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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36 views

On boundary values of ${\cal C}^\infty_c$ functions

The issue is about defining compact-supported smooth functions over the real semi-axis, such that they are null at the origin. My professor said it can be done in two equivalent ways: $f \in {\cal C}^...
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41 views

Show that $f^{(n)} (0) = 0,$ for all $n \geq 1.$ [duplicate]

Define a function $f : \mathbb R \longrightarrow \mathbb R$ by $$f(x)=\begin{cases}e^{-1/x^2}&x\neq 0\\ 0&x=0 \end{cases}$$ Show that $f^{(n)} (0) = 0,$ for all $n \geq 1.$ I can prove that $...
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44 views

How can I smoothly transform one periodic function into another if the period time is allowed to differ?

How can I smoothly transform one periodic function into another if the period time is allowed to differ? Let us visit the very simplest example I can think of : Two audio signals pure sine waves. Note ...
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How can we construct a compactly supported function which is equal to $1$ on a given interval?

I know that for any open cover $(\Omega_i)_{i\in I}$ of $\mathbb R$ we can find a $C^\infty$-partition of unity subordinated to $(\Omega_i)_{i\in I}$. Moreover, if $\eta$ is a mollifying kernel$^1$ on ...
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Can two functions with this property intersect at a “bad” set of points?

A groud set of $[0, 1]$ is a set $A \subseteq [0, 1]$ such that $A$ can be written as a finite union of open intervals and single points. For example, $[1/3, 1/3]$ is a groud set since it is equal to $...
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25 views

Smoothness of parameter-dependent integral implies smoothness of integrand

Let $f:\mathbb{R}^d\times\mathbb{R}^n\to\mathbb{R}$ be continuous. Suppose that $$ x\mapsto\int_{\mathbb{R}^n}\varphi(y)f(x,y)\,dy $$ is continuously differentiable for each $\varphi\in\mathcal{C}_c^\...
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Smoothing function to combine a polynomial and a constant

click to see the image I have below function: $y=0$ for $x<=0.12$ and $y=80((x/0.12)^8-1)$ for $x>0.12$ I need to smooth out the transition between these two functions. Can I find out the best ...
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On analytic test functions

In what follows, ${\cal C}^\infty_c(U)$ is intended to be the set of all smooth functions with compact support defined on a non-empty open subset $U \subseteq \mathbb R^n$. Wikipedia: There $[\dots]$ ...
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92 views

Bound on a certain difference of expectations involving Lipschitz functions

Suppose $X$ is a random variable with full support on the interval $[-1,1]$, with $\mathbb{E}[X]=0$. (You may assume the existence of a p.d.f. for $X$ if it helps.) For any $f:[-1,1]\rightarrow\mathbb{...
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doubt about smooth curves and pushforward

I'd like to clarify two things about pushforwards and smooth curves. $$\textbf{Question 1}$$ Suppose we have a smooth function $F:M\to N$, where $M$ and $N$ are two differentiable manifolds of ...
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64 views

Is ${\cal C}^\infty$ complete?

Given $a \in \mathbb R \backslash \{ 0 \}$, is the metric space $ X \equiv\{ f \in {\cal C}^{\infty} ([-a, a]) \}$ equipped with the uniform norm $||.||_\infty \displaystyle := \underset {X}{\sup}|f| $...
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Can any coordinate function be completed into an orthogonal coordinates system?

Let $U$ be an open neighbuorhood $U \subset \mathbb{R}^2$, and let $g$ be a smooth Riemannian metric on $U$. Let $x$ be a smooth function on $U$, with nonvanishing derivative $dx \neq 0$. Let $p \in U$...
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On the existence of homotopy between continuous and differentiable [duplicate]

Let be $M$ and $N$ smooth manifolds. Take $f\colon M\rightarrow N$ a continuous maps, so exists smooth $g$ from $M$ to $N$ such that $g$ is homotope to $f$, how can it proved?
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Extrema of arbitrary smooth functions in $L^2$

This isn't a very precisely worded question, but here goes: Is there a sense in which generic smooth functions in $L^2$ will have multiple maxima? Do arbitrary smooth functions in $L^2$ almost always ...
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Approximating $C^2(\mathbb{R}^d;\mathbb{R}) \cap C^1_b(\mathbb{R}^d;\mathbb{R})$ by $C^2_b(\mathbb{R}^d;\mathbb{R})$?

