Abstract
In classical analysis, functional transforms make it possible to investigate problems from a different perspective and sometimes simplify their analysis. In convex analysis, the most suitable notion of a transform is the Legendre transform, which maps a function to its (Fenchel) conjugate. This transform is studied in detail in this chapter. In particular, it is shown that the conjugate of an infimal convolution is the sum of the conjugates. The key result of this chapter is the Fenchel–Moreau theorem, which states that the proper convex lower semicontinuous functions are precisely those functions that coincide with their biconjugates.
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Bauschke, H.H., Combettes, P.L. (2011). Conjugation. In: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9467-7_13
Download citation
DOI: https://doi.org/10.1007/978-1-4419-9467-7_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9466-0
Online ISBN: 978-1-4419-9467-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)