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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How to show $C^\infty_0$ dense in $C_0^1$ with respect to $W^{1,p}$-norm

I want to prove that $C^\infty_0$ is dense in $C_0^1$ with respect to $\|f\|^p=\|f\|_{L^p}^p+\|\nabla f\|_{L^p}^p$ or a reference for that proof. Or alternatively is $C_0^1$ contained in $W_0^{1,p}$, ...
Perelman's user avatar
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Sobolev space $H^{1/2}$ and smooth functions on $S^1$

As far as I know some $\Omega \subset \mathbb{R}^n$, $C^{\infty}_0(\Omega)$ is dense in $W^{k,p}(\Omega)$ and this also holds for a Lipschitz domain $\Omega$, in that case, $C^{\infty}(\overline{\...
User1's user avatar
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2 votes
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Some confusion about $f_{*p}$ when $f\in C^{\infty}(\mathfrak{g}^*)$ and $p\in \mathfrak{g}^*$

Let $G$ be a Lie group with its Lie algebra $\mathfrak{g}$. Let $f\in C^{\infty}(\mathfrak{g}^*)$ and $p\in \mathfrak{g}^*$. We know that $(\mathfrak{g}^*)^*\cong \mathfrak{g}$ and for every $p\in \...
Mahtab's user avatar
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What does it mean for an inclusion map to be smooth?

Sorry for the elementary question, but I do not understand what does it mean for an inclusion map to be "smooth"? My understanding is that an inclusion map will send an element $x$ into ...
Y2H's user avatar
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1 answer
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Smooth functions with vanished derivatives at the boundary

I need to find a function $f$ in $[a,b]$ which satisfy that $$ f(a) = f_1, \\ f(b) = f_2, \\ f^{(j)}(a) = 0, \quad j = 1,...,k, \\ f^{(j)}(b) = 0, \quad j = 1,...,k, $$ where $f^{(j)}$ is the j-th ...
luyipao's user avatar
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Smooth function times continuous is Lipschitz?

Is a smooth compactly supported function multiplied by a continuous but not necessarily differentiable function already lipschitz continuous? right now Im just interested in the answer. If you know ...
Perelman's user avatar
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Parameterization of area between $y=x+\cos x$ and $y=x+\sin x$

In the $xy$-plane a function is given: $f(x,y)=x+y$. Let $A$ be the area that is in the $xy$-plane and is encapsulated by $x=0$, $x=\frac{\pi}{4}$, $y=x+\cos(x)$ and $y=x+\sin(x)$. a) Make a ...
Zert44's user avatar
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2 votes
0 answers
41 views

Which function space on $\mathbb{R}^n$ is identified with $C^\infty(S^n)$?

Let $\mathbb{R}^n$ be $n$-dimensional Euclidean space and $S^n$ be $n$-sphere. Then, it is well-known that $S^n$ is the one-point compactification of $\mathbb{R}^n$. Now consider $C^\infty(S^n)$, the ...
Keith's user avatar
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On the smooth functions and differentiable manifolds [duplicate]

I'm new on stackexchange community (this is my second question on these topics) and this one is linked to my first (About differentiability of economics functions). Indeed, I asked on the economic ...
Economos's user avatar
3 votes
1 answer
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Is it really true that $\mathcal{S}(\mathbb{R}^n)$ is identified with smooth functions on $S^n$ vanishing at a fixed point?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space on $\mathbb{R}^n$ and $C^\infty(S^n)$ be the space of smooth functions on $n$-sphere. Now fix a point $x \in S^n$ and define \begin{equation} C^\...
Keith's user avatar
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Density of $C_c^\infty(\Omega)$ in $L^p(\Omega)$ for a bounded open set $\Omega$ - any detailed proof?

Let $\Omega$ be an open, bounded domain in $\mathbb{R}^n$ (NOT necesasrily with smooth boundary). Now, let $C_c^\infty(\Omega)$ be the space of smooth functions on $\Omega$ with compact support. Then ...
Keith's user avatar
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17 votes
3 answers
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Can a smooth curve have a segment of straight line?

Setting: we are given a smooth curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^n$ Informal Question: Is it possible that $\gamma$ is a straight line on $[a,b]$, but not a straight line on $[a,b]^c$? ...
John Frank's user avatar
4 votes
2 answers
241 views

What structure do smooth maps and diffeomorphisms preserve?

Context: Continuous maps between topological spaces are structure preserving in the following sense: Given two topological spaces $(X,\tau_X),(Y,\tau_Y)$ (where $(X,\tau_X)$ is the topological space ...
frelg's user avatar
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What smooth convex functions $f \colon (0, \infty) \to [0, \infty)$ satisfy $f(1) = 0$ and $f(x) = x f\left(\frac{1}{x}\right)$?

