All Questions
Tagged with real-analysis fa.functional-analysis
1,447 questions
6
votes
1
answer
314
views
Generators of a convex cone defined by a differential inequality
Consider the cone of continuously twice differentiable functions mapping positive reals to itself (i.e., $f\in C^2(\mathbb R_{++})$ and $f\colon \mathbb R_{++}\to\mathbb R_{++}$) that satisfy
\begin{...
- 61
-1
votes
1
answer
136
views
An elementary question about integration by parts! [closed]
Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
9
votes
1
answer
459
views
Why should the map $-\Delta^{-1}$ be continuous?
I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$
in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
- 91
0
votes
1
answer
629
views
Fourier Transform of sub-Gaussian distributions
The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?
Let $x \in \mathbf{R}^n$ denote some sub-...
- 445
5
votes
2
answers
429
views
Does the truncated Hausdorff moment problem admit absolutely continuous solutions?
Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
- 311
5
votes
1
answer
101
views
Does minimum of an analytic map restricted to analytic curves implies minimum?
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be an analytic function such that its restriction to any arbitrary analytic curve $\gamma$ passing through the origin $0\in \mathbb{R}^n$ attains a local ...
- 53
1
vote
1
answer
201
views
Existence of a certain norm on space of measurable functions
Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can ...
- 13
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
- 623
8
votes
3
answers
637
views
Method to compute fundamental solutions which are distributions
The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...
4
votes
1
answer
484
views
Question about normalization factors in the direct integral of operators
So the original question I wanted to ask was this one:
I'm currently a bit puzzled about the normalization for the Gelfand transform $U$:
So if we have a periodic Schrödinger operator $H$, then we ...
- 95
0
votes
1
answer
697
views
How much do we know about this "local" Hardy-Littlewood maximal function?
The "local" Hardy-Littlewood maximal function is given by $$(M_\phi f)(x)= \sup_{0<\epsilon<1}|\phi_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$...
- 171
0
votes
1
answer
268
views
Linear operator has one-dimensional kernel
Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether ...
- 329
1
vote
0
answers
137
views
Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?
I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?
For example, let $f_i $ be the eigenfunctions of an operator $D$, ...
- 3,804
4
votes
1
answer
264
views
Density of the max set of a non-differentiable function
For $f : [0;1] \to \mathbb{R}$, let $M_f := \{x \in [0;1] \mid f(x)$ is a local strict maximum of $f\}$. It is easy to see that for any $f$, $M_f$ is at most countable. It is also easy to see that ...
- 15.9k
4
votes
1
answer
225
views
Multivariate Zero-Bias Transform
The zero-bias transform for a univariate random variable $W$ is defined as a random variable $W^*$ satisfying
\begin{align}
\mathbb{E} [ W \cdot f(W )] = \mathbb{E} [ f' (W^*)]
\end{align}
for any ...
- 1,127
0
votes
0
answers
75
views
Dense Egoroff theorem
Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...
- 5,405
0
votes
0
answers
308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 617
1
vote
1
answer
161
views
Continuity of image of resolvent operator with respect to resolvent parameter
Suppose $D$ is a first-order differential operator on a manifold $M$ and that the inverse $(D+t)^{-1}:H^0(M)\rightarrow H^1(M)$ exists for all $t > 0$, where $H^i(M)$ is the $i^\text{th}$ Sobolev ...
- 1,903
1
vote
1
answer
194
views
Uniformly approximating a function of vanishing variation by functions of vanishing gradient
Let us say that a bounded smooth function $f:\mathbb{R}\rightarrow\mathbb{R}$ has vanishing variation at infinity (or satisfies "property $A$" for short) if, for any $r\neq 0$, we have
$$\lim_{x\...
- 1,903
2
votes
1
answer
140
views
An inequality about embedding of cube into metric spaces
A k-cube in $X$ is a function $\psi:\{-1,1\}^k\to (X,d)$.
An edge of a cube is a pair of points $\{\psi(\epsilon_1),\psi(\epsilon_2)\}$ in $X$ such that $\epsilon_1$ and $\epsilon_2 $ differ in ...
- 1,245
2
votes
1
answer
497
views
Are the partial derivatives of a function increasing in both variables measurable?
Let $f$ be a function from $[0,1]\times[0,1]$ to $\mathbb{R}$ that is nondecreasing in both variables, i.e. $f(x_1,y_1)\le f(x_2,y_2)$ whenever
$x_1\le x_2$ and $y_1\le y_2$. It is known that the ...
