# Questions tagged [smooth-functions]

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.

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### Morse-Bott inequalities in $\mathbb{CP}^n$

I want to prove the following statement (which I heavily believe is true): Let $g:\mathbb{CP}^n\to \mathbb{R}$ be a non-constant Morse-Bott function and denote by $\text{Icrit}(g)$ the set of ...
• 359
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### Mollifier function of $f(x)=|x|$ in Scilab.

Hello. I am trying to graph the different approximations of the function $f(x)=|x|$ by convolutions with the standard Mollifier function. However, my code doesn't output the correct graph. What is ...
• 2,876
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### 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 ...
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### Let $f \in \mathcal C_0$ and $\varepsilon >0$. Is there an explicit construction of $g\in\mathcal C_c^\infty$ such that $\|f-g\|_\infty <\varepsilon$?

Let $\mathcal C_0 (\mathbb R^d)$ be the space of real-valued continuous functions on $\mathbb R^d$ that vanish at infinity. Let $\mathcal C_c^\infty (\mathbb R^d)$ be the space of real-valued smooth ...
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1 vote
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### Arbitrary parametrized functions determined by n points

An arbitrary polynomial of degree $n$ is determined by $n+1$ points. In some sense, one can think of the polynomial as having $n+1$ parameters - the $n+1$ coefficients associated to it. Is there a ...
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• 3,892
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### Proof by density Sobolev Spaces?

When reading about Sobolev Spaces, some results are proven using the density of smooth function. So they prove the results on smooth function and then conclude by taking limits within the integral. ...
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### How the author proves that this function is smooth on the boundary $\partial W$?

I'm reading a proof of Lemma 8.2. from this lecture note. Lemma 8.3. Let $M$ be an $m$-manifold, $W$ an open set in $M$, and $f: W \to \mathbb{R}$ a smooth function. Suppose that $x \in W$. Then ...
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### There is a smooth function $g: M \longrightarrow \mathbb{R}$ which agrees with $f$ on some neighbourhood of $x$ in $W$

I'm trying to prove Lemma 8.2. from this lecture note. Lemma 8.3. Let $M$ be an $m$-manifold, $W$ an open set in $M$, and $f: W \to \mathbb{R}$ a smooth function. Suppose that $x \in W$. Then there ...
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### There is a smooth function $\theta: M \to[0,1]$ such that $\theta =0$ on $M \setminus W$ and that $\theta = 1$ on some neighbourhood of $x$

I'm trying to prove Lemma 8.2. from this lecture note. Lemma 8.2. Let $M$ be an $m$-manifold and $W$ an open set in $M$. Let $x \in W$. Then there is a smooth function $\theta: M \longrightarrow[0,1]$...
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### Is this a typo in the existence proof of a special smooth function?

I'm reading below lemma from this lecture note. Lemma 8.1. There is a smooth function, $\theta_0: \mathbb{R}^n \longrightarrow[0,1] \subseteq \mathbb{R}$, with $\theta_0(x)=1$ whenever $\|x\| \leq 1$,...
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