Suppose $f:\mathbb{R}^d \to \mathbb{R}$ is a twice continuously differentiable function with bounded first derivative. Can we approximate $f$ with a sequence $(f_n)_{n\in\mathbb{N}}$ such that all $...
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Surjective, open and submersion all have in common: image has non-empty interior. What about injective and immersion?

From these: Does open map imply dimension of domain is greater than or equal to dimension of range? (please don't use sard's theorem) Dimension of domain is greater than/less than/equal to ...
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Is this function smooth?

I want to show that this function is smooth (i.e. $f \in C^\infty[\mathbb{R}^2,\mathbb{R}]$) but I am unsure how. Is the below smooth? $f(x,y)=\left(x^{2}+y^{2}\right)^{0.5}\cdot\left(\cos\left(10\...
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32 views

Do progressively more differentiable functions (once differentiable, then $C^1$, then twice differentiable, etc.) form an infinite nested hierarchy?

I know that the Baire category theorem can be used to show that The space of functions $\mathbb{R} \to \mathbb{R}$ differentiable at at least one point is a meager subset of the space of all ...
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1answer
51 views

If $x^\alpha f$ is in $L^1$, for some $\alpha>0$, then $f$ vanishes at infinity.

How I got to the question: In Folland p.249, Theorem 8.22, there is a list of properties about the Fourier Transform on $\mathbb{R}^n$. Letter $d$ property says If $f$ and $x^\alpha f$ are in $L^1$ ...
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1answer
63 views

Can one or more element of a partition of unity be identically zero?

Probably a silly or obvious question! But for the sake of clarity, let's take the open subset of $\mathbb{R}$ $(-1,1)$ and its open cover given by all the open sets $U_a = (-a,a)$ with $0<a<1$. ...
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Define parameters to quantify smoothness of the curves

In the picture, the two short curves seem to have more and sharper peaks than the longest one. The longest curve still had some waggles at the end of stage. Could anyone share some ideas about ...
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1answer
37 views

Tangent vector to a curve $\gamma:(-\pi,\pi)\to \mathbb P^2(\mathbb R)$

Let $\gamma: (-\pi,\pi)\to \mathbb P^2(\mathbb R)$ such that $\gamma(t)=(1:\sin t:\cos t)$. I have to find the tangent vector $v\in T_{p_o} \mathbb P^2(\mathbb R)$ to $\Gamma=\gamma((-\pi,\pi))$ with $...
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Name for Set Of Functions Smoothly Interpolating Between Points

Suppose I have a continuous differentiable surface such as the following: $$f(x,z)=\sin(e^{2\pi+x^2+z^2}), \space\space\space x,z\in\mathbb{R}$$ As $x$ and $z$ become large, the function oscillates ...
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30 views

Extending smooth functions on submanifolds

If $M$ is a smooth manifold and $S\subset M$ is an embedded submanifold that is not closed in $M$, then can we construct a smooth function $f\in C^{\infty}(M)$ that is non-zero everywhere on $S$ and ...
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163 views

Does open map imply dimension of domain is greater than or equal to dimension of range? (please don't use sard's theorem)

Note 1: same question as this, but please don't use sard's theorem or anything similar. preferably, no measure theory. Note 2: Didier's comment: I think another proof can use the constant rank ...
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1answer
66 views

Does open map imply dimension of domain is greater than or equal to dimension of range?

