Test function

  • Thread starter squenshl
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  • #1
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Homework Statement


Investigate whether [tex]\psi[/tex](x) = [tex]\phi[/tex](c(x - [tex]\eta[/tex])) & [tex]\psi[/tex](x) = [tex]\phi[/tex](x2) are test functions.

Homework Equations





The Attempt at a Solution


The first function is smooth but has no contact support as it is only 0 at x = [tex]\eta[/tex] so this is not a test function.
The second function is smooth but is not 0 at any interval, so this is not a test function.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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But you did not say what are the assumptions about [tex]\phi[/tex]. Without knowing them one can't answer these questions.
 
  • #3
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Sorry, assume that [tex]\phi[/tex](x) is a test function on (-[tex]\infty[/tex],[tex]\infty[/tex])
 
  • #4
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They are both test functions I think.
 
Last edited:
  • #5
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You see, there are many kinds of test function spaces. You probably have learned about just one. To see that some function is a test function in the sense you know it, you need to check the precise definition of your test function space. Which conditions a given function must satisfy to be a test function? Differentiable? How many times? Compact support? Or vanishing sufficiently fast at infinity?
 

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