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

DOI:
https://doi.org/10.1090/S0002-9947-09-04698-4

Published electronically:
April 17, 2009

MathSciNet review:
2529920

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-09-04698-4

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.