Update: Answer given uses Sard's Theorem. For no Sard's Theorem, see here: Does open map imply dimension of domain is greater than or equal to dimension of range? (don't use sard's theorem) ...
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1answer
61 views

Doubt about pushforwards

Given the smooth function $\psi:M\to N$, where $M$ and $N$ are two differentiable manifolds, we have introduced the pushforward map $\psi_{*,p}:T_pM\to T_{\psi(p)}N$ with $p\in M$ and the image of ...
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A weak notion of local convexity of the objective function for feasable points satisfying suficient condition

Let us suppose that does hold the sufficient condition for the minimization problem \begin{equation*} \begin{array}{c c} \text{minimize}_{x\in\mathbb{R}^n} & f(x) \\ \text{subject to} & \...
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Prove that $ \left\vert u\left( x,t\right) \right\vert \leq\frac{C}{t} $

Let $u$ be a solution to \begin{align*} u_{tt} & =\Delta u\text{, }x\in\mathbb{R}^{3}\text{, }t>0,\\ u\left( x,0\right) & =g\left( x\right) ,u_{t}\left( x,0\right) =h\left( x\right)...
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1answer
165 views

Proving that inverse of a smooth function is smooth

Suppose I have a smooth function $g: \mathbb{R}^n \to \mathbb{R}^t$ and write the variables as $(x,y)$ where $x \in \mathbb{R}^t$. Suppose the Jacobian matrix of $g(\cdot, y)$ is invertible at $y = 0$ ...
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Finite group acting on a smooth manifold $M$ by diffeomorphisms. Fixed Point set is Smooth Manifold [duplicate]

Let $M$ be a smooth manifold and let $G$ be a finite group acting on $M$ by diffeomorphisms. Show that the set of fixed points $M^G := \{m \in M \mid g . m = m, \forall g \in G\}$ is a smooth manifold ...
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Question regarding “affinely radial test functions”

We shall call the function $f \colon \mathbb{R}^n \to \mathbb{R}$ affinely radial, if there exists $x \in \mathbb{R}^n$ such that the value of $f$ at a given point depends solely on the distance of ...
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1answer
28 views

Smooth approximations to the ramp function

I am looking for $C^1$ and $C^2$ continuous approximations to the ramp function, f(x), that satisfy the condition $f(x)=0$, $x\leq0$ (essentially smoothing out the discontinuity in the first ...
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How to smooth out 2 corners in a piecewise function?

I have been experimenting with this a lot, but it eventually proves itself to be trickier than I expected. Let \begin{equation} f(z) = \begin{cases} 1, &z<0 \\ \frac{1}{2}...
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36 views

Invariance of rank under diffeomorphism

So I am kind of confused by this: We have $U \subset \mathrm{R}^m$ and $V \subset \mathrm{R}^n$ as open sets. If $f: U \rightarrow V$ is a diffeomorphism then we essentially have $m=n$. Same holds ...
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1answer
65 views

Embedding Real Projective Space in Sphere

Let $n > 1$ be an integer and set $N =\frac{n(n+1) }{2}$. Let $F : R^n \setminus \{0\} → R^N$ be defined by $(x_i)_i \to (x_ix_j)_{i \leq j}$ , where the $x_i x_j$ are ordered lexicographically. (...
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1answer
34 views

Smooth Map between the smooth manifolds $GL_n(\mathbb{R})$ and $S_n(\mathbb{R})$

First of all I am sorry for the bad notation, but I shall denote the smooth manifold of real symmetric $n \times n$ matrices by $S_n(\mathbb{R})$. I shall denote the transpose of a matrix $A$ by $A^T$ ...
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1answer
26 views

Proof involving the sum of $L-$smooth functions

Consider a set of functions $\{f_i\}_{i=1}^n$ where each $f_i$ is $L_i$ smooth, where $L_i$ will be the smallest possible number such that $$\|\nabla f(x) -\nabla f(y) \| \leq L_i \|x-y\|$$ for any $x,...
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Doubt in understanding Differentials

I am studying smooth manifolds, and recently I got to know about tangent spaces and the differentials. Suppose $f:M \to N$ is a smooth map, then it induces a differential between the tangent spaces $...
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Is the set $S$ of all smooth functions on $B_1$ whose normal derivative vanishes along the boundary $\partial B_1$ dense in $C^0 (B_1, \Bbb C)\ $?

