Namespaces
Variants
Actions

Difference between revisions of "Riesz inequality"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822501.png" /> be an [[Orthonormal system|orthonormal system]] of functions on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822502.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822503.png" /> almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822504.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822505.png" />.
+
<!--
 +
r0822501.png
 +
$#A+1 = 23 n = 0
 +
$#C+1 = 23 : ~/encyclopedia/old_files/data/R082/R.0802250 Riesz inequality
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822507.png" />, then its Fourier coefficients with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822508.png" />,
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r0822509.png" /></td> </tr></table>
+
Let  $  \{ \phi _ {n} \} $
 +
be an [[Orthonormal system|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
 
satisfy the Riesz inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225010.png" /></td> </tr></table>
+
$$
 +
\| \{ 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
  
b) For any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225013.png" />, there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225015.png" /> as its Fourier coefficients and satisfying the Riesz inequality
+
$$
 +
\| f \| _ {q}  \leq  M  ^ {2/p-} 1 \| \{ c _ {n} \} \| _ {p} ,\ \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225017.png" /></td> </tr></table>
+
\frac{1}{p}
 +
+
 +
\frac{1}{q}
 +
= 1.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225019.png" />, then the [[Conjugate function|conjugate function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225020.png" /> and the Riesz inequality
+
If $  f \in L _ {p} [ 0, 2 \pi ] $,
 +
$  1 < p \leq  \infty $,  
 +
then the [[Conjugate function|conjugate function]] $  \overline{f}\; \in L _ {p} [ 0, 2 \pi ] $
 +
and the Riesz inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225021.png" /></td> </tr></table>
+
$$
 +
\| \overline{f}\; \| _ {p}  \leq  A _ {p} \| f \| _ {p}  $$
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225022.png" /> is a constant depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225023.png" />.
+
holds, where $  A _ {p} $
 +
is a constant depending only on $  p $.
  
 
Assertion 1) was for the first time proved by F. Riesz [[#References|[1]]]; particular cases of it were studied earlier by W.H. Young and F. Hausdorff. Assertion 2) was first proved by M. Riesz [[#References|[2]]].
 
Assertion 1) was for the first time proved by F. Riesz [[#References|[1]]]; particular cases of it were studied earlier by W.H. Young and F. Hausdorff. Assertion 2) was first proved by M. Riesz [[#References|[2]]].
Line 23: Line 67:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Riesz,  "Ueber eine Verallgemeinerung der Parsevalschen Formel"  ''Math. Z.'' , '''18'''  (1923)  pp. 117–124</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Riesz,  "Sur les fonctions conjuguées"  ''Math. Z.'' , '''27'''  (1927)  pp. 218–244</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Riesz,  "Ueber eine Verallgemeinerung der Parsevalschen Formel"  ''Math. Z.'' , '''18'''  (1923)  pp. 117–124</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Riesz,  "Sur les fonctions conjuguées"  ''Math. Z.'' , '''27'''  (1927)  pp. 218–244</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For 2) see also [[Interpolation of operators|Interpolation of operators]] (it is a consequence of the Marcinkiewicz interpolation theorem and the weak type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082250/r08225024.png" /> of the conjugation operator) and [[#References|[a3]]].
+
For 2) see also [[Interpolation of operators|Interpolation of operators]] (it is a consequence of the Marcinkiewicz interpolation theorem and the weak type $  ( 1, 1) $
 +
of the conjugation operator) and [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Butzer,  R.J. Nessel,  "Fourier analysis and approximation" , '''1''' , Birkhäuser  (1971)  pp. Chapt. 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Hausdorff,  "Eine Ausdehnung des Parsevalschen Satzes über Fourier-reihen"  ''Math. Z.'' , '''16'''  (1923)  pp. 163–169</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1975)  pp. Chapt. VI, §5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Butzer,  R.J. Nessel,  "Fourier analysis and approximation" , '''1''' , Birkhäuser  (1971)  pp. Chapt. 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Hausdorff,  "Eine Ausdehnung des Parsevalschen Satzes über Fourier-reihen"  ''Math. Z.'' , '''16'''  (1923)  pp. 163–169</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1975)  pp. Chapt. VI, §5</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


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

[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 $ ( 1, 1) $ 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
How to Cite This Entry:
Riesz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_inequality&oldid=48564
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article