All Questions
Tagged with fa.functional-analysis integration
176 questions
0
votes
1
answer
157
views
Weak convergence of $f(x,e^{itx})$
This is the desired result (what I want to prove):
$$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$
Given that $f\in C([a,b]\times\{e^{i\...
- 233
1
vote
0
answers
146
views
integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 501
1
vote
2
answers
117
views
If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
5
votes
0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
3
votes
1
answer
79
views
Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$
Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
- 407
5
votes
2
answers
517
views
Functions whose product with every $L^1$ function is $L^1$
Let $\mu$ be a probability measure and $f$ a measurable function whose
product with any integrable function is integrable: $$
\int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$
...
- 3,599
0
votes
1
answer
80
views
Orthogonal space of polynomials
Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and
$$
\int_0^{+\infty} f(t) t^n ...
8
votes
1
answer
537
views
Reference request: Expository paper on the use of functional analysis in differential and integral equations
Some textbooks on functional analysis do not hint that a major raison d'être of the subject is its use in the study of differential and integral equations. The reader could go all the way through ...
0
votes
0
answers
44
views
Integral of gradient of a function times a vector fields, null whatever the function, implies null divergence and tangential limits conditions
I'm reposting this question from math.stackexchange, as I haven't got answers so far.
At the beginning of "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de
...
- 101
3
votes
0
answers
105
views
Maximal-type inequality for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$
Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere
$$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$
Let ...
- 840
2
votes
1
answer
321
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
- 4,074
1
vote
1
answer
120
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...
- 141
7
votes
3
answers
909
views
Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
- 331
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
1
vote
0
answers
87
views
$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?
Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
- 309
1
vote
0
answers
75
views
Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
- 155
2
votes
1
answer
188
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 75
0
votes
0
answers
77
views
Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
- 39
0
votes
1
answer
79
views
Convergence in sequential Lebesgue spaces
Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
- 1,101
0
votes
1
answer
135
views
When integrating by part produces a singularity
I'm currently interesting in the following operator:
$$O[f](x):=f(x)-2xe^{x^2}\int_x^{+\infty}dt \, e^{-t^2} f(t)$$ for all $x\in\mathbb{R}$ and $f$ smooth and decaying at infinity fast enough with ...
- 41
6
votes
1
answer
230
views
Integration along fibres of continuous map on compact Hausdorff spaces
Let $p:Z\to X$ be a continuous surjective map between compact Hausdorff spaces.
Does there exist a family $m=(m_x)_{x\in X}$ of Radon probability measures on $Z$, such that
the support of $m_x$ is ...
user473423
1
vote
1
answer
410
views
Takesaki lemma: existence Gelfand-Pettis integral
Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras").
In another post, it was explained ...
- 175
1
vote
0
answers
77
views
Integration over a finite-dimensional subspace of Hilbert space
Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
- 503
7
votes
2
answers
398
views
Quantifying difficulty of integrals versus inverses
Recently, I have been discussing inverses with a tenth grade class and integrals with an eleventh/twelfth grade class, and this has led me to the following wonder:
Wonder. Is there a "reasonable&...
- 7,831
1
vote
0
answers
70
views
Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 39
2
votes
1
answer
272
views
Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space
Problem Statement
Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
- 53
1
vote
1
answer
89
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II
This is a follow-up to this previous question, but under stronger assumptions.
Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real
scalar field). Let $\tilde ...
- 12.5k
7
votes
2
answers
464
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.5k
1
vote
1
answer
210
views
On a property of complex exponentials
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
- 4,135
0
votes
1
answer
73
views
A solution satisfying an integral inequality is bounded [closed]
Let $y$ be a positive function and $c>0$. If $y$ satisfies the following integral inequality
\begin{equation}
y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2}
\end{equation}
...
- 11
1
vote
1
answer
177
views
Finding $W^{1,\infty}$ solutions to an integral equation by fixed point
Let $f \in L^\infty((0,1)\times(0,1))$ and consider the integral equation
\begin{align*}
u(x, y)= \int_0^y \int_0^{x-y} u(x-y+\tau, s+\tau) f(s+\tau, \tau) d s d \tau -\int_0^y f(x-y+s, s) d s.
