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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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1answer
86 views

Find linear transformation whose kernel is given

Question: given $V=C^∞(-∞,∞)$ i.e the vector space of real-valued continuous functions with continuous derivatives of all orders on $(-∞,∞)$ and $W=F(-∞,∞)$ the vector space of real-valued functions ...
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18 views

Constructing a uniform convergence sequence

Let $f : \overline{\Omega} \subset \mathbb{R}^{N} \to \mathbb{R}$ be a $C^{2}(\overline{\Omega})$ function. Can we always construct a sequence $f_{n}$ such that $f_{n} \to f$ uniformly in $\overline{\...
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1answer
71 views

How to check the smoothness of the following function?

I was solving a problem which boiled down to checking the smoothness of the following function. $$ \tilde{p} (z) = \begin{cases} \frac{1}{p(1/z)} &\quad z\neq 0 \\ ...
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33 views

Properties of Complex Function $f(z)=e^{-\frac{1}{z^{1/3}}}$

This post will be about a part of an example from my complex analysis book. Problem: They claim that there exist a function $J_+(z)$ holomorphic in the upper half plane $\operatorname{im}z>0$, ...
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0answers
38 views

Smooth extension of function on $\mathbb{R}^p$

I have the following specific problem: For an application I use the following function $g : \Bbb R^{Nd} \to [0,\infty)$, $$(x^1, \ldots, x^N) \mapsto \begin{cases} \prod\limits_{\substack{(a,b)\in\{...
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2answers
71 views

$\partial U$ is a smooth curve (i.e., the image of a smooth path $\mathbf{c}$ with $\mathbf{c}' \not= 0$)?

In a section discussing global maxima/minima, my textbook says the following: Simply stated, $\mathbf{x}_0$ and $\mathbf{x}_1$ are points where $f$ assumes its largest and smallest values. As ...
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18 views

Construct a $C^2$-continuous function for symmetric weighting

Let $\alpha>1$. I am looking for a $C^2$-continuous function $w:[0,\infty)\to\mathbb{R}$ which satisfies the following: \begin{align*} w(x)=1, &\ \ \ \ \ \ x\in[0,\frac{1}{\alpha}],\\ w(x)=1-w(...
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1answer
40 views

Proof that the product of locally integrable function $g$ with any test function is zero implies $g=0$ a.e

How to prove that If for $g\in L_{p}$ $p\in(1,2]$ and for all $\phi \in C_{0}^{\infty}$ we have $\int_{\mathbb{R}} g \phi d\mu =0 $ then $g=0$ a.e where $\mu $ is the Lebesgue measure. I was ...
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20 views

What conditions can the shape parameters satisfy to make a generalized Gaussian function a smooth function?

Generalized Gaussian function (GGF): $$g(x_j;\mu,\theta,q):=\frac{q}{2\theta\Gamma(1/q)}\exp\left\{-(\frac{|x_j-\mu|}{\theta})^q\right\},$$ where $\mu, \theta>0, q>0$ are mean, scale parameter ...
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11 views

Extending function with local maximum in a bounded smooth domain.

Let $\Omega \subset \mathbb{R}^{N}$ be a bounded $C^{2}$-domain. Take $V$ as a subset of $\Omega$ and define a smooth function $\phi$ such that $\phi \in C^{2}(\overline{V})$ and attains a local ...
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1answer
83 views

Decreasing smooth approximation of an indicator function with bounded derivative

Consider an indicator function $\chi_{[a,b]}$ on an interval $[a,b]$. My question - is there a decreasing sequence $f_\delta\in C^1(V)$ (where $[a,b]\subset V$ and V is open), such that $f_\delta\...
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1answer
39 views

Can I find an infinitely differentiable function of of bounded moments closest to triangle wave?

Based on this question regarding existance of closest function in Schwarz class, where answer was negative. What if we add a new constraint. Not only infinitely differentiable compact support but with ...
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2answers
42 views

Which function in the Schwarz class of functions is “closest” to triangle wave in $L^2$ sense?

