2
votes
0answers
76 views

A Tricky Weak Derivatives question

I recently came across the following statement and am having trouble proving it correct. I wonder if you could help. Given a weak derivative, $x'$, there exists an absolutely continuous ...
5
votes
1answer
82 views

Derivative of $\frac{x}{f(x)}\frac{df}{dx}$

Suppose we have a function $f(x):\mathbb R^+\to\mathbb R^+$ that satisfies: 1) $0\leq\frac{df}{dx} \leq 1$ 2) $f(0) = 0$, then do we have $$\frac{d}{dx}\left(\frac{x}{f(x)}\frac{df}{dx} ...
0
votes
0answers
30 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
0
votes
1answer
39 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
4
votes
0answers
23 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
1
vote
1answer
54 views

Domain of densely-defined second derivative operator, and its factorization

Let $$-d_x^2: \{f \in L^2[0,1];f \in AC^1[0,1] , f(0)=f(1)\} \rightarrow L^2[0,1]$$ be the second derivative operator. Here $AC^1[0,1]$ is the space of functions whose first derivative is absolutely ...
1
vote
1answer
45 views

A Particular Frechet Derivative and Interpretation

I would like find the Fréchet derivative of the following functional: $$ \begin{align} F : C[0,1] &\rightarrow \mathbb{R}\\ w &\mapsto \frac{\int_0^1 xw(x)f(x) \, dx}{\int_0^1 w(x)f(x) \, dx}. ...
2
votes
1answer
29 views

Differentiability of product/composition of function

How will be the product and composition of two functions, where one is differentiable and another is just continuous, behave?I mean to say, if the product or composition is differentiable, then what ...
1
vote
0answers
32 views

Differentiate a log of $L^p$ norm, don't understand this result

I'm reading this paper. In it, the authors show this lemma: And then they prove this lemma My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log ...
0
votes
1answer
25 views

Check complex differentiability

I am trying to take a derivative w.r.t $z\in\mathbb{C}$ of the following map: $z\mapsto \sum_{j=0}^{\infty}\lambda_{j} (T(\psi+zh))_{j}$ where $(\lambda_{j})$ is a bounded sequence, $T$ is a ...
2
votes
1answer
80 views

Sobolev Spaces and Derivative

I need help on the problem 8.9 at page 238 of the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis. Set $I=(0,1)$. Let $u \in W^{2,p}(I)$ with ...
0
votes
0answers
20 views

Operator differentiability

I was wondering, what techniques can one use to prove that an operator (let's say acting on real analytic functions and taking values in a Banach space) is infinitely differentiable? I know that, for ...
1
vote
1answer
38 views

Revisiting the product rule for derivatives

Let $E=C^{\infty}(\mathbb R, \mathbb R)$ Consider a linear transformation on $E$: $\delta$ such that $\forall f, g \in E, \delta(fg) =g\delta(f) +f\delta(g)$ Prove that there is some ...
0
votes
2answers
47 views

Proving that the function f is constant, mean value theorem, derivatives

Having the following inequality, for a real-valued function $f$ which is twice differentiable: $f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$. and assuming that $f$ is bounded, ...
0
votes
1answer
31 views

derivate of indicator function

What is the derivative of the indicator function: \begin{equation} f(x)=\begin{cases} 1 & x^{\min} x\leq x^{\max}\\ -\infty &\mbox{otherwise}? \end{cases} \end{equation} thank you
4
votes
1answer
38 views

Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
1
vote
1answer
64 views

Frechet/Gateaux differentiability of an integral operator L^2 --> R

Let $f: R \rightarrow R$ be a continuously differentiable function on the real numbers (if needed also infinitely many often differentiable). Define the Operator $F : L^2([0,1]) \rightarrow R$ for $x ...
0
votes
0answers
74 views

k-times differentiable functions on [0,1]

Is $C^k[0,1]$ (the set of all k-times differentiable function, not necessarily continuously) complete with respect to the norm $\|f\|_\infty + \|f'\|_\infty +\cdots+\|f^{(k)}\|_\infty$? I know the ...
2
votes
1answer
129 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
3
votes
2answers
49 views