\end{...
user139844
5
votes
0
answers
198
views
If $\lim_{t\to +\infty} \int_{0}^{\pi} f(x)\exp(e^{xt}) \, dx=0$ then $f=0$ a.e?
Question, Let $f \in L^1(\alpha, \beta)$ , $\beta>0$ and
$$
F(x)= \int_{\alpha}^{\beta} f(t)\exp(e^{xt}) \, dt
$$ such that $\displaystyle \lim_{x\to +\infty}F(x)=0$. Does this imply that $f$ is ...
- 1,503
1
vote
0
answers
199
views
Integration with respect to Haar measures normalised over a subspace
Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work
$\int_{\mathcal{U}(d)} \frac{\...
- 111
2
votes
1
answer
309
views
Bounded operator on $L^2(\Bbb R^2)$
Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$:
$$
f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...
- 501
0
votes
0
answers
120
views
How to prove an equality involving Laguerre polynomials
Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$.
How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...
- 501
1
vote
1
answer
240
views
Integration by parts for indicator of a sphere to indicator of a ball
Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
2
votes
1
answer
125
views
For which value of $C(f)$ would the following inequality hold?
I am wondering what would be the value of $C(f)$ for the following inequality to hold? E.g., $C(f)$ could be some quantity related to the Lipschitz constant or the size of the domain.
$$\left(\int f(x,...
- 23
1
vote
1
answer
134
views
Do these two pairs coincide at small time?
Let $\alpha, \beta:\mathbb R_+\to [0,1]$ be continuous and decreasing functions s.t. $\alpha(0)=1=\beta(0)$ and $\alpha, \beta$ are continuously differentiable on $(0,\infty)$ satisfying for some $c&...
- 1,334
0
votes
0
answers
66
views
Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 463
2
votes
0
answers
155
views
Second differential of total variation
I am trying to give meaning to the notion of second differential of total variation.
For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by
$$TV(u)=...
- 319
4
votes
1
answer
405
views
Why is it difficult to define a direct integral of Banach spaces or Banach algebras?
In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or ...
- 1,054
4
votes
1
answer
339
views
Mikusiński's approach to Bochner integrals; replace absolute by unconditional?
In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions:
Defn. Let $X$ be a Banach space. ...
- 39k
1
vote
3
answers
579
views
Squeezing more convergence from the convergence in all $L^p$ spaces
Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
- 5,407
2
votes
0
answers
118
views
What is the justification for using Wiener integrals to integrate over a space of differentiable functions?
In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
- 33
3
votes
0
answers
277
views
Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
- 67
6
votes
1
answer
346
views
Best approximation of L1 function by Lipschitz function
Fix constant $L,C>0$ and $k\geq 1$ and let $f\in W^{1,k}(\mathbb{R}^d,\mathbb{R}^n)$ with $\|f\|_{W^{1,k}}\leq C$.
Is there a known estimate on the distance
$$
\|f - \operatorname{Lip}_L(\mathbb{R}^...
- 5,405
2
votes
1
answer
173
views
Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$
Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by
$$
R[f](w,b) := ...
- 6,853
3
votes
2
answers
247
views
Bochner integral over convex sets lies in the convex set?
Let $(\Omega,\Sigma,\mu)$ be a probability space, $E$ be a separable Banach space, $f:\Omega\rightarrow E$ be Borel-measurable and consider the Bochner-integral
$$
\bar{\mu}:=\int_{\omega\in \Omega}\, ...
- 5,405
3
votes
1
answer
140
views
On an asymptotic integral
Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that
$$
\int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}.
$$
Does it follow that $\phi$ is a ...
- 4,135
0
votes
1
answer
125
views
Characterizing subsets of integrable functions
Let us consider the space $L^1(0,1;\mathbb{R})$ of real-valued, Lebesgue integrable functions defined on the interval $(0,1)$ (where we only distinguish functions which are not equal almost everywhere)...
- 820