Would it be possible to calculate which function in the Schwarz class of infinitely differentiable functions with compact support is closest to triangle wave? Let us measure closeness as $$<f-g,f-...
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50 views

How to show that $\exists$ $\phi\in C_c^{\infty}(\Omega)$ s.t. $\int_{\Omega_1}D\phi(x)dx$ has unit norm? [duplicate]

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 $\...
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1answer
59 views

A more wide class than smooth functions

I will call a function $f$ almost smooth when for every point $a$ there are numbers $y_0$, $y_1$, $y_2$ ... $$f(x) = y_0 + \frac{y_1}{1!}(x-a) + \dots + \frac{y_n}{n!}(x-a) ^n + O((x-a)^{n+1})$$ for ...
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55 views

Prove that a function is smooth easily

In this answer I claimed that the function $f (x) = x^{n + 1} \sin \exp \frac{1}{x}$ is smooth in all points except zero. But how to prove this claim easily (without calculating the derivative ...
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1answer
78 views

$C^n$ function with big/small O notation

Can the statement that $f$ is a $C^n$ function (say for $\mathbb{R}\to\mathbb{R}$ functions) be written in terms of big/small O? Maybe it is equivalent to existence of numbers $f(a),\dots,f^{(n)}(a)$ ...
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1answer
57 views

Function that are not smooth because $f(U) \not \subset V$

In the book of Lee (page 35 of second edition) on smooth manifolds, for a map $f : M \rightarrow N$ between manifolds to be smooth, we have to prove in particular that for each point p in $M$ there is ...
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1answer
54 views

Sigmoid functions similar to atan but with different upper and lower bound

Do there exist smooth sigmoid functions similar to atan but with different upper and lower bounds? This function must go through origin. Something like on the picture: Function If not, why? Where can ...
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1answer
55 views

existence of a smooth function with bounded derivative

I am studying a proof in a differential geometry book where the author use a a smooth function satisfying some properties. Here is what it is claimed. $\forall 0 < \epsilon < \pi$ there exists a ...
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1answer
47 views

Smooth Urysohn function having $0$ as regular value

Let $M$ be a manifold and $C_0,C_1$ be two disjoint closed subsets of $M$ , then smooth Urysohn Lemma says that there exists a smooth function $f:M\rightarrow [0,1]$ such that $f(C_0)=\{0\},f(C_1)=\{1\...
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1answer
143 views

Smooth and not analytic

Can someone show me, without reference to Taylor series, why a complex function can be smooth but not analytic? I do not understand it intuitively or visually either. I would like an explanation ...
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1answer
51 views

Proof verification on the openness of the space of matrices of full rank

Let be $m<n$ and $M_m(m\times n, \mathbb{R})$ be the set of the matrices $m \times n$ of full rank $m$. I want to show that $M_m(m\times n,\mathbb{R})$ is an open subset of $M(m\times n,\mathbb{R}...
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1answer
181 views

Existence of a smooth map from the Circle to the Square

I know that there is no diffeomorphism from the unit circle, $S^1$ to the square of side length 2 centered at 0. However, can we construct a bijective map from $f : \mathbb{R}^2 \rightarrow \mathbb{R}...
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0answers
79 views

C^infty approximation of Heaviside step function: definite integral of a bump function

I'm wondering if there is an analytical form for a $C^\infty$ approximation with a compact support of a Heaviside step function $f(x) = I_{x \geq 0}$. In attempting to construct one, I'm taking a bump ...
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1answer
42 views

Existence of a smooth extension vanishes at infinity

in my research I saw this argument used in an article and I don't know if it's true (I believe yes). Let $\Omega \subset \mathbb{R}^3$ be an open, bounded and simply connected domain and let $\...
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1answer
36 views

