Difference between revisions of "A-integral"
From Encyclopedia of Mathematics
(Importing text file) |
|||
| (3 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | One of the generalizations of the Lebesgue integral, given by E. Titchmarsh [[#References|[1]]] for the integration of functions conjugate to summable ones. A measurable function | + | {{TEX|done}} |
| + | One of the generalizations of the [[Lebesgue integral]], given by E. Titchmarsh [[#References|[1]]] for the integration of functions conjugate to summable ones. A measurable function $f(x)$ is called $A$-integrable over $[a,b]$ if | ||
| − | + | $$m\{x\colon|f(x)|>n\}=O\left(\frac1n\right)$$ | |
and if | and if | ||
| − | + | $$I=\lim_{n\to\infty}\int\limits_a^b[f(x)]_ndx$$ | |
exists, where | exists, where | ||
| − | + | $$[f(x)]_n=\begin{cases}f(x)&\text{if }|f(x)|\leq n,\\0&\text{if }|f(x)|>n.\end{cases}$$ | |
| − | The number | + | The number $I$ is called the $A$-integral. It is denoted by |
| − | + | $$(A)\quad\int\limits_a^bf(x)dx.$$ | |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.G. Titchmarsh, "On conjugate functions" ''Proc. London Math. Soc.'' , '''29''' (1928) pp. 49–80</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.A. Vinogradova, "Generalized integrals and Fourier series" ''Itogi Nauk. Mat. Anal. 1970'' (1971) pp. 65–107 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.G. Titchmarsh, "On conjugate functions" ''Proc. London Math. Soc.'' , '''29''' (1928) pp. 49–80</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.A. Vinogradova, "Generalized integrals and Fourier series" ''Itogi Nauk. Mat. Anal. 1970'' (1971) pp. 65–107 (In Russian)</TD></TR></table> | ||
Latest revision as of 11:01, 31 March 2017
One of the generalizations of the Lebesgue integral, given by E. Titchmarsh [1] for the integration of functions conjugate to summable ones. A measurable function $f(x)$ is called $A$-integrable over $[a,b]$ if
$$m\{x\colon|f(x)|>n\}=O\left(\frac1n\right)$$
and if
$$I=\lim_{n\to\infty}\int\limits_a^b[f(x)]_ndx$$
exists, where
$$[f(x)]_n=\begin{cases}f(x)&\text{if }|f(x)|\leq n,\\0&\text{if }|f(x)|>n.\end{cases}$$
The number $I$ is called the $A$-integral. It is denoted by
$$(A)\quad\int\limits_a^bf(x)dx.$$
References
| [1] | E.G. Titchmarsh, "On conjugate functions" Proc. London Math. Soc. , 29 (1928) pp. 49–80 |
| [2] | I.A. Vinogradova, "Generalized integrals and Fourier series" Itogi Nauk. Mat. Anal. 1970 (1971) pp. 65–107 (In Russian) |
How to Cite This Entry:
A-integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A-integral&oldid=18741
A-integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A-integral&oldid=18741
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article