Examples of Functions

Alright so I am trying to find examples of functions that are differentiable at a point, but not continuous there. Also a function continuous at no point; a function continuous only at one point. ...
1
vote
1answer
45 views

A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
0
votes
2answers
72 views

One-sided total derivative

Given a function from half space into euclidean space: $f:\mathbb{H}^m\to\mathbb{R}^n$ Suppose its one-sided limit exists at a specific point: $\lim_{\mathbb{H}^m\owns v\to 0}\frac{1}{\lVert ...
4
votes
0answers
63 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
0
votes
0answers
42 views

Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
1
vote
1answer
53 views

Mean Value related problem.

I'm working on a function $f : \left[a,a+h\right] \rightarrow \mathbb{R}$. I know that $f$ satisfies the conditions of the Mean Value Theorem thus I have $\theta \in \left(0,1\right)$ such that ...
0
votes
2answers
68 views

Proof using Fermat's Theorem on stationary points?

It seems intuitive that, if a function differentiable on [a,b] is such that f'(a) < 0 < f'(b) then there exists some c in the open interval (a,b) such that f'(c)=0, but I can't prove it ...
1
vote
1answer
37 views

Prove a functional to be differentiable

Consider the functional $I:C[a,b]\rightarrow\mathbb{R}$ given by $$I(x)=\int_{b}^{a}x(t)dt.$$ Prove that $I$ is differentiable and find its derivative at $x_0\in C[a,b]$. The answer just says that it ...
0
votes
1answer
41 views

Example: Differentiable, NOT locally lipschitz?

Everybody Good Evening, I'm wondering: Is being differentiable (Gateaux or Frechet) at ONE point enough for being locally lipschitz? If not, can you provide a counterexample? Thx in advance!
2
votes
1answer
47 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
0
votes
1answer
59 views

What is the Frechet derivative of $(u^+)^q$?

I know that if we define $E[u]=\int_\Omega u^+dx$, where $\Omega$ is compact in $R^n$ and $u\in H_0^1(\Omega)$, $u^+:=\max\{u,0\}$, then $E[u]$ is not Frechet differentiable. However, if now I define ...
0
votes
1answer
26 views

Differentiability in $\mathbb R^n$

Let $U\in \mathbb{R}^n$ be open, and let $f:U\to \mathbb{R}^m$, and let $a\in U$. Let $\|\cdot\|'$ be a norm on $\mathbb{R}^n$, and let $\|\cdot\|''$ be a norm on $\mathbb{R}^m$. Prove that $f$ is ...
0
votes
1answer
76 views

Frechet Derivative: Why bounded (linear) operator?

Why do we want the frechet derivative to be a bounded linear operator? (This meant more as a collecting ideas - I know bounded operators behave fine but that would exclude alot of examples such as the ...
1
vote
1answer
136 views

Implicit function theorem and derivative (proof of splitting lemma)

I have this theorem with a part of the proof: $\quad$ Let $V$ be a Hilbert space, $U$ an open neighborhood of $u\in V$, and let $\varphi\in C^2(U,\mathbf R)$. Define implicity the linear operator ...
0
votes
1answer
90 views

Partial derivative w.r.t an integration

For example, I have a functional $$J(f)=\int \frac{f(x)}{1+x^2}dx.$$ How to calculate $\frac{\partial J}{\partial f(x)}$? Does it equal to $\int \frac{1}{1+x^2}dx$? It seems that the question is ...
0
votes
2answers
50 views

Second order quasilinear PDE

Some quick question about PDE's. Only recently started studying PDE's so this might be trivial. The second-order quasilinear elliptic equation is given by: $ -\sum_{i=1}^{n} \frac{\partial}{\partial ...
2
votes
1answer
45 views

