Distributional representations of Nκ(∞)‐functions. Issue 10 (12th February 2015)

• Record Type: Journal Article
• Authors: Langer, Matthias; Woracek, Harald
• Is Part Of: Mathematische Nachrichten. Volume 288:Issue 10(2015)
• Page Start: 1127
• Page End: 1149
• Publication Date: 2015-02-12
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201300280
• Languages: English
• ISSNs: 0025-584X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>The subclasses <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjdc5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:media:mana201300280:mana201300280-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi mathvariant="script">N</mml:mi><mml:mi>κ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>∞</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula> of the classes <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjd49" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:media:mana201300280:mana201300280-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="script">N</mml:mi><mml:mi>κ</mml:mi></mml:msub></mml:math></alternatives></inline-formula> of generalized Nevanlinna functions appear in the context of Pontryagin space models, where they correspond to model relations having a particular spectral behaviour. Applications are found, for instance, in the investigation of differential expressions with singular coefficients. We study representations of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjd3r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:media:mana201300280:mana201300280-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi mathvariant="script">N</mml:mi><mml:mi>κ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>∞</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>‐functions as Cauchy‐type integrals in a distributional sense and characterize the class of distributions occurring in such representations. We make explicit how the Pontryagin space model of an <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjd26" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:media:mana201300280:mana201300280-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi mathvariant="script">N</mml:mi><mml:mi>κ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>∞</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>‐function is related to the multiplication operator in the <italic>L</italic><sup>2</sup>‐space of the measure which describes the action of the representing distribution away from infinity. Moreover, we determine the distributional representations of a pair of functions associated with a symmetric generalized Nevanlinna function.</p> </abstract>