope for is a simple, linear procedure which exposes this underlying difficulty and solves the problem in a least-squares sense. To accomplish this, we first construct an explicit analytic approximation of the desired function and recast the continuation problem in terms of a 'residual function' defined on the measurement window D itself. The resulting procedure is robust in the presence of noise, and we demonstrate its performance with some numerical experiments.",
author = "A. Dienstfrey and Leslie Greengard",
year = "2001",
month = "10",
doi = "10.1088/0266-5611/17/5/305",
language = "English (US)",
volume = "17",
pages = "1307--1320",
journal = "Inverse Problems",
issn = "0266-5611",
publisher = "IOP Publishing Ltd.",
number = "5",

}

TY - JOUR

T1 - Analytic continuation, singular-value expansions, and Kramers-Kronig analysis

AU - Dienstfrey, A.

AU - Greengard, Leslie

PY - 2001/10

Y1 - 2001/10

N2 - We describe a systematic approach to the recovery of a function analytic in the upper half-plane, C+, from measurements over a finite interval on the real axis, D ⊂ R. Analytic continuation problems of this type are well known to be ill-posed. Thus, the best one can hope for is a simple, linear procedure which exposes this underlying difficulty and solves the problem in a least-squares sense. To accomplish this, we first construct an explicit analytic approximation of the desired function and recast the continuation problem in terms of a 'residual function' defined on the measurement window D itself. The resulting procedure is robust in the presence of noise, and we demonstrate its performance with some numerical experiments.

AB - We describe a systematic approach to the recovery of a function analytic in the upper half-plane, C+, from measurements over a finite interval on the real axis, D ⊂ R. Analytic continuation problems of this type are well known to be ill-posed. Thus, the best one can hope for is a simple, linear procedure which exposes this underlying difficulty and solves the problem in a least-squares sense. To accomplish this, we first construct an explicit analytic approximation of the desired function and recast the continuation problem in terms of a 'residual function' defined on the measurement window D itself. The resulting procedure is robust in the presence of noise, and we demonstrate its performance with some numerical experiments.

UR - http://www.scopus.com/inward/record.url?scp=0035472967&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035472967&partnerID=8YFLogxK

U2 - 10.1088/0266-5611/17/5/305

DO - 10.1088/0266-5611/17/5/305

M3 - Article

AN - SCOPUS:0035472967

VL - 17

SP - 1307

EP - 1320

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 5

ER -