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 $ X $ 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 $ X ^ {*} $. For example, when $ 1 \leq p < \infty $ and $ 1 / p + 1 / q = 1 $, there is an isometric isomorphism between the spaces $ ( L _ {p} [ a , b ] ) ^ {*} $ and $ L _ {q} [ a , b ] $, under which corresponding elements $ x ^ {*} $ and $ g $ are related by

$$ x ^ {*} ( f ) = \ \int\limits _ { a } ^ { b } g ( x) f ( x) d x . $$

If one considers some class $ X $ of $ 2 \pi $- periodic summable functions on $ [ - \pi , \pi ] $, then the conjugate class is defined to be the class of functions conjugate to the functions in $ X $. For example, the class conjugate to $ L _ {p} [ - \pi , \pi ] $( $ 1 < p < \infty $) coincides with the class of functions $ f $ in $ L _ {p} [ - \pi , \pi ] $ for which

$$ \int\limits _ {- \pi } ^ \pi f ( x) d x = 0 . $$

The class conjugate to $ \mathop{\rm Lip} \alpha $, $ 0 < \alpha < 1 $, coincides with the class of functions in $ \mathop{\rm Lip} \alpha $ for which $ \int _ {- \pi } ^ \pi f ( x) d x = 0 $.

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)