Monotonous sequence of functions

In mathematics, a monotonic function sequence is a special function sequence of real-valued functions . A function sequence is called monotonically increasing if the function values ​​for each argument form a monotonically increasing sequence and monotonically decreasing if they form a monotonically decreasing sequence. Monotonic function sequences are one of the many monotonic terms in mathematics and can be viewed as a special case of monotonic mapping .

definition

Are for real-valued functions, then the sequence of functions is called ${\displaystyle f_{k}\colon D\mapsto \mathbb {R} }$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle (f_{k})_{k\in \mathbb {N} }}$

• monotonically increasing on , when all is ${\displaystyle D}$ ${\displaystyle f_{k}(x)\leq f_{k+1}(x)}$ ${\displaystyle x\in D}$
• monotonously noticeable when is for everyone , and ${\displaystyle D}$ ${\displaystyle f_{k}(x)\geq f_{k+1}(x)}$ ${\displaystyle x\in D}$
• monotonic if it is either monotonically decreasing or monotonically increasing.

example

Consider the sequence of functions as an example . she is ${\displaystyle f_{k}(x)=x^{k}}$

• falling monotonously , since it is equivalent to and ${\displaystyle D=[0,1]}$ ${\displaystyle f_{k+1}(x)\leq f_{k}(x)}$ ${\displaystyle f_{k+1}(x)-f_{k}(x)\leq 0}$

${\displaystyle x^{k+1}-x^{k}=x^{k}(x-1)\leq 0}$

since in is always for and is always less than zero for . The sequence of functions is thus also monotonous . ${\displaystyle x^{k}}$ ${\displaystyle [0,1]}$ ${\displaystyle x\in [0,1]}$ ${\displaystyle (x-1)}$ ${\displaystyle x\in [0,1]}$ ${\displaystyle [0,1]}$

• increasing monotonically , since then is always greater than 1 and the term is always positive, i.e. is ${\displaystyle [1,\infty )}$ ${\displaystyle x^{k}}$ ${\displaystyle (x-1)}$

${\displaystyle x^{k+1}-x^{k}=x^{k}(x-1)\geq 0}$.

The sequence of functions is thus also monotonous . However, it is not monotonous because it has no clear monotony behavior on this larger interval, but only on the smaller sub-intervals and . ${\displaystyle [1,\infty )}$ ${\displaystyle [0,\infty )}$ ${\displaystyle [0,1]}$ ${\displaystyle [1,\infty )}$

• not monotonous . It is always positive, but it is ${\displaystyle [-1,0]}$ ${\displaystyle (1-x)}$

${\displaystyle x^{k}={\begin{cases}\geq 0&{\text{ falls }}k{\text{ gerade }}\\\leq 0&{\text{ falls }}k{\text{ ungerade }}\end{cases}}}$.

Thus, changes to the signs of the time, it can therefore apply no monotony. ${\displaystyle x^{k}-x^{k+1}}$ ${\displaystyle x\neq 0}$

use

Monotonous function sequences are used as a prerequisite in some theorems of analysis, such as Dini's theorem and especially in integration theory, for example, in the theorem of monotonic convergence and in the proof of Fatou's lemma .

literature

• Jürgen Elstrodt: Measure and integration theory . 6th edition. Springer, Berlin / Heidelberg / New York 2009, ISBN 978-3-540-89727-9 .