Vanish at infinity

<h2 id="definitions">Definitions</h2>
<p>A function on a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> is said to vanish at infinity if the function approaches 
  
    
      
        0
      
    
    {\displaystyle 0}
  
 as the input grows without bounds (that is, 
  
    
      
        f
        (
        x
        )
        →
        0
      
    
    {\displaystyle f(x)\to 0}
  
 as 
  
    
      
        ‖
        x
        ‖
        →
        ∞
      
    
    {\displaystyle \|x\|\to \infty }
  
). Or,
</p>

lim
          
            x
            →
            −
            ∞
          
        
        f
        (
        x
        )
        =
        
          lim
          
            x
            →
            +
            ∞
          
        
        f
        (
        x
        )
        =
        0
      
    
    {\displaystyle \lim _{x\to -\infty }f(x)=\lim _{x\to +\infty }f(x)=0}

<p>in the specific case of functions on the real line.
</p><p>For example, the function 
</p>

f
        (
        x
        )
        =
        
          
            1
            
              
                x
                
                  2
                
              
              +
              1
            
          
        
      
    
    {\displaystyle f(x)={\frac {1}{x^{2}+1}}}

<p>defined on the <a href="/facts/Real_line/6eq42xX5">real line</a> vanishes at infinity.
</p><p>Alternatively, a function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 on a <a href="/facts/Locally_compact_space/hesalT7q">locally compact space</a> 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 vanishes at infinity, if given any <a href="/facts/Positive_number/z0xollg1">positive number</a> 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
, there exists a <a href="/facts/Compact_space/d0cgXJH7">compact</a> subset 
  
    
      
        K
        ⊆
        Ω
      
    
    {\displaystyle K\subseteq \Omega }
  
 such that 
</p>

|
        
        f
        (
        x
        )
        
          |
        
        <
        ε
      
    
    {\displaystyle |f(x)|<\varepsilon }

<p>whenever the point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 lies outside of 
  
    
      
        K
        .
      
    
    {\displaystyle K.}
  
<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> In other words, for each positive number 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
, the set 
  
    
      
        
          {
          
            x
            ∈
            Ω
            :
            ‖
            f
            (
            x
            )
            ‖
            ≥
            ε
          
          }
        
      
    
    {\displaystyle \left\{x\in \Omega :\|f(x)\|\geq \varepsilon \right\}}
  
 has compact closure. 
For a given locally compact space 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of such functions
</p>

f
        :
        Ω
        →
        
          K
        
      
    
    {\displaystyle f:\Omega \to \mathbb {K} }

<p>valued in 
  
    
      
        
          K
        
        ,
      
    
    {\displaystyle \mathbb {K} ,}
  
 which is either 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 or 
  
    
      
        
          C
        
        ,
      
    
    {\displaystyle \mathbb {C} ,}
  
 forms a 
  
    
      
        
          K
        
      
    
    {\displaystyle \mathbb {K} }
  
-<a href="/facts/Vector_space/M1pxMLx2">vector space</a> with respect to <a href="/facts/Pointwise/dt4Mrs6w">pointwise</a> <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplication</a> and <a href="/facts/Addition/apFWJBFB">addition</a>, which is often denoted 
  
    
      
        
          C
          
            0
          
        
        (
        Ω
        )
        .
      
    
    {\displaystyle C_{0}(\Omega ).}
  
 
</p><p>As an example, the function
</p>

h
        (
        x
        ,
        y
        )
        =
        
          
            1
            
              x
              +
              y
            
          
        
      
    
    {\displaystyle h(x,y)={\frac {1}{x+y}}}

<p>where 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
      
    
    {\displaystyle y}
  
 are <a href="/facts/Real_number/R02gw5Pb">reals</a> greater or equal 1 and correspond to the point 
  
    
      
        (
        x
        ,
        y
        )
      
    
    {\displaystyle (x,y)}
  
 on 
  
    
      
        
          
            R
          
          
            ≥
            1
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {R} _{\geq 1}^{2}}
  
 vanishes at infinity.
</p><p>A <a href="/facts/Normed_space/rmDljfyE">normed space</a> is locally compact if and only if it is finite-dimensional so in this particular case, there are two different definitions of a function "vanishing at infinity". 
The two definitions could be inconsistent with each other: if 
  
    
      
        f
        (
        x
        )
        =
        ‖
        x
        
          ‖
          
            −
            1
          
        
      
    
    {\displaystyle f(x)=\|x\|^{-1}}
  
 in an infinite dimensional <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>, then 
  
    
      
        f
      
    
    {\displaystyle f}
  
 vanishes at infinity by the 
  
    
      
        ‖
        f
        (
        x
        )
        ‖
        →
        0
      
    
    {\displaystyle \|f(x)\|\to 0}
  
 definition, but not by the compact set definition.
</p>
<h2 id="rapidly-decreasing">Rapidly decreasing</h2>
<p class="note">Main article: <a href="/facts/Schwartz_space/B035faw7">Schwartz space</a></p>
<p>Refining the concept, one can look more closely to the rate of vanishing of functions at infinity. One of the basic intuitions of <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> is that the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> interchanges <a href="/facts/Smooth_function/mffL4ch2">smoothness</a> conditions with rate conditions on vanishing at infinity. Using <a href="/facts/Big_O_notation/weFFjSWg">big <i>O</i> notation</a>, the rapidly decreasing test functions of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">tempered distribution</a> theory are <a href="/facts/Smooth_function/mffL4ch2">smooth functions</a> that are 
</p>

O
        
          (
          
            
              |
            
            x
            
              
                |
              
              
                −
                N
              
            
          
          )
        
      
    
    {\displaystyle O\left(|x|^{-N}\right)}

<p>for all 
  
    
      
        N
      
    
    {\displaystyle N}
  
, as 
  
    
      
        
          |
        
        x
        
          |
        
        →
        ∞
      
    
    {\displaystyle |x|\to \infty }
  
, and such that all their <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a> satisfy the same condition too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distribution theory</a> of tempered distributions will have the same property.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Infinity/cF506uD7">Infinity</a> – Mathematical concept</li>
<li><a href="/facts/Projectively_extended_real_line/9epfpLw2">Projectively extended real line</a> – Real numbers with an added point at infinity</li>
<li><a href="/facts/Zero_of_a_function/mlK3dhoT">Zero of a function</a> – Point where function's value is zero</li></ul>
<h2 id="citations">Citations</h2>

<ul><li><a href="/facts/Edwin_Hewitt/vY1ff529">Hewitt, E</a> and Stromberg, K (1963). <i>Real and abstract analysis</i>. Springer-Verlag.{{cite book}}:  CS1 maint: multiple names: authors list (link)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Function vanishing at infinity - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-15. <a href="https://www.encyclopediaofmath.org/index.php/Function_vanishing_at_infinity" target="_blank">https://www.encyclopediaofmath.org/index.php/Function_vanishing_at_infinity</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"vanishing at infinity in nLab". ncatlab.org. Retrieved 2019-12-15. <a href="https://ncatlab.org/nlab/show/vanishing+at+infinity" target="_blank">https://ncatlab.org/nlab/show/vanishing+at+infinity</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Vanish at infinity open-in-new

Vanish at infinity