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One of the generalizations of the Lebesgue integral, proposed by A. Denjoy (1919) and studied in detail by T.J. Boks (1921). A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168001.png" /> on a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168002.png" /> is periodically extended (with period length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168003.png" />) to the entire straight line. For an arbitrary subdivision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168004.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168005.png" />, for an arbitrary selection of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168007.png" />, and an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b0168008.png" />, the following sum is constructed:
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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/b/b016/b016800/b0168009.png" /></td> </tr></table>
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If, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680011.png" /> converges in measure to a definite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680012.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680013.png" /> is said to be the Boks integral (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680015.png" />-integral) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680017.png" />. Thus, the Boks integral is an integral of Riemann type and is a generalization of the Riemann integral.
+
One of the generalizations of the Lebesgue integral, proposed by A. Denjoy (1919) and studied in detail by T.J. Boks (1921). A real-valued function  $  f $
 +
on a segment  $  [a, b] $
 +
is periodically extended (with period length  $  b - a $)  
 +
to the entire straight line. For an arbitrary subdivision  $  a = x _ {0} < x _ {1} < \dots < x _ {n} = b $
 +
of  $  [a, b] $,  
 +
for an arbitrary selection of points  $  \overline \xi \; = \{ \xi _ {i} \} _ {1}  ^ {n} $,
 +
$  \xi _ {i} \in [x _ {i-1 }  , x _ {i} ] $,
 +
and an arbitrary  $  t $,
 +
the following sum is constructed:
  
The Boks integral represents a considerable extension of the Lebesgue integral: Any summable function is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680018.png" />-integrable and these integrals coincide, but there exist non-summable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680019.png" />-integrable functions; in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680020.png" /> is the function conjugate with a summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680021.png" />, then it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680022.png" />-integrable and the coefficients of the series conjugate with the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680023.png" /> are the coefficients of the respective Fourier series (in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680024.png" />-integration) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680025.png" /> (A.N. Kolmogorov). The theory of the Boks integral was not further developed, since the [[A-integral|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016800/b01680026.png" />-integral]] proved to be more convenient for the integration of functions conjugate with summable functions.
+
$$
 +
I(t)  =  \sum _ { i=1 } ^ { n }
 +
f( \xi _ {i} +t)
 +
[x _ {i} -x _ {i-1} ].
 +
$$
 +
 
 +
If, for  $  \rho = \max _ {i} (x _ {i} - x _ {i-1 }  ) \rightarrow 0 $,
 +
$  I(t) $
 +
converges in measure to a definite limit  $  I $,
 +
the number  $  I $
 +
is said to be the Boks integral ( $  B $-
 +
integral) of  $  f $
 +
over  $  [a, b] $.
 +
Thus, the Boks integral is an integral of Riemann type and is a generalization of the Riemann integral.
 +
 
 +
The Boks integral represents a considerable extension of the Lebesgue integral: Any summable function is $  B $-
 +
integrable and these integrals coincide, but there exist non-summable $  B $-
 +
integrable functions; in particular, if $  g $
 +
is the function conjugate with a summable function $  f $,  
 +
then it is $  B $-
 +
integrable and the coefficients of the series conjugate with the Fourier series of $  f $
 +
are the coefficients of the respective Fourier series (in the sense of $  B $-
 +
integration) of $  g $(
 +
A.N. Kolmogorov). The theory of the Boks integral was not further developed, since the [[A-integral| $  A $-
 +
integral]] proved to be more convenient for the integration of functions conjugate with summable functions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.J. Boks,  "Sur les rapports entre les méthodes de l'intégration de Riemann et de Lebesque"  ''Rend. Circ. Mat. Palermo (2)'' , '''45'''  (1921)  pp. 211–264</TD></TR><TR><TD valign="top">[2]</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">  T.J. Boks,  "Sur les rapports entre les méthodes de l'intégration de Riemann et de Lebesque"  ''Rend. Circ. Mat. Palermo (2)'' , '''45'''  (1921)  pp. 211–264</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


One of the generalizations of the Lebesgue integral, proposed by A. Denjoy (1919) and studied in detail by T.J. Boks (1921). A real-valued function $ f $ on a segment $ [a, b] $ is periodically extended (with period length $ b - a $) to the entire straight line. For an arbitrary subdivision $ a = x _ {0} < x _ {1} < \dots < x _ {n} = b $ of $ [a, b] $, for an arbitrary selection of points $ \overline \xi \; = \{ \xi _ {i} \} _ {1} ^ {n} $, $ \xi _ {i} \in [x _ {i-1 } , x _ {i} ] $, and an arbitrary $ t $, the following sum is constructed:

$$ I(t) = \sum _ { i=1 } ^ { n } f( \xi _ {i} +t) [x _ {i} -x _ {i-1} ]. $$

If, for $ \rho = \max _ {i} (x _ {i} - x _ {i-1 } ) \rightarrow 0 $, $ I(t) $ converges in measure to a definite limit $ I $, the number $ I $ is said to be the Boks integral ( $ B $- integral) of $ f $ over $ [a, b] $. Thus, the Boks integral is an integral of Riemann type and is a generalization of the Riemann integral.

The Boks integral represents a considerable extension of the Lebesgue integral: Any summable function is $ B $- integrable and these integrals coincide, but there exist non-summable $ B $- integrable functions; in particular, if $ g $ is the function conjugate with a summable function $ f $, then it is $ B $- integrable and the coefficients of the series conjugate with the Fourier series of $ f $ are the coefficients of the respective Fourier series (in the sense of $ B $- integration) of $ g $( A.N. Kolmogorov). The theory of the Boks integral was not further developed, since the $ A $- integral proved to be more convenient for the integration of functions conjugate with summable functions.

References

[1] T.J. Boks, "Sur les rapports entre les méthodes de l'intégration de Riemann et de Lebesque" Rend. Circ. Mat. Palermo (2) , 45 (1921) pp. 211–264
[2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Boks integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boks_integral&oldid=46098
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article