Difference between revisions of "Uniform Convergence of Sequences of Functions"
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+ | ==Sequences of Functions== | ||
<td><strong>Definition:</strong> An <strong>Infinite Sequence of Functions</strong> <math>(f_n(x))_{n=1}^{\infty} = (f_1(x), f_2(x), ..., f_n(x), ...)</math> is a sequence of functions with a common domain. The <strong><math>n^{\mathrm{th}}</math> Term</strong> of the sequence is the function <math>f_n(x)</math>.</td> | <td><strong>Definition:</strong> An <strong>Infinite Sequence of Functions</strong> <math>(f_n(x))_{n=1}^{\infty} = (f_1(x), f_2(x), ..., f_n(x), ...)</math> is a sequence of functions with a common domain. The <strong><math>n^{\mathrm{th}}</math> Term</strong> of the sequence is the function <math>f_n(x)</math>.</td> | ||
<p><em>We can define a finite sequence of functions analogously. A finite sequence of functions is denoted <math>(f_n)_{n=1}^{\infty}</math>.</em></p> | <p><em>We can define a finite sequence of functions analogously. A finite sequence of functions is denoted <math>(f_n)_{n=1}^{\infty}</math>.</em></p> | ||
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<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" class="image" /></div> | <div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" class="image" /></div> | ||
+ | ==Pointwise Convergence of Functions== | ||
<tr> | <tr> | ||
<td><strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. Then <math>(f_n)_{n=1}^{\infty}</math> is said to be <strong>Pointwise Convergent</strong> to the the function <math>f</math> written <math>\lim_{n \to \infty} f_n(x) = f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists a <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.</td> | <td><strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. Then <math>(f_n)_{n=1}^{\infty}</math> is said to be <strong>Pointwise Convergent</strong> to the the function <math>f</math> written <math>\lim_{n \to \infty} f_n(x) = f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists a <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.</td> | ||
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<p>This sequence clearly converges to <math>f(0) = 0</math>. So, we conclude that <math>\displaystyle{\lim_{n \to \infty} f_n(x) = f(x)}</math> for all <math>x \in [0, 1]</math>. Hence the sequence <math>(f_n(x))_{n=1}^{\infty}</math> is pointwise convergent on all of <math>[0, 1]</math>.</p> | <p>This sequence clearly converges to <math>f(0) = 0</math>. So, we conclude that <math>\displaystyle{\lim_{n \to \infty} f_n(x) = f(x)}</math> for all <math>x \in [0, 1]</math>. Hence the sequence <math>(f_n(x))_{n=1}^{\infty}</math> is pointwise convergent on all of <math>[0, 1]</math>.</p> | ||
− | + | ==Uniform Convergence of Sequences of Functions== | |
<p>Recall from the <a href="/pointwise-convergence-of-sequences-of-functions">Pointwise Convergence of Sequences of Functions</a> page that we say the sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> with common domain <math>X</math> is convergent to the limit function <math>f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.</p> | <p>Recall from the <a href="/pointwise-convergence-of-sequences-of-functions">Pointwise Convergence of Sequences of Functions</a> page that we say the sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> with common domain <math>X</math> is convergent to the limit function <math>f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.</p> | ||
<p>Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!</p> | <p>Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!</p> |
Revision as of 11:20, 27 October 2021
Contents
Sequences of Functions
Definition: An Infinite Sequence of Functions is a sequence of functions with a common domain. The Term of the sequence is the function .
We can define a finite sequence of functions analogously. A finite sequence of functions is denoted .
We can also denote an infinite sequence of functions as simply . We can also use curly brackets to denote a sequence of functions such as or simply .
For example, consider the following sequence of functions:
This is a sequence of diagonal straight lines that pass through the origin and whose slope is increasing. The following illustrates a few of the functions in this sequence:
For another example, consider the following sequence of functions:
This is a sequence of the simplest degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:
Pointwise Convergence of Functions
Definition: Let be a sequence of functions with common domain . Then is said to be Pointwise Convergent to the the function written if for all and for all there exists a such that if then .
For example, consider the following sequence of functions defined on :
We claim that is pointwise convergent to . The following image shows the first six functions in the sequence given above. It should be intuitively clear that the sequence converges to the limit function .
To show this, fix and assume that and let be given. Then since we have that:
Choose such that which can be done by the Archimedean property. Then and so for we have that:
Therefore for . Now, for , notice that:
This sequence clearly converges to . So, we conclude that for all . Hence the sequence is pointwise convergent on all of .
Uniform Convergence of Sequences of Functions
Recall from the <a href="/pointwise-convergence-of-sequences-of-functions">Pointwise Convergence of Sequences of Functions</a> page that we say the sequence of functions with common domain is convergent to the limit function if for all and for all there exists an such that if then .
Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!
Definition: Let be a sequence of functions with common domain . Then is said to be Uniformly Convergent to the the limit function written Failed to parse (syntax error): {\displaystyle \lim_{n \to \infty} f_n(x) = f(x) \: \mathit{uniformly \: on} \: X} or Failed to parse (syntax error): {\displaystyle f_n \to f \: \mathit{uniformly \: on} \: X} if for all there exists a such that if then for all .
Graphically, if the sequence of functions are all real-valued and uniformly converge to the limit function , then from the definition above, we see that for all there exists an such that for all we have that the following inequality holds for all :
The following graphic illustrates the concept of uniform convergence of a sequence of functions <math>(f_n(x))_{n=1}^{\infty}<math>:
Licensing
Content obtained and/or adapted from:
- Sequences of Functions, mathonline.wikidot.com under a CC BY-SA license
- [1] under a CC BY-SA license
- [2] under a CC BY-SA license