Riesz inequality
Let be an orthonormal system of functions on an interval
and let
almost everywhere on
for any
.
a) If ,
, then its Fourier coefficients with respect to
,
![]() |
satisfy the Riesz inequality
![]() |
b) For any sequence with
,
, there exists a function
with
as its Fourier coefficients and satisfying the Riesz inequality
![]() |
If ,
, then the conjugate function
and the Riesz inequality
![]() |
holds, where is a constant depending only on
.
Assertion 1) was for the first time proved by F. Riesz [1]; particular cases of it were studied earlier by W.H. Young and F. Hausdorff. Assertion 2) was first proved by M. Riesz [2].
References
[1] | F. Riesz, "Ueber eine Verallgemeinerung der Parsevalschen Formel" Math. Z. , 18 (1923) pp. 117–124 |
[2] | M. Riesz, "Sur les fonctions conjuguées" Math. Z. , 27 (1927) pp. 218–244 |
[3] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |
[4] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |
Comments
For 2) see also Interpolation of operators (it is a consequence of the Marcinkiewicz interpolation theorem and the weak type of the conjugation operator) and [a3].
References
[a1] | P.L. Butzer, R.J. Nessel, "Fourier analysis and approximation" , 1 , Birkhäuser (1971) pp. Chapt. 8 |
[a2] | F. Hausdorff, "Eine Ausdehnung des Parsevalschen Satzes über Fourier-reihen" Math. Z. , 16 (1923) pp. 163–169 |
[a3] | E.M. Stein, G. Weiss, "Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1975) pp. Chapt. VI, §5 |
Riesz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_inequality&oldid=48564