I want to find all smooth (that is, infinitely differentiable) convex functions $f \colon (0, \infty) \to [0, \infty)$ with $f(1) = 0$ satisfying the functional equation $$ f(x) = x f\left(\frac{1}{x}\...
ViktorStein's user avatar
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Smooth vector bundles have smooth sections that are not constantly zero

Say $E$ is a smooth vector bundle, then is it necessarily true that $E$ has smooth sections that aren't constantly zero, namely $\Gamma^\infty(E) \neq \{0\}$ for $0$ the zero section of $E$. My ...
Jeff's user avatar
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0 answers
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Convolution of Schwartz and $C^\infty$ function with bounded increasing [duplicate]

I have the following problem: Let $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in C^\infty(\mathbb{R}^n)$ such that exists $\alpha>0$ $$ \left|g(x)\right|\leq \frac{1}{1+|x|^\alpha} \quad \forall x\in\...
matdlara's user avatar
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Can the construction of this 2D Curve of Constant Width be adapted to a 3D Surface of Constant Width?

A Surface of Constant Width is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the ...
Lawton's user avatar
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1 answer
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I wonder why the author wrote $\phi_i:\mathbb{R}^n\to\mathbb{R}$ instead of $\phi_i:A\to\mathbb{R}$. ("Analysis on Manifolds" by James R. Munkres.)

I am reading "Analysis on Manifolds" by James R. Munkres. Theorem 16.3 (Existence of a partition of unity). Let $\mathcal{A}$ be a collection of open sets in $\mathbb{R}^n$; let $A$ be ...
佐武五郎's user avatar
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Prove that if every smooth function on a subset of a manifold can be extended to a smooth function on the whole manifold, then the subset is closed.

Consider a nonempty subset $A$ of a smooth manifold $M$ and suppose that every smooth function on $A$ can be extended to a smooth function on $M$. We want to show that $A$ is closed. How might we ...
utx7563yu's user avatar
1 vote
0 answers
66 views

How do we know that there is a smooth function that is constant on two disjoint non-empty compact sets in $\mathbb{R}^n$?

Let $A$ and $B$ be non-empty compact sets in $\mathbb{R}^n$ such that $A\cap B\neq\emptyset.$ Prove that there is a smooth function $f$ on $\mathbb{R}^n$ with values ranging (inclusively) between 0 ...
utx7563yu's user avatar
3 votes
2 answers
73 views

Example of smooth compactly supported function

Is there an easy example of a smooth compactly supported function (ie. a test function) that equals $e^x$ on the interval $[-1,1]$? This is in reference to the following stack exchange question here, ...
HtmlProg's user avatar
7 votes
2 answers
126 views

How to compute $\displaystyle\lim_{n\to\infty} \frac1{n+1}\sum_{k=1}^n \left|X+\frac kn\right|-\left|X-\frac kn\right|$?

I was just playing around with the real absolute value, trying to build something smooth (for no particular reason). After some experimentation I got to the sequence $(f_n)_n$ given by $$f_n(X) := \...
Alma Arjuna's user avatar
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1 answer
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Multiplying tangent vectors by positive numbers (Gullemin-Pollack 1.8.2)

This is a question from Differential Topology by Guillemin and Pollack: Let $g$ be a smooth, everywhere-positive function on $X$. Check that the multiplication map $T(X)\rightarrow T(X)$, $(x,v)\...
Hempelicious's user avatar
4 votes
0 answers
50 views

Do smooth curves uniquely identify the smooth structure? [duplicate]

(introduction) First of all, it is a bit tricky to start this question by defining the terms, because the idea of "smooth structure" is, I think, a bit too spread out to contain its essence ...
cnikbesku's user avatar
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1 answer
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Unambiguous definition of smoothness in the parameterization of a curve

Given the parameterization of a curve in $\mathbb{R}^n$, $$\boldsymbol \gamma (t) = (x_1(t), x_2(t), \ldots, x_n(t))$$ I can not find a univocal definition of smoothness. This answer requires existing ...
BowPark's user avatar
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1 vote
2 answers
91 views

Can a smooth matrix function $M_{ij}(t)$ change rank?

Let $M_{ij}(t)$ be smooth functions of a real parameter $t$, such that $\mathbb M(t) = (M_{ij}(t))$ is a $n\times m$ real matrix. By smooth I mean that $M_{ij}(t)$ is continuous and differentiable. ...
a06e's user avatar
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1 vote
2 answers
104 views

How Can I solve this problem on Hopf Map?