Derivative of function that includes norm

I was solving the problem: find the derivative of a function f : H → R, $f (x) = \sin ||x||^3$ (H is Hilbert space). I got the answer $f'(x)=3\cos||x||^3 x||x||$. Is this correct or I am doing ...
0
votes
1answer
119 views

The gradient of the standard mollifier

Please check my proof for the following result: I want to prove a result for $D\eta_{\epsilon}$ the gradient of the standard mollifier $\eta$. The function $\eta$ is defined as follows: Let ...
0
votes
0answers
83 views

What do they mean by this and how do they prove it? : $\widehat{-2\pi i x f(x)} = \frac{d}{d\xi}\hat{f}(\xi)$

Where $f$ is rapidly decreasing, $f: \Bbb{R}$ to itself. $$\widehat{-2\pi i x f(x)} = \frac{d}{d\xi}\hat{f}(\xi)$$ where $\hat{f}$ is the Fourier transform of $f$. Do they mean differentiation ...
0
votes
1answer
30 views

Methods of computing the derivative of vector norms

I am very new to norms. Except the basic definitions and properties of the norm, I don't know too much about it. Now, I am very interested in computing the derivative of the norms. So, I am wondering ...
0
votes
0answers
100 views

Questions about the Gateaux derivative

We defined that a function is Gateaux differentiable, if all directional derivatives exist. I just wanted to check, whether I got a few things right: Now I wanted to ask, whether it is true that if ...
3
votes
1answer
162 views

Frechet derivative of compact operator is compact

... this seems to be a well known fact as mentioned in this and in this manuscript. However, I was not able to find a proof or to prove it by myself. So my question is: How to prove this? Any hint ...
0
votes
1answer
43 views

Do monotone operators have positive Frechet derivatives?

If a scalar function $f\colon \mathbb R \to \mathbb R$ is monotone and differentiable, then $f'\geq 0$. Monotonicity is generalized for an operator $A\colon V \to V^*$, where $V$ is a Banach spaces ...
1
vote
1answer
31 views

Regarding the propagation of uncertainty

Not sure if I should post this here or in the statistic section. I have a question regarding the propagation of uncertainty. In this book , a dilution ratio is expressed as a function of the sample ...
2
votes
1answer
115 views

$L^p$ derivative vs normal derivative.

Let $f, g : \mathbb{R} \rightarrow \mathbb{C}$ be Lebesgue measurable functions, and let $1 \leq p < \infty$. If $f, g \in L^p$ and $$ \large \lim_{y \rightarrow 0} \normalsize \left\| ...
1
vote
1answer
149 views

Cauchy Derivative Estimates for entire functions with a bound.

The problem statement: Assume $f$ is an entire function and that there is an $n \in \mathbb{N}$ and a $C < \infty$ such that for $z \in C$ $$|f(z)| \le C ( 1+|z|^n)$$ Also assume that $f$ is never ...
3
votes
0answers
73 views

Question on derivative

I have this : And i don't understand (3.5) . i.e : why $\displaystyle\frac{d}{dt} G_t(\eta(t)u)=(G'_t(\eta),\eta ')+\partial_tG_t(\eta))$ Please Thank you .
1
vote
1answer
128 views

Product and Quotient rule for Fréchet derivatives

Does anyone know whether the product/quotient rule for Fréchet derivatives still hold? For example, consider the evaluation operator: $$\rho_x : (C[a,b],\|\cdot\|_\infty) \rightarrow ...
2
votes
0answers
135 views

Space of functions that are everywhere differentiable

Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
2
votes
1answer
158 views

Question regarding Gâteaux Derivative

This is a question on an assignment for a grad engineering class that I cannot seem to figure out. The statement is as follows: Consider $X$ the space of continuous functions on the interval [0,1]. ...
4
votes
4answers
178 views

Proving $C_c^\infty(\Omega)$ is not complete

Let $\Omega \subseteq \mathbb{R}^n$ be a domain. How can I prove that $C_c^\infty(\Omega)$ (with the usual topology) is not sequentially complete? I don't think I have ever seen a proof of this ...