For questions about infinitely or arbitrarily differentiable (smooth) functions of one or several variables. To be used especially for real-valued functions; for complex-valued functions, the tag holomorphic-functions is more appropriate.
Questions tagged [smooth-functions]
762 questions
2
votes
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Prove that function $F$ is smooth
Let $F: \mathbb R^n \times \mathbb R^{m^2} \to \mathbb R^{m^2}$, $F_{kl}(x,y)=\sum_{ij} y_{ik}y_{jl}\beta_{ij}(x,y)+\alpha_{kl}(x)+y_{kl}$. Prove that for smooth function $\alpha_{kl}(x),\beta_{kl}(x,y)$, where $\alpha_{kl}(x)$ is vanishing in zero,…
William8649
- 77
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How to show that $\exists$ $\phi\in C_c^{\infty}(\Omega)$ s.t. $\int_{\Omega_1}D\phi(x)dx$ has unit norm?
Let $\Omega\subset \mathbb R^n$ be open, connected and $\Omega=\Omega_1 \cup \Omega_2$ where $\Omega_1\cap \Omega_2=\emptyset$, $\mu(\Omega_1)>0,\mu(\Omega_2)>0.$ Then show that there exists $\phi\in C_c^{\infty}(\Omega)$ such that $\displaystyle…
Mini_me
- 2,165
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vote
2 answers
Problem about the construction of a smooth function
I need to construct a function $f\in C^\infty(\mathbb{R})$, such that
$$f(x)=\left\{\begin{aligned}
&0,\quad x\leqslant 0,\\
&x,\quad x\geqslant 1.
\end{aligned}\right.$$
I already know that we can construct some smooth functions using something…
Exjudger
- 31
1
vote
2 answers
Determine whether a function is smooth?
Suppose we have 4 functions $f,g,h,j$ all of them are functions from $R^n$ to $R$, in particular, $f,g$ are smooth. I’m curious, when given following condition: $$fh+gj=0$$can we claim that $h$ and $j$ are also smooth?
GK1202
- 609
0
votes
0 answers
Smoothness of integral of a smooth function
If I have a function $l(w,x)$, defined as $\mathbb R^n \times \mathbb R^m \mapsto \mathbb R$ that is continuously differentiable and $L$-smooth with respect to $w$, $\|\nabla l(w,x) - \nabla l(w',x)\| \leq L$ for all $w,w' \in \mathbb R^n$, is the…
Interception
- 43
0
votes
1 answer
Application of Mollifier function.
According to Wikipedia https://en.wikipedia.org/wiki/Mollifier, one of the uses of Mollifer functions is to smooth a function. How could you smooth with a mollifer function the function $f(x)=|x|$ at the origin (the "corner")? (I would like to be…
eraldcoil
- 3,508
0
votes
1 answer
Why is the Log-Sum-Exp function 1-smooth
This may be a trivial question. I tried to find the proof for the smoothness of the log-sum-exp function $f: \mathbb{R}^N \to \mathbb{R}$, defined as
$$f(x) = \log \sum_i \mathrm{exp}(x_i).$$
I have seen on multiple papers stating that the…
Lyapunov1729
- 167
0
votes
1 answer
Derivation on $ {\cal C}^\infty_0 (\mathbb R)$
It can be proven that
$X \equiv \{f \in {\cal C}^\infty_0(\mathbb R) \ | \ \exists \ g \in {\cal C}^\infty_0(\mathbb R)$ that verifies $ g' = f \}$
is isomorphic to $Y \equiv \{ f \in {\cal C}^\infty_0 (\mathbb R)\ | \ \int_\mathbb R f \ dx =…
ric.san
- 121
0
votes
1 answer
If $f:M\times M\to\Bbb R$ is $C^\infty$ and $M$ a compact smooth manifold, is $\bar f(p)=\int_M f(p,q)dq$ smooth?
Let $M$ be a compact smooth manifold subset of some $\Bbb R^n$ and $f:M\times M\to\Bbb R$ a $C^\infty$ function.
I need to check wether $\bar f(p)=\int_M f(p,q)dq$ is smooth, but I have no clue where to start, or to come up with a contradiction.…
Garmekain
- 3,124
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- 26
0
votes
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Continuous on a disk, smooth on a disk minus a closed interval. Does it extend?
Suppose $u:\mathbb{D}\to\mathbb{R}$ is a continuous function, where $\mathbb{D}$ is the open unit disk. Let $I$ be a closed segment in $\mathbb{D}$. To make things simpler we can suppose $I=[a,b]$ is a closed interval in $\mathbb{R}\cap\mathbb{D}$.…
J. Moeller
- 2,934
-1
votes
1 answer
On properties of the support of ${\cal C}^{\infty}_c$ functions
Consider the bump function $f \in {\mathcal C}^\infty_c(]-1, 1[)$:
$$f(x) = \exp \left( \frac {1}{x^2-1} \right) $$
There is a canonical way to extend it so that the extension $\tilde f \in {\mathcal C}^\infty_c(\mathbb R)$.
In general,
the…
ric.san
- 121
-1
votes
1 answer
Is $C^\infty\subset L_{1,\text{loc}}(\mathbb{R})$?
Is $C^\infty\subset L_{1,\text{loc}}(\mathbb{R})$? Or, can we connect differentiability to integrability (such as between $C^1$ and $L_{1,\text{loc}}(\mathbb{R})$)? I think this correct, but I don't have any rigorous arguments.
acarturk
- 320
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