Hi, I am not mathematician. (I am computer science)
I need your help to construct a compact support gaussian function
It is ok for an interval [a, b]?
[tex]f(x) = \exp(-0.5\frac{(x-\mu)^2}{\sigma^2})[x/tex]
if x \in (a, b)
and
otherwise
And how I can to prove this
I am not familiar with "compact support gaussian function" considered as one term. I found only one occurrence of this phrase on the web. Strictly speaking, this is not a Gaussian function because the latter has infinite support. You can give any definition you like; the question is in the properties of the defined object. Do you need to guarantee any particular properties of this function?
Yes Emakarov, I want to use the gaussian function as membership function of fuzzy numbers.
An definition of fuzzy numbers from [1] is that
fuzzy numbers are functions [tex] f:R\to [0,1] [\tex]
that satisfy four conditions, one condition is that the support has to be closed and bounded (compact).
[1] Salih Aytar, Serpil Pehlivan, Musa A. Mammadov, The core of a sequence of fuzzy numbers, Fuzzy Sets and Systems, Volume 159, Issue 24, 16 December 2008, Pages 3369-3379, ISSN 0165-0114
Depending on the definition of support, you may need to change the condition x ∈ (a, b) to x ∈ [a, b] because (a, b) is not closed.
Then your function definitely satisfies this condition. Note, however, that if , then , but , so may serve as a probability density on , but not on [a, b].