Boks 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
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) |
Boks integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boks_integral&oldid=46098