I want to solve this: Let $\pi : S^3\to S^2$ be the Hopf map. Show that for all $a\in (-1,1)$, the pre-image $\pi ^{-1} (Z_a)\subset S_3$ is the common solution set of the equations $|z|^2 + |w|^2 = 1$...
Mr Prof's user avatar
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0 answers
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If $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0,$ let $g(t)=\frac{f(t)}{t},$ will $g$ be in $C^{\infty}(\mathbb{R})$?

Suppose $f\in C^{\infty}(\mathbb{R})$ satisfies $f(0)=0$ (and $f^{\prime}(0)\ne 0,$ I don't know whether this condition is necessary or not so I put it in parentheses), define $g:\mathbb{R}\to\mathbb{...
Tiffany's user avatar
  • 572
0 votes
1 answer
38 views

Uniform convergence and extending a smooth function

I am currently reading this paper about extending a smooth function and I'm a bit confused. Here is the context. I am trying to prove the following theorem: Theorem. Let $f:\mathbb R^+ \rightarrow \...
Joseph Kwong's user avatar
0 votes
1 answer
46 views

Are all multivariate elementary functions $C^\infty$ on any open ball for which they are defined?

Are all multivariate elementary functions $C^\infty$ on any open ball for which they are defined? I believe they are, and can find no counterexample, but do not know how to prove this. (All the ...
SRobertJames's user avatar
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0 answers
60 views

Let $f: \mathbb C \to \mathbb C$ be a proper smooth map with continuous extension $F : \mathbb C P^1 \to \mathbb C P^1$. Must $F$ be smooth?

We know that, if $p : \mathbb C \to \mathbb C$ is a nontrivial polynomial, then the function $P: \mathbb C P^1 \to \mathbb C P^1$ defined as $P([z:1]) = [p(z):1]$ and $P([1:0]) = [1:0]$ is a smooth ...
Squirrel-Power's user avatar
0 votes
1 answer
47 views

Let $F: \mathbb R^m \to \mathbb R^n$ be smooth, $F(0) = 0$. Let $H(t,x) = \frac{F(tx)}{t}$ if $t \neq 0$ and $DF|_0(x)$ otherwise. Show $H$ is smooth

This question is taken from my differential topology lecture notes. Let $F : \mathbb R^m \to \mathbb R^n$ be a smooth map such that $F(0) = 0$ and define $H: \mathbb R^{m+1} \to \mathbb R^n$ as $$H(t,...
Squirrel-Power's user avatar
2 votes
1 answer
71 views

Let $F: \mathbb R^n \to \mathbb R^m$ be smooth and such that for all $a \in \mathbb R$, $x \in \mathbb R^n$, $F(ax) = aF(x)$. Prove $F$ is linear

Let $F: \mathbb R^n \to \mathbb R^m$ be smooth and such that for all $a \in \mathbb R$, $x \in \mathbb R^n$, $F(ax) = aF(x)$. Prove $F$ is linear. I am sorry to say this, but I am really stuck on ...
Squirrel-Power's user avatar
1 vote
1 answer
28 views

Restricted Strong Smoothness vs Strong Smoothness

This paper defines the Restricted Strong Smoothness as follows: A differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ is said to be Restricted Strongly Smooth (RSS) with modulus $L_s>0$ or is $...
Saeed's user avatar
  • 175
1 vote
0 answers
72 views

Smoothness of Fourier transform

I am trying to understand, because the Fourier transform of the function $f(x) = e^{ -\sqrt{ \lvert x \rvert } }$ is smooth. My question: Under which conditions is the Fourier transform of an $L^1$ or ...
TheFibonacciEffect's user avatar
0 votes
1 answer
50 views

Can smooth approximations always preserve injectivity?

Most generaly, does every continuous injective mapping $f:M\rightarrow N$ (these are smooth manifolds) have a smooth and regular injective approximation of arbitrary precision, given that $\dim(N)>\...
cnikbesku's user avatar
  • 383
3 votes
1 answer
53 views

Question about external derivative

I am new on computing external derivative, wedge product and pull-backs so I am having issues to understund some things about those things. For example, an excercise of my class notes is to prove that ...
Superdivinidad's user avatar
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0 answers
55 views

How to calculate the derivative of composition map on smooth manifold?