Choosing Coordinate Charts — Smooth Functions on Manifolds

I'm trying to prove the following claim from Lee's Intro to Smooth Manifolds. $\textbf{Claim:}$ For smooth manifolds M and N, show that $F:M \rightarrow N$ is smooth if and only if $F^*(C^{\infty}(N))...
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2answers
103 views

Differential of a smooth map over a manifold

So the definition of differential of a smooth map between manifolds, $f:M\longrightarrow N$, at a point $p\in M$ is: $f_{*p}:T_{p}M\longrightarrow T_{f(p)}N$, where $f_{*p}(X)\in T_{f(p)}N$ and for ...
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3answers
87 views

if $d_f \neq 0$ then f is a submersion

I often see in my course of differential geometry that $d_f$ the differential of a smooth function $f$ is sumersive iff $d_f \neq 0$. Is this a true statement, if yes, how could we prove that? I ...
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79 views

Derivations and the tangent space on $\mathbb{S}^2$

Let $\mathbb{S}^2 = {(x, y, z) \in \mathbb{R}^3 |x^2 + y^2 + z^2 = 1}$ be the unit sphere, and let $p \in \mathbb{S}^2$. Let also $D$ be a 1’st order differential operator on $\mathbb{S}^2$ at $p$, i....
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46 views

Show that rational function of $C^k$ polynomials is $C^k$

Let $E \subseteq \mathbb{R}^n$ be an open set. Suppose $$P,Q: E \to \mathbb{R}$$ are polynomial functions in several variables. For example $$P(x,y) = x^2y + x + y^2 +1$$ Suppose $Q$ ...
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2answers
92 views

Show that inversion is a smooth operator

Consider the operator $$I: GL_n \to Gl_n : A \mapsto A^{-1}$$ which sends an invertible matrix to its inverse. Identifying $Mat_n \cong \mathbb{R}^{n^2}$ through the map $$\phi:A ...
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1answer
135 views

Inverse function theorem: derive stronger form

Let $E \subseteq \mathbb{R}^n$ be an open set, and $f: E \subseteq \mathbb{R}^n \to \mathbb{R}^n$ be a $C^r$ function. If $Df(x)$ is invertible for some $x \in E$, there are open sets $U,V$ of $...
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1answer
57 views

Confused by conditions for smoothness for a complex curve. Tangent exists vs smoothness

Dealing with curves in the complex plane... curves of the form $$z(t) = x(t) + iy(t)$$ I'm looking at page 2 of the notes here. The conditions for smooth curves. https://sites.oxy.edu/ron/math/312/...
2
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1answer
73 views

Is it possible $\cos \phi (s)$ and $\sin \phi (s)$ to be nth order differentiable and not $\phi (s)$?

Pressley's Differential Geometry book supposes that the angle between the tangent vector to a curve and any fixed vector is smooth (i.e. all orders of differentiablity exits and are continuous) and in ...
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1answer
53 views

smoothness of a map that comes from a smooth submersion

Let $f:N\rightarrow M$ be a surjective submersion. Let $g:W\rightarrow M$ be a smooth map. Let $w\in W$. We have $g(w)\in M$. As $f$ is surjective, there exists $n\in N$ such that $f(n)=g(w)$. ...
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35 views

Smooth but not convex

In Theorem 4.2. of the following lecture http://users.ece.utexas.edu/~cmcaram/EE381V_2012F/Lecture_4_Scribe_Notes.final.pdf it is shown that when the objective function is smooth and not necessarily ...
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2answers
145 views

If a composition of functions is smooth and one of them is smooth, then the other is smooth

Show that a map $\xi$ between smooth manifolds $M$ and $N$ is smooth if and only if $f ◦\xi$ is a smooth function on $M$ whenever $f$ is a smooth function on $N$. One implication is clear because I ...
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2answers
66 views

Clarification on definition of smooth map between smooth manifolds

Let $M$ and $N$ be $n$-dimensional smooth manifolds. A map $F: M \to N$ is smooth if for each $p \in M$ there exists smooth charts $(U, \varphi)$ containing $p$ and $(V, \psi)$ containing $F(p)$ such ...
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2answers
60 views

How do I show that this set is dense in the function space?