1
vote
0
answers
74
views
Nonlinear maps in Riesz Thorin theorem
The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...
user127450
1
vote
0
answers
93
views
Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
- 33
1
vote
1
answer
87
views
Function is almost everywhere 1 w.r.t. sequence of regular Borel probability measures
Let $\epsilon>0$ be given. Let $Y$ be a compact, Hausdorff space and let $U\subseteq Y$ be an open subset. Assume that $(\mu_n)_{n\in\mathbb{N}}$ is a sequence of regular Borel probability measures ...
- 11
3
votes
1
answer
1k
views
A calculus question related to the nonnegative definite functions
I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$...
- 1,649
4
votes
1
answer
670
views
A generalization of a theorem of Grothendieck
In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$.
Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$.
Assume that $S$ is a subvector space of ...
- 356
0
votes
1
answer
303
views
Approximation of a $C^{\infty}_c$ function with tensor products of a constant tensor rank
I asked the following question a few days ago:
Approximation of a $C^{\infty}_c$ function by tensor products
However, I then realised that I actually need a stronger result in my proof.
As in the ...
- 357
3
votes
2
answers
435
views
A possible norm on a subspace of $C^\infty([0,1])$?
I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...
- 3,388
7
votes
1
answer
497
views
Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?
Is it true that
$$
||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ?
$$
where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
- 193
1
vote
1
answer
210
views
Eigenvalues of the double layer potential
Consider the double layer potential $K: L^2(S^2)\to L^2(S^2)$
$$(Kf)(x)=\int_{S^2}f(y)\frac{\partial}{\partial v_y}E(x,y)dS_y,$$
where $E(x,y)=||x-y||^{-1}$ and $\frac{\partial}{\partial v_y}$ means ...
- 171
4
votes
2
answers
958
views
Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
- 619
3
votes
1
answer
167
views
Recovering residue using local real information
Let $f(z)$ be defined by a Laurent series at z = 0 with real coefficients. In particular, $f(x) \in \mathbb{R}$ for $x \in \mathbb R$.
Compute the residue of $f(z)$ at z = 0 using just the ...
- 31
1
vote
1
answer
287
views
Interpolation between $L^1$ and $L^2$ spaces
I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:
Let $f \in \mathbb{R}^n$ be such that
$$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
- 175
1
vote
2
answers
931
views
A question on the Lebesgue differentiation theorem
In the paper [Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234] it is considered the limit
$$
\lim_{\...
- 2,715
3
votes
0
answers
280
views
Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
- 1,385
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
4
votes
2
answers
2k
views
mean value theorem for operators
This might be a trivial question but I am not very familiar with the subject matter. I was wondering if some sort of mean value theorem works for operators on function spaces. Say $F: \mathcal{S_1} \...
- 85
3
votes
1
answer
228
views
The sheaf of generalized functions on compact subsets
For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual ...
- 1,270
4
votes
1
answer
555
views
Construct smooth functions with prescribed derivatives
To be specific, suppose we are given a sequence of smooth functions $\{f_k\}_{k\geq 0}$ on flat torus $\mathbf{T}^2$(you may think of it as doubly periodic functions on $\mathbf{R}^2$ and smooth).
...
- 151
2
votes
2
answers
953
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
- 201
2
votes
1
answer
149
views
The infinite set of $SBV$ function?
Let $u\in SBV(\Omega)$ where by $SBV$ we denote the special bounded variation function and $\Omega\subset \mathbb R^N$ is open bounded.
Let's identify $u$ by its approximation representative (see ...
- 679
4
votes
0
answers
298
views
Operator topologies
Let $L(H)$ be the space of bounded operators on some Hilbert space.
We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT).
...
- 41
1
vote
1
answer
294
views
A problem about the quotient space of an extended Dirichlet space
Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
- 369
0
votes
1
answer
770
views
About weak derivatives [closed]
I have a question about weak derivatives.
Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...
- 721
1
vote
1
answer
392
views
Integral kernel smooth
Assume that $Tf(x):=\int_{\mathbb{R}^n} K(x,y)f(y) dy$ is an operator such that $T \in L( H^{-k}, H^k)$ is continuous for any $k$, where $H^k$ is the $k-th$ order Sobolev space on $\mathbb{R}^n$.
...
- 11
2
votes
0
answers
341
views
Trace class operators convergent series
On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as
$$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
- 305
3
votes
0
answers
214
views
Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
5
votes
1
answer
219
views
Uniqueness from orthogonality relation?
This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...
- 128k
-2
votes
1
answer
158
views
About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246
7
votes
1
answer
609
views
$H^s$ norm of a solution of a nonlinear Schrödinger equation
I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...
- 71