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J$C
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It is clear that there are functions in L2 that are not in L1, but what about the other way? And what effect does considering L2(R) versus L2([a,b]) have?
Thanks.
Thanks.
That's the key.maze said:Squaring a number greater than 1 makes it larger, whereas squaring a number less than 1 makes it smaller. Thus squaring a function makes the tail smaller, but makes singularities larger.
Very nice, but once again don't give out complete answers.maze said:D H's point about behavior at infinity actually got me thinking a little bit, and I think you actually can find an example that depends on the behavior at infinity.
No. Maze's first example is exactly what I had in mind in post #4. In fact, any function of the form [itex]f(x) = c/x^a,\, a\in\,[1/2,1)[/itex] will be a member of L1 but not L2 for any closed domain that includes 0.J$C said:Perhaps the closed condition would preclude all of the singularity counter examples and in fact L1[a,b]=L2[a,b].
Sure.J$C said:So a function such as 1/sqrt(x) is considered a member of L1[0,1] even though the function is not defined at 0?
J$C said:So a function such as 1/sqrt(x) is considered a member of L1[0,1] even though the function is not defined at 0?
And yes, Maze's infinite domain solution is quite slick.
D H said:Sure.
L1, L2, etc are defined in terms of integrability, so the function needs to be integrable over the domain -- which means that it must be defined almost everywhere over the domain.
Boundedness is not required (except for L-infinity, of course).
The main difference between functions in L1 and L2 is the programming language they are written in. L1 refers to the first language or primary language that a person learns, while L2 refers to a second language that a person learns. This means that functions in L1 are written in the language that a person is most familiar with, while functions in L2 are written in a language that a person is still learning.
Yes, functions in L1 can be translated into L2. However, the process may not be straightforward as there may be differences in syntax and structure between the two languages. It's important to have a good understanding of both languages in order to accurately translate functions from L1 to L2.
Some common functions in L1 that may not exist in L2 include functions related to a specific programming language or technology, such as built-in functions for string manipulation or file input/output. Additionally, L1 may have functions that are specific to the culture or language, such as functions for handling date and time formats or currency conversion.
Functions in L1 and L2 can interact with each other through various methods, such as using libraries or APIs. Libraries are collections of pre-written functions that can be imported into a program and used by both L1 and L2 functions. APIs (Application Programming Interfaces) are interfaces that allow different programming languages to communicate and exchange data, allowing functions in L1 and L2 to interact with each other.
No, it is not necessary to learn both L1 and L2 to use functions effectively. However, having a good understanding of both languages can be beneficial in situations where functions in L1 need to be translated or when working with functions from different languages. It's important to have a strong foundation in at least one language and to continuously learn and improve in order to effectively use functions in any language.