Hausdorff-Young inequalities

Estimates of the Fourier coefficients of functions in $ L _ {p} $; established by W.H. Young [1] and F. Hausdorff [2]. Let $ \phi _ {n} $ be an orthonormal system of functions on $ [ a, b] $, let $ | \phi _ {n} ( t) | \leq M $ for all $ t \in [ a, b] $ and for all $ n = 1, 2 \dots $ and let $ 1 < p \leq 2 $, $ 1/p + 1/p ^ \prime = 1 $. If $ f \in L _ {p} $, then

$$ \tag{1 } \left ( \sum _ {n = 1 } ^ \infty | c _ {n} ( f ) | ^ {p ^ \prime } \right ) ^ {1/p ^ \prime } \leq \ M ^ {( 2 - p) / p } \left ( \int\limits _ { a } ^ { b } | f ( t) | ^ {p} dt \right ) ^ {1/p} , $$

where $ c _ {n} ( f ) $ are the Fourier coefficients of $ f $. If $ \sum _ {n = 1 } ^ \infty | a _ {n} | ^ {p} $ converges, there exists a function $ g $ such that

$$ \tag{2 } \left ( \int\limits _ { a } ^ { b } | g ( t) | ^ {p ^ \prime } dt \right ) ^ {1/p ^ \prime } \leq \ M ^ {( 2 - p) / p } \left ( \sum _ {n = 1 } ^ \infty | a _ {n} | ^ {p} \right ) ^ {1/p} . $$

For $ g $ one may take $ \sum _ {n = 1 } ^ \infty a _ {n} \phi _ {n} $, and this series converges in $ L _ {p ^ \prime } $.

The Hausdorff–Young inequalities (1) and (2) are equivalent. For $ p > 2 $ they do not hold. Moreover, if $ b _ {n} \geq 0 $ and if $ \sum _ {n = 1 } ^ \infty b _ {n} ^ {2} < \infty $, then there exists a continuous function $ f $ such that its Fourier coefficients $ c _ {n} ( f ) $ in the trigonometric system satisfy the condition $ | c _ {n} ( f ) | > b _ {n} $. A qualitative statement of the Hausdorff–Young inequality (if $ f \in L _ {p} $, $ 1 \leq p \leq 2 $, then $ \{ c _ {n} ( f ) \} \in l _ {p ^ \prime } $) for unbounded orthonormal systems of functions does not hold, in general. An analogue of the Hausdorff–Young inequalities is valid for a broad class of function spaces.

References

[1]	W.H. Young, "On the determination of the summability of a function by means of its Fourier constants" Proc. London Math. Soc. (2) , 12 (1913) pp. 71–88
[2]	F. Hausdorff, "Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen" Math. Z. , 16 (1923) pp. 163–169
[3]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[4]	S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[5]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[6]	K. de Leeuw, J.P. Kahane, Y. Katznelson, "Sur les coefficients de Fourier des fonctions continues" C.R. Acad. Sci. Paris , 285 (1977) pp. 1001–1003
[7]	S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)

Comments

Taking for $ g $ the series $ \sum _ {1} ^ \infty a _ {n} \phi _ {n} $ gives $ a _ {n} = c _ {n} ( g) $ for all $ n \geq 1 $.