Conjugate class of functions
A concept in the theory of functions which is a concrete instance of duality in functions spaces. Thus, if a class of functions is regarded as a Banach space or a topological vector space, then the conjugate class is defined as the class of functions isometrically isomorphic to the dual space
. For example, when
and
, there is an isometric isomorphism between the spaces
and
, under which corresponding elements
and
are related by
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If one considers some class of
-periodic summable functions on
, then the conjugate class is defined to be the class of functions conjugate to the functions in
. For example, the class conjugate to
(
) coincides with the class of functions
in
for which
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The class conjugate to ,
, coincides with the class of functions in
for which
.
References
[1] | M. Fréchet, C.R. Acad. Sci. , 144 (1907) pp. 1414–1416 |
[2] | F. Riesz, C.R. Acad. Sci. , 144 (1907) pp. 1409–1411 |
[3] | I. [I. Privalov] Priwaloff, Bull. Soc. Math. France , 44 (1916) pp. 100–103 |
[4] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |
[5] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |
Conjugate class of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_class_of_functions&oldid=46467