Given a smooth manifold $N$ of 4 dimension. By smooth manifold here, we mean that $N$ is a manifold with a smooth atlas $\,\mathcal A=\Big\{\big(U_i,\varphi_i:\, U_i\to\mathbb R^4\big) \big|\,i\in I \...
PermQi's user avatar
  • 507
1 vote
0 answers
52 views

Convergence of a series of the product of two functions when one of them is shifted by one

Suppose $f(x)$ is a smooth function and $g_n(x)$ is a sequence of functions such that $$ \sum_{n=1}^\infty f^{(n)}(x) g_n(x) $$ converges for $x \in [a,b]$, where $f^{(n)}(x)$ denotes the $n$-th ...
Megatron's user avatar
  • 111
0 votes
1 answer
107 views

Showing that the isomorphism of the general linear group of a vector space with the group of invertible matrices is smooth

This is Example 7.3(e) from John Lee's Introduction to Smooth Manifolds. If $V$ is any real or complex vector space, $GL(V)$ denotes the set of invertible linear maps from $V$ to itself. It is a group ...
nomadicmathematician's user avatar
3 votes
1 answer
92 views

Simple proof for complete, metric vector space

I have the vector space $C_c^\infty([a, b])=\{f \in C_c^\infty : supp(f) \subset [a, b] \}$ and a metric on this space with $d(\varphi, \psi)=\sum_{n=0}^{\infty} 2^{-n} \frac{\lVert \varphi-\psi \...
Lopsio's user avatar
  • 65
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0 answers
78 views

I want you to check if I understand the case 1 in the proof of Theorem 3-11. (Partitions of unity in "Calculus on Manifolds" by Michael Spivak)

I am reading "Calculus on Manifolds" by Michael Spivak. I am now reading the case 1 in the proof of Theorem 3-11 about partitions of unity. Unfortunately I could not understand the case 1 in ...
佐武五郎's user avatar
0 votes
1 answer
30 views

What does "by (2) for each $x$ this sum is finite in some open set containing $x$" mean? ("Calculus on Manifolds" by Michael Spivak.)

3-11 Theorem. Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then there is a collection $\Phi$ of $C^\infty$ functions $\varphi$ defined in an open set containing $A$, with the ...
佐武五郎's user avatar
1 vote
1 answer
89 views

It is very easy to find a $C^\infty$ partition of unity for $A$. Is this really true? I feel strange. "Calculus on Manifolds" by Michael Spivak.

The following definition (theorem) is from "Calculus on Manifolds" by Michael Spivak. Let $W$ be an arbitrary open set containing $A$. Is $\Phi:=\{1:W\ni x\mapsto 1\in\mathbb{R}\}$ a $C^\...
佐武五郎's user avatar
2 votes
1 answer
78 views

Can a function be differentiable but not strongly differentiable (Knuth)?

Donald Knuth defined $f$ is strongly differentiable at $x$ if $$f(x+\epsilon) = f(x) + \epsilon f'(x) + \mathcal{O}(\epsilon^2)$$ for sufficiently small $\epsilon$. What differentiable functions are ...
SRobertJames's user avatar
  • 4,360
1 vote
2 answers
65 views

Smooth convex function with unique minimizer is not strictly convex?

Is smooth convex function with unique minimizer strictly convex? Here the strictly convexity of a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ means $$f((1-\lambda)x+\lambda y)<(1-\lambda)f(x)+\...
Dat Ba Tran's user avatar
0 votes
1 answer
35 views

Differential of a map $\Gamma : M \times N \to P$

I have a smooth map $\Gamma : X\times Y \to Z$ between manifolds with or without boundary, and I am interested in the maps $\Gamma_y : X \to Z$ given by $\Gamma_y(x) = \Gamma(x,y)$. For a given $y\in ...
Nick F's user avatar
  • 1,219
0 votes
0 answers
34 views

When is $f(z)$ analytic in the neigbourhood of $z$ for some complex $z$ close to the real line?

Let $f(x)$ be real-valued, 1-periodic, bounded and $C^{\infty}$ for real $x$. When is $f(z)$ analytic in the neigbourhood of $z$ for some complex $z$ close to the real line ? Notice this is similar to ...
mick's user avatar
  • 16k
2 votes
0 answers
52 views

Is the dual of the space of semi-regular distributions complete

Consider $C^\infty(X)$, equipped with its Fr'echet structure, and $\mathcal D'(Y)$, equipped with the strong topology, where $X,Y$ are compact manifolds. Is the strong dual of $C^\infty(X)\hat\otimes \...
Or Kedar's user avatar
  • 921
1 vote
1 answer
83 views

Pullback of a Partition of Unity

I'm trying to prove that the collection of function $\{F^*\rho_\alpha\}$ is a partition of unity on $N$ subordinate to the open cover $\{F^{-1}(U_\alpha)\}$ of $N$. Here $\{\rho_\alpha\}$ is a ...
一団和気's user avatar

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