Consider the smooth manifold $\mathbb{S}^3$ embedded in $\mathbb{R}^4$, note that $$\widetilde{T}:= \frac{1}{\sqrt{2}}\mathbb{T}^2 = \left\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4; \ x_1^2 + x_2^2 = x_3^2 ...
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1answer
27 views

Example of not quasi-regular mapping

Example : (1) If $f : \mathbb{R}^2\rightarrow \mathbb{R}^2,\ f(x)=C\cdot x$ is dilation, then it is bi-Lipschitz map. (2) More generally, we consider an inversion $f:\mathbb{R}^2-\{o\}\rightarrow \...
2
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1answer
66 views

How to show a real valued function of several variables is analytic?

Let $f:\Omega\subset\Bbb{R}^m\to\Bbb{R}^n$ be a given function. In general, how can I show that $f$ is smooth (infinitely differentiable) and analytic. I know this is a bit vaguely stated question, ...
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0answers
71 views

A Smooth Function with Peculiar Properties

Suppose $f:\mathbb R^n\to\mathbb R$ is a smooth function and define $E_f := \{x\in\mathbb R^n\;|\;f(x)=0\text{ and }\nabla f(x)\ne 0\}$. Can we find $f$ such that $E_f$ has positive $n$-dimensional ...
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0answers
43 views

For what functions is $\lim_{n\to \infty}|f^{(n)}(x)|=0$? (Where $f^{(n)}(x)$ is the $n$th derivative of $f$)

I recently came across a formula that involved the assumption that $\lim_{n\to \infty}|f^{(n)}(x)|=0$, and at first I thought that it would be true for every function, but then I remembered that ...
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1answer
98 views

Several Questions on Smooth Urysohn's Lemma

The smooth version of Urysohn's Lemma is Let $A,B$ be two disjoint closed subsets in $\mathbb{R}^n$ with one of them compact, then there exists a smooth function $f: \mathbb{R}^n\to [0,1]$ such ...
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2answers
48 views

Straight Lines are Strict Minimizers of Arclength in Euclidean Space

Let $a$ and $b$ be two points in an euclidean space $\mathbb{R^n}$ Then,I denote by $\Omega(a,b)$ set of all maps $\gamma : [0,1] \to \mathbb{R}^n$ such that $\gamma(0) = a,\gamma(1) = b$ and the ...
2
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1answer
36 views

Find a sequence $(\phi_n)_n \subset C^{\infty}_c(\mathbb{R}^N)$ which converges in both $L^p(\nu)$ and $L^q(\mu)$ to $1_E$

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Cr}{C^{\infty}_c(\R^N)}$ Suppose we have two non-zero Borel measures on $\R^N$, labeled $\nu$ and $\mu$, and we have $1 \leq p, q < \infty$. Let $E \...
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2answers
106 views

Are charts for smooth manifolds homeomorphisms or diffeomorphisms?

I will link the following lecture notes, because it makes no sense to keep pasting from them. When reading them, there are two things I do not understand. The author introduces smooth manifolds by ...
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1answer
64 views

A weird definition of regular function

Consider the following definition, where immersions are defined: Later the author states: The problem: In the definition, a regular function is defined to have constant rank in particular. But later ...
6
votes
2answers
86 views

Does an integral inequality imply a pointwise inequality?

$\newcommand{\R}{\mathbb{R}}$ Let $(f_n)_n \subset L^1(\R^N)$. Suppose that for any nonnegative function $\phi \in C_c^{\infty}(\R^N)$, we have: $$ 0 \leq \liminf_{n \rightarrow \infty} \int f_n \, \...