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A restriction on the behaviour of increase of a function in an integral metric. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597101.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597103.png" /> satisfies the Lipschitz integral condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597104.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597105.png" /> with constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597106.png" /> if
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<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/l/l059/l059710/l0597107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597108.png" />. In this case one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l0597109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971012.png" />. For the case of a periodic function (with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971013.png" />) the Lipschitz integral condition is defined similarly, only in inequality (*) the upper limit of integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971014.png" /> must be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059710/l05971015.png" />.
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A restriction on the behaviour of increase of a function in an integral metric. A function  $  f $
 +
in a space  $  L _ {p} ( a , b ) $
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with  $  p \geq  1 $
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satisfies the Lipschitz integral condition of order  $  \alpha > 0 $
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on  $  [ a, b ] $
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with constant  $  M > 0 $
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if
 +
 
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$$ \tag{* }
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\left \{
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\int\limits _ { a } ^ { b- }  h
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| f ( x + h ) - f ( x) |  ^ {p} \
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d x
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\right \}  ^ {1/p}  \leq  M h  ^  \alpha
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$$
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for all $  h \in ( 0 , b - a ) $.  
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In this case one writes $  f \in  \mathop{\rm Lip} _ {M} ( \alpha , p ) $,  
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$  f \in H _ {p}  ^  \alpha  ( M) $
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or $  f \in  \mathop{\rm Lip} ( \alpha , p ) $,  
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$  f \in H _ {p}  ^  \alpha  $.  
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For the case of a periodic function (with period $  b - a $)  
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the Lipschitz integral condition is defined similarly, only in inequality (*) the upper limit of integration $  b - h $
 +
must be replaced by $  b $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.V. Besov,  V.P. Il'in,  S.M. Nikol'skii,  "Integral representations of functions and imbedding theorems" , Wiley  (1978)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.V. Besov,  V.P. Il'in,  S.M. Nikol'skii,  "Integral representations of functions and imbedding theorems" , Wiley  (1978)  (Translated from Russian)</TD></TR></table>

Latest revision as of 22:17, 5 June 2020


A restriction on the behaviour of increase of a function in an integral metric. A function $ f $ in a space $ L _ {p} ( a , b ) $ with $ p \geq 1 $ satisfies the Lipschitz integral condition of order $ \alpha > 0 $ on $ [ a, b ] $ with constant $ M > 0 $ if

$$ \tag{* } \left \{ \int\limits _ { a } ^ { b- } h | f ( x + h ) - f ( x) | ^ {p} \ d x \right \} ^ {1/p} \leq M h ^ \alpha $$

for all $ h \in ( 0 , b - a ) $. In this case one writes $ f \in \mathop{\rm Lip} _ {M} ( \alpha , p ) $, $ f \in H _ {p} ^ \alpha ( M) $ or $ f \in \mathop{\rm Lip} ( \alpha , p ) $, $ f \in H _ {p} ^ \alpha $. For the case of a periodic function (with period $ b - a $) the Lipschitz integral condition is defined similarly, only in inequality (*) the upper limit of integration $ b - h $ must be replaced by $ b $.

References

[1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3] O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)
How to Cite This Entry:
Lipschitz integral condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_integral_condition&oldid=47671
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article