Difference between revisions of "Majorant and minorant"
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Two functions, the values of the first being not less and the values of the second being not greater than the corresponding values of a given function (for all admissible values of the independent variable). | Two functions, the values of the first being not less and the values of the second being not greater than the corresponding values of a given function (for all admissible values of the independent variable). | ||
For functions represented by a power series, a majorant is e.g. the sum of a power series with positive coefficients which are not less than the absolute values of the corresponding coefficients of the given series. | For functions represented by a power series, a majorant is e.g. the sum of a power series with positive coefficients which are not less than the absolute values of the corresponding coefficients of the given series. | ||
| − | A majorant (minorant) of a subset | + | A majorant (minorant) of a subset $X$ of an ordered set $E$ is an element $y\in E$ such that $y\geq x$ ($x\geq y$) for every $x\in X$. |
| − | In the theory of integral and differential equations, a majorant (minorant) or majorant function (minorant function) for some function | + | In the theory of integral and differential equations, a majorant (minorant) or majorant function (minorant function) for some function $f$ is a continuous function whose [[Dini derivative|Dini derivative]] at each point $t$ is not less (not greater) than $f(t)$ and is different from $-\infty$ $(+\infty)$. The difference between any majorant and any minorant is a non-decreasing function. Any summable function on an interval has absolutely-continuous majorants and minorants which are arbitrarily close to its indefinite [[Lebesgue integral|Lebesgue integral]]. The notion of majorant and minorant can be generalized to the case of additive set functions and also to the case when the derivatives are taken in some generalized sense. |
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | The terms "majorant" and "minorant" are rarely used in the sense 3) in English; instead the terms "upper | + | The terms "majorant" and "minorant" are rarely used in the sense 3) in English; instead the terms "upper bound" and "lower bound" are in common use. |
See also [[Lower bound|Lower bound]]; [[Upper and lower bounds|Upper and lower bounds]]. | See also [[Lower bound|Lower bound]]; [[Upper and lower bounds|Upper and lower bounds]]. | ||
Latest revision as of 10:06, 15 April 2014
Two functions, the values of the first being not less and the values of the second being not greater than the corresponding values of a given function (for all admissible values of the independent variable).
For functions represented by a power series, a majorant is e.g. the sum of a power series with positive coefficients which are not less than the absolute values of the corresponding coefficients of the given series.
A majorant (minorant) of a subset $X$ of an ordered set $E$ is an element $y\in E$ such that $y\geq x$ ($x\geq y$) for every $x\in X$.
In the theory of integral and differential equations, a majorant (minorant) or majorant function (minorant function) for some function $f$ is a continuous function whose Dini derivative at each point $t$ is not less (not greater) than $f(t)$ and is different from $-\infty$ $(+\infty)$. The difference between any majorant and any minorant is a non-decreasing function. Any summable function on an interval has absolutely-continuous majorants and minorants which are arbitrarily close to its indefinite Lebesgue integral. The notion of majorant and minorant can be generalized to the case of additive set functions and also to the case when the derivatives are taken in some generalized sense.
References
| [1] | N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French) |
| [2] | S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) |
Comments
The terms "majorant" and "minorant" are rarely used in the sense 3) in English; instead the terms "upper bound" and "lower bound" are in common use.
See also Lower bound; Upper and lower bounds.
Majorant and minorant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Majorant_and_minorant&oldid=14263