Riesz inequality

Let $ \\{ \phi _ {n} \\} $ be an orthonormal system of functions on an interval $ [ a, b] $ and let $ | \phi _ {n} | \leq M $ almost everywhere on $ [ a, b] $ for any $ n $.

a) If $ f \in L _ {p} [ a, b] $, $ 1 < p \leq 2 $, then its Fourier coefficients with respect to $ \\{ \phi _ {n} \\} $,

$$ c _ {n} = \int\limits _ { a } ^ { b } f \overline \phi \; _ {n} dx $$

satisfy the Riesz inequality

$$ \| \\{ c _ {n} \\} \| _ {q} \leq M ^ {2/p-} 1 \| f \| _ {p} ,\ \ \frac{1}{p} \+ \frac{1}{q} = 1. $$

b) For any sequence $ \\{ c _ {n} \\} $ with $ \| \\{ c _ {n} \\} \| _ {p} < \infty $, $ 1 < p \leq 2 $, there exists a function $ f \in L _ {q} [ a, b] $ with $ c _ {n} $ as its Fourier coefficients and satisfying the Riesz inequality

$$ \| f \| _ {q} \leq M ^ {2/p-} 1 \| \\{ c _ {n} \\} \| _ {p} ,\ \ \frac{1}{p} \+ \frac{1}{q} = 1. $$

If $ f \in L _ {p} [ 0, 2 \pi ] $, $ 1 < p \leq \infty $, then the conjugate function $ \overline{f}\; \in L _ {p} [ 0, 2 \pi ] $ and the Riesz inequality

$$ \| \overline{f}\; \| _ {p} \leq A _ {p} \| f \| _ {p} $$

holds, where $ A _ {p} $ is a constant depending only on $ p $.

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[edit]

[1] |  F. Riesz, "Ueber eine Verallgemeinerung der Parsevalschen Formel" Math. Z. , 18 (1923) pp. 117–124

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[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[edit]

For 2) see also Interpolation of operators (it is a consequence of the Marcinkiewicz interpolation theorem and the weak type $ ( 1, 1) $ of the conjugation operator) and [a3].

#### References[edit]

[a1] |  P.L. Butzer, R.J. Nessel, "Fourier analysis and approximation" , 1 , Birkhäuser (1971) pp. Chapt. 8

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[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