Let $B_1 : = \{z \in \Bbb C\ |\ |z| \leq 1 \},$ and let $C^0(B_1,\Bbb C)$ be the space of continuous complex valued functions on $B_1$ equipped with the uniform convergence topology. Let $S$ be the ...
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1answer
24 views

Diffeomorphism of $\mathbb{R}^2$ with itself determined by three pairings of points

The narrow version of the question is whether given distinct points $a,b,c,c' \in \mathbb{R}^2$ there exists a diffeomorphism $\mathbb{R}^2 \to \mathbb{R}^2$ such that $a\mapsto a,\ b\mapsto b,\ c\...
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1answer
30 views

Proof of continuity of a map from product of two spaces to $\mathbb R^n.$

$\mathbf {The \ Problem \ is}:$ Given , $U$ is an open subset of $\mathbb R^{m+n}$ and $f : U \to \mathbb R^n$ be a $C^1$ map ,then show that the map $F : (p,v) (\in U ×\mathbb R^{m+n}) \mapsto Df_p (...
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Prove that an injective immersion of closed manifolds is an embedding

Definition: An embedding of manifolds is a differentiable map $f:N^n\hookrightarrow M^m$ which has an injective pushforward. An immersion of manifolds $f:N^n\looparrowright M^m$ is a differentiable ...
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Show that if $f $ is $L$-smooth, then $g(x) := f(x) - \frac{m}{2} \Vert x \Vert^2$ is $(L-m)$-smooth

A continuously differentiable function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is $L$-smooth if $\nabla f$ is $L$-Lipschitz, i.e., for all $x,y \in \mathbf{dom}\,f$, $$ \Vert \nabla f(x) - \nabla f(...
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1answer
28 views

Lagrange Remainder the taylor expansion of $e^{-1/x^2}$

I know $e^{-1/x^2}$ and $f(x) = 0$ if $x = 0$, is an example of a smooth function that cannot be represented all the coefficients are $0$, but I am curious about the Lagrange remainder. Doesn't the ...
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67 views

When a distribution on $\mathbb{R}^n$ is smooth of compact support

The following statement is from the book " The analysis of linear partial diffential operators I by Lars Hörmander page 252 If $v \in \mathcal{E}'(\mathbb{R^n})$ ( $v$ is a distribution on $\...
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2answers
38 views

Prove that if $\int_0^1 f(yx) dx = g(y)$, so $f(x)=g(x)+xg'(x)$ for any $x$.

I'm trying to solve this question: $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and $g: \mathbb{R} \to \mathbb{R}$ is of $C^1$ class, such that $g(y) = \int_0^1 f(yx) dx$. Prove that $f(x)=...
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28 views

A smooth action of a Lie group on a smooth manifold. Circle Example

I want to check that I understand this definition correctly and that my example makes sense. Let $\varphi:G\to Diff(M)$ be a action of a Lie group $G$ on a smooth manifold $M$. We say that $\varphi$ ...
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1answer
44 views

Q: How to smooth out a corner line in a piecewise function of 2 variables?

Let \begin{equation}h(x,y)= \begin{cases} 0, x \leq 0 \\ x(10-5y), x>0 \end{cases}\end{equation} be a piecewise smooth funcion, that is continuous everywhere, and smooth everywhere except at $(0,y)...
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21 views

**Smooth** Vs **Regular** function.

What is the difference between a smooth and a regular function $f:\mathbb{R^n} \rightarrow \mathbb{R}$? Or in other way, can you please post both definitions?

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