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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The kernel average for two convex functions and its application to the extension and representation of monotone operators


Authors: Heinz H. Bauschke and Xianfu Wang
Journal: Trans. Amer. Math. Soc. 361 (2009), 5947-5965
MSC (2000): Primary 52A41, 47N10; Secondary 47H05
Published electronically: April 17, 2009
MathSciNet review: 2529920
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Abstract: We provide and analyze an average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also known as selfdual Lagrangian) which can be used for finding an explicit maximal monotone extension of the given monotone operator. This completely settles one of the open problems posed by Fitzpatrick in the setting of reflexive Banach spaces.


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Additional Information

Heinz H. Bauschke
Affiliation: Department of Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia, Canada V1V 1V7
Email: heinz.bauschke@ubc.ca

Xianfu Wang
Affiliation: Department of Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia, Canada V1V 1V7
Email: shawn.wang@ubc.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04698-4
PII: S 0002-9947(09)04698-4
Keywords: Arithmetic average, autoconjugate, convex function, epigraphical average, Fenchel conjugate, Fitzpatrick function, maximal monotone operator, monotone operator, proximal average, proximal mapping, selfdual Lagrangian
Received by editor(s): May 10, 2007
Received by editor(s) in revised form: October 2, 2007
Published electronically: April 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.