smooth function as a sum of rect or tri functions
$begingroup$
At the beginning it looks to me easy but i am cannot find a mathematical prove for this.
I am trying to prove that smooth function cannot be written as a sum of rect functions (or tri functions)
for example let say that f(x) is a smooth function i want to prove that it cannot be written as:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:rectleft(ncdot :gleft(xright)right)right)$
and the same for the tri function:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:trileft(ncdot :gleft(xright)right)right)$
Thanks
edit:
The definition of the rect and tri functions:
Rectangular function::
$operatorname{rect}(t) = Pi(t) = begin{cases}
0 & text{if } |t| > frac{1}{2}, \
frac{1}{2} & text{if } |t| = frac{1}{2}, \
1 & text{if } |t| < frac{1}{2}.
end{cases}$
Triangular_function:
$begin{align}
operatorname{tri}(x) = Lambda(x) &overset{underset{mathrm{def}}{}}{=} max(1 - |x|, 0) \
&= begin{cases}
1 - |x| qquad & |x| < 1 \
0 qquad & mathrm{otherwise} \
end{cases}
end{align}$
sequences-and-series functions
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
$endgroup$
add a comment |
$begingroup$
At the beginning it looks to me easy but i am cannot find a mathematical prove for this.
I am trying to prove that smooth function cannot be written as a sum of rect functions (or tri functions)
for example let say that f(x) is a smooth function i want to prove that it cannot be written as:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:rectleft(ncdot :gleft(xright)right)right)$
and the same for the tri function:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:trileft(ncdot :gleft(xright)right)right)$
Thanks
edit:
The definition of the rect and tri functions:
Rectangular function::
$operatorname{rect}(t) = Pi(t) = begin{cases}
0 & text{if } |t| > frac{1}{2}, \
frac{1}{2} & text{if } |t| = frac{1}{2}, \
1 & text{if } |t| < frac{1}{2}.
end{cases}$
Triangular_function:
$begin{align}
operatorname{tri}(x) = Lambda(x) &overset{underset{mathrm{def}}{}}{=} max(1 - |x|, 0) \
&= begin{cases}
1 - |x| qquad & |x| < 1 \
0 qquad & mathrm{otherwise} \
end{cases}
end{align}$
sequences-and-series functions
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
$endgroup$
1
$begingroup$
I tried to prove it in the Fourier domain and use the Support of them and say that the support of the Fourier transform of rect is not finite but it is not enough. also i go to Gibbs phenomenon but it prove that a rect functions cannot be sum of smooth function
$endgroup$
– Ophir Yaniv
Jan 16 at 16:55
add a comment |
$begingroup$
At the beginning it looks to me easy but i am cannot find a mathematical prove for this.
I am trying to prove that smooth function cannot be written as a sum of rect functions (or tri functions)
for example let say that f(x) is a smooth function i want to prove that it cannot be written as:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:rectleft(ncdot :gleft(xright)right)right)$
and the same for the tri function:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:trileft(ncdot :gleft(xright)right)right)$
Thanks
edit:
The definition of the rect and tri functions:
Rectangular function::
$operatorname{rect}(t) = Pi(t) = begin{cases}
0 & text{if } |t| > frac{1}{2}, \
frac{1}{2} & text{if } |t| = frac{1}{2}, \
1 & text{if } |t| < frac{1}{2}.
end{cases}$
Triangular_function:
$begin{align}
operatorname{tri}(x) = Lambda(x) &overset{underset{mathrm{def}}{}}{=} max(1 - |x|, 0) \
&= begin{cases}
1 - |x| qquad & |x| < 1 \
0 qquad & mathrm{otherwise} \
end{cases}
end{align}$
sequences-and-series functions
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
$endgroup$
At the beginning it looks to me easy but i am cannot find a mathematical prove for this.
I am trying to prove that smooth function cannot be written as a sum of rect functions (or tri functions)
for example let say that f(x) is a smooth function i want to prove that it cannot be written as:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:rectleft(ncdot :gleft(xright)right)right)$
and the same for the tri function:
$ fleft(xright):=:sum _{n=0}^{infty }left(a_n:trileft(ncdot :gleft(xright)right)right)$
Thanks
edit:
The definition of the rect and tri functions:
Rectangular function::
$operatorname{rect}(t) = Pi(t) = begin{cases}
0 & text{if } |t| > frac{1}{2}, \
frac{1}{2} & text{if } |t| = frac{1}{2}, \
1 & text{if } |t| < frac{1}{2}.
end{cases}$
Triangular_function:
$begin{align}
operatorname{tri}(x) = Lambda(x) &overset{underset{mathrm{def}}{}}{=} max(1 - |x|, 0) \
&= begin{cases}
1 - |x| qquad & |x| < 1 \
0 qquad & mathrm{otherwise} \
end{cases}
end{align}$
sequences-and-series functions
sequences-and-series functions
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
edited Jan 16 at 16:48
Ophir Yaniv
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
asked Jan 16 at 16:21
Ophir YanivOphir Yaniv
62
62
1
$begingroup$
I tried to prove it in the Fourier domain and use the Support of them and say that the support of the Fourier transform of rect is not finite but it is not enough. also i go to Gibbs phenomenon but it prove that a rect functions cannot be sum of smooth function
$endgroup$
– Ophir Yaniv
Jan 16 at 16:55
add a comment |
1
$begingroup$
I tried to prove it in the Fourier domain and use the Support of them and say that the support of the Fourier transform of rect is not finite but it is not enough. also i go to Gibbs phenomenon but it prove that a rect functions cannot be sum of smooth function
$endgroup$
– Ophir Yaniv
Jan 16 at 16:55
1
1
$begingroup$
I tried to prove it in the Fourier domain and use the Support of them and say that the support of the Fourier transform of rect is not finite but it is not enough. also i go to Gibbs phenomenon but it prove that a rect functions cannot be sum of smooth function
$endgroup$
– Ophir Yaniv
Jan 16 at 16:55
$begingroup$
I tried to prove it in the Fourier domain and use the Support of them and say that the support of the Fourier transform of rect is not finite but it is not enough. also i go to Gibbs phenomenon but it prove that a rect functions cannot be sum of smooth function
$endgroup$
– Ophir Yaniv
Jan 16 at 16:55
add a comment |
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I tried to prove it in the Fourier domain and use the Support of them and say that the support of the Fourier transform of rect is not finite but it is not enough. also i go to Gibbs phenomenon but it prove that a rect functions cannot be sum of smooth function
$endgroup$
– Ophir Yaniv
Jan 16 at 16:55