Investigating Test Functions: \psi(x) & \phi(x2)

In summary, the conversation discusses the investigation of two functions, \psi(x) = \phi(c(x - \eta)) and \psi(x) = \phi(x2), to determine if they are test functions. The first function does not have contact support and is therefore not a test function. The second function is smooth but does not have any interval where it is equal to 0, also making it not a test function. However, assuming \phi(x) is a test function on (-\infty,\infty), both functions are considered test functions. The conversation also highlights the importance of understanding the precise definition and conditions of a test function space in order to determine if a function qualifies as a test function.
  • #1
squenshl
479
4

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.
 
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  • #2
But you did not say what are the assumptions about [tex]\phi[/tex]. Without knowing them one can't answer these questions.
 
  • #3
Sorry, assume that [tex]\phi[/tex](x) is a test function on (-[tex]\infty[/tex],[tex]\infty[/tex])
 
  • #4
They are both test functions I think.
 
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  • #5
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?
 

Related to Investigating Test Functions: \psi(x) & \phi(x2)

1. What are test functions in scientific experiments?

Test functions are mathematical functions used in scientific experiments to model and analyze complex systems. They are usually defined as smooth, infinitely differentiable functions that satisfy certain properties, such as compact support and rapid decay. These functions are used as approximations for more complex functions in order to simplify calculations and make them more manageable.

2. How are test functions used in investigating functions?

Test functions are used in investigating functions by providing a simplified representation of the system being studied. They are often used as a basis for approximating more complex functions and can help in understanding the behavior of these functions under different conditions. By using test functions, scientists can analyze and predict the behavior of a system without having to deal with the complexities of the actual function.

3. What is the difference between \psi(x) and \phi(x2) in test functions?

\psi(x) and \phi(x2) are both test functions, but they differ in their properties and uses. \psi(x) is typically used in analyzing functions that are defined over a continuous domain, while \phi(x2) is used for functions that are defined over a discrete domain. Additionally, \psi(x) is usually an even function, while \phi(x2) is an odd function.

4. How do test functions help in scientific research?

Test functions are essential tools in scientific research as they provide a simplified representation of complex systems. By using test functions, scientists can analyze and understand the behavior of a system without having to deal with the complexities of the actual function. This allows for more efficient and accurate research, as well as the development of new theories and models.

5. Are there any limitations to using test functions in scientific experiments?

While test functions are useful in simplifying complex systems, they also have limitations. One of the main limitations is that they are only approximations of the actual function and may not accurately represent the behavior of the system in all cases. Additionally, the choice of test function can also affect the results of the experiment, so it is important to carefully select an appropriate test function for the specific research being conducted.

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