Hausdorff-Young inequalities

Estimates of the Fourier coefficients of functions in ; established by W.H. Young [1] and F. Hausdorff [2]. Let be an orthonormal system of functions on , let for all and for all and let , . If , then

(1)

where are the Fourier coefficients of . If converges, there exists a function such that

(2)

For one may take , and this series converges in .

The Hausdorff–Young inequalities (1) and (2) are equivalent. For they do not hold. Moreover, if and if , then there exists a continuous function such that its Fourier coefficients in the trigonometric system satisfy the condition . A qualitative statement of the Hausdorff–Young inequality (if , , then ) 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 the series gives for all .