Majorant

Majorant (from Fr. majorer - increase) - a term that is used in mathematics to refer to several concepts that generalize the concept of a supremum or an exact upper bound . Most often used in proving the convergence of integrals and series.

Content

Majorant of an ordered set

The concept of majorants of an ordered set is introduced to define the supremum of a set. Let M be a subset of an ordered set. Then a majorant of the set M is an element no less than any element of M. The supremum of the set M is the minimum of all majorants of the set M. ^[1]

Majorant functions

A majorant of a function is a function whose values are not less than the corresponding values of a given function on the considered interval of an independent variable. The integrability of the majorants of a sequence of integrable functions is a sufficient condition for the existence of an integral on the limit of the sequence. ^[2]

Example

Let integrable functions $f_{n}(x),n=1,2,...$ ${\ displaystyle f_ {n} (x), n = 1,2, ...}$ have a limit $\lim _{n\to \infty }f_{n}(x)=f(x)$ ${\ displaystyle \ lim _ {n \ to \ infty} f_ {n} (x) = f (x)}$ and there is an integrable majorant $g(x)\geq |f_{n}(x)|,n=1,2,...$ ${\ displaystyle g (x) \ geq | f_ {n} (x) |, n = 1,2, ...}$ Then we can go to the limit under the sign of the integral: ^[3]

\lim _{n\to \infty }\int \limits _{-\infty }^{+\infty }f_{n}(x)dx=\int \limits _{-\infty }^{+\infty }\lim _{n\to \infty }f_{n}(x)dx

{\ displaystyle \ lim _ {n \ to \ infty} \ int \ limits _ {- \ infty} ^ {+ \ infty} f_ {n} (x) dx = \ int \ limits _ {- \ infty} ^ { + \ infty} \ lim _ {n \ to \ infty} f_ {n} (x) dx}

Majority Majors

A majorant of a series is a number series , all members of which, starting from a certain number, are not less than the absolute value of the corresponding members of this series. If the initial series depends on the argument, for example, is a power law or trigonometric , then indicate the interval at which the inequality holds. To construct a majorant of matrix series, the matrix norm is used.

As a majorant, they usually use simple, well-converging series - one-dimensional and multi - dimensional geometric progression and series with factorial in the denominator of the terms. The convergence of the majorant implies the convergence of the original series. For series that are functions, building a majorant is the main tool for proving convergence.

Examples are the proofs of the Hadamard series theorem , the Abel lemma for series of several complex variables, and the proof of the pointwise convergence of a trigonometric series. ^[4] ^[5]

Class majorant

The concept of a majorant can be introduced on any set if a numerical function is given on it. A majorant of a class or subset is an element whose value of the function on which is the supremum of the values of the function on this class or subset. Similar definitions are introduced to simplify the presentation. ^[6]

Notes

↑ Limited sets. Majorants and minorants.
↑ Mathematical Encyclopedic Dictionary. - M .: “Owls. Encyclopedia " , 1988. - S. 847.
↑ MAJORANT AND MINORENT
↑ Summary of installation lectures for the state exam in mathematics in the field of “Applied Mathematics and Physics” (St. Petersburg State University)
↑ Comprehensive analysis
↑ MAJORANTS AND MINORENTS OF THE GRAPH CLASS WITH A FIXED DIAMETER AND NUMBER OF TOPS. T. I. Fedoryaeva

[1] Limited sets. Majorants and minorants.

[ReferenceA-2] Mathematical Encyclopedic Dictionary. - M .: “Owls. Encyclopedia " , 1988. - S. 847.

[3] MAJORANT AND MINORENT

[4] Summary of installation lectures for the state exam in mathematics in the field of “Applied Mathematics and Physics” (St. Petersburg State University)

[5] Comprehensive analysis

[6] MAJORANTS AND MINORENTS OF THE GRAPH CLASS WITH A FIXED DIAMETER AND NUMBER OF TOPS. T. I. Fedoryaeva