1
vote
0answers
47 views

$n$th derivative of $f(x)$ using limit definition

After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative: Let $F^n=f(x+nh)$ ...
0
votes
0answers
17 views

Duality relation with respect to differential operators

I have the following differential operator $$L^{\pm}h(x,y)=\pm\frac{x}{2}\frac{\partial h}{\partial x}(x,y)\pm\frac{y}{2}\frac{\partial h}{\partial y}(x,y)+\frac{1}{2}\frac{\partial^2 h}{\partial ...
0
votes
0answers
13 views

Coordinate Change Operator

Let $ f: \mathbb{R} \rightarrow \mathbb{R} $ be analytic. Recall that for $ h \in \mathbb{R} $, the translated function $ \tilde{f} (x) = f(x+h) $ can be formally written as $ \tilde{f} = e^{ h ...
0
votes
1answer
31 views

Inverse of an operator on two functions

I have the following operator, defined for two twice-differentiable functions $f,g$: $X(f,g):=\frac{(g')^3+fg'f''+g'(f')^2-ff'g''}{g'f''-f'g''}$ This operator has the following property: A curve ...
0
votes
1answer
37 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
1
vote
0answers
13 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
2
votes
2answers
51 views

Is this operator bounded ??

Let $X$ be the Banach space $X:=\{ f\in C(\mathbb{R},\mathbb{R}),\sup_{t\in \mathbb{R}}|e^{-s^2}f(s)|<+\infty \}$ equipped with the norm $$|f|_X=\sup_{t\in \mathbb{R}}|e^{-s^2}f(s)|$$ I want to ...
2
votes
2answers
99 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
7
votes
1answer
241 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
5
votes
1answer
79 views

Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
0
votes
1answer
60 views

Question about step in proof of Schauder's theorem

The statement is the following: Let $X,Y$ be Banach spaces and $T:X \rightarrow Y$ be a continuous linear operator. Then is $T'$ compact iff $T$ is compact. I have already understood the implication ...
0
votes
1answer
53 views

Minkowski functional and strange theorem

I have a theorem that says the following: Let X be a normed space and $U\subset X$ a convx subset with $0 \in \text{int(U)}$, then we have: $U$ is absorbing and if $\{x;||x|| < \epsilon\} \subset ...
0
votes
1answer
28 views

Definition of a norm infinity

I have $u:\mathbb{R}^3\times(0,\infty)\longrightarrow\mathbb{R}$ and $g:\mathbb{R}^3\longrightarrow\mathbb{R}$. Which means the following?: $\|u(\cdot,t)\|_{L^\infty(\mathbb{R}^3)}$, and ...
1
vote
1answer
101 views

taylor series for a function of matrices

Say I have a function $(A+B)^{-1}$ where $A$, $B$ are matrix-valued functions of some vector $x$. Can I then expand this function around $x=0$ as: $$(A+B)^{-1} = (A[0]+B[0])^{-1} - (A[0]+B[0])^{-2} ...
1
vote
1answer
123 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 ...
1
vote
1answer
123 views

Factoring a time derivative operator outside of an integral in space

I'm trying to integrate $$\int_a^b \frac{d}{dt} \left[ \frac{du}{dx}\right]dx.$$ Assume $u$ is a sufficiently smooth function of both $t$ and $x$. Since the integral operator is in space only, can ...
4
votes
2answers
210 views

Interchanging closed operators and integrals

I am dealing with a problem in Evans PDE without measure theory knowledge... We have contraction semigroup $\{S_t\}_{t \geq 0}$ on real Banach space $X$, i.e family of bounded linear operators from $ ...
5
votes
2answers
459 views

Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
2
votes
0answers
351 views

Adjoint of the infinitesimal generator of a stochastic process

I need help seeing that $$ \mathcal{L}^* g = -\frac{\partial (bg)}{\partial x} + \frac{1}{2}\frac{\partial^2(\sigma^2g)}{\partial x^2} $$ is the adjoint operator of $$ \mathcal{L} = b\frac{\partial ...
6
votes
1answer
339 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
5
votes
3answers
601 views

Square root of differential operator

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$. How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number) For ...
1
vote
3answers
272 views

what is/are the spectrum of operators and their applications

this is an educational question. can someone please explain with some simple examples: (1) what is/are the spectrum of operator (2) where it is useful For providing examples of spectrum of ...
0
votes
1answer
244 views

Linear operator categories

Let's consider linear operators on the set of complex-valued functions to the same set. I wonder to which categories such operators can be classified. All linear operators I encountered so far fall ...
1
vote
3answers
523 views

Scale Operator $Uf(x)=f(kx)$

I am looking for an operator $U$, that can do this to a function: $$Uf(x)=f(2x).$$ In particular I am happy if there is an $U$ for the general case: $Uf(x)=f(kx)$. Does such an operator exist for ...