Question

Explain Why these sequences of functions can not be dominated by an integrable function g: 1 $|x| ? n, a) $f_n(x) = \begin{cases} \frac{1}{n}, & |x| ? n, \\ 0 & |x| > n \end{cases}$ b) $f_n(x) = n\phi(nx)$, where $\phi$ is a smooth compactly supported function c) $f_n(x) = \phi(x - n)$, where $\phi$ is a smooth compactly supported function.

          Explain Why these sequences of functions can not be dominated by an integrable function g:

1
$|x| ? n,
a) $f_n(x) = \begin{cases} \frac{1}{n}, & |x| ? n, \\ 0 & |x| > n \end{cases}$
b) $f_n(x) = n\phi(nx)$, where $\phi$ is a smooth compactly supported function
c) $f_n(x) = \phi(x - n)$, where $\phi$ is a smooth compactly supported function.

Explain Why these sequences of functions can not be dominated by an integrable function g:

1
|x| ? n,
a)fn(x) = (1)/(n), |x| ? n,
0 |x| > n b)fn(x) = nϕ(nx), whereϕis a smooth compactly supported function
c)fn(x) = ϕ(x - n), whereϕis a smooth compactly supported function.

Added by Karen B.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

04/21/2022

Step 1

In case (b), f(x) is not dominated by g because for every Show more…

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Possibly using dominated convergence theorem (not sure) Explain Why these sequences of functions can not be dominated by an integrable function g: 1 Ix|<n, a) fr(x)= n 0 Ix|>n b) f(x) = n(n x), where is a smooth compactly supported function c) fn(x) = (x - n), where is a smooth compactly supported function.