All Questions
12,776 questions
0
votes
1
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221
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Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
- 1
-1
votes
1
answer
1k
views
Non trivial zeros of Riemann zeta function [closed]
Question Define $f(z)=(s-1)\zeta(s)$ where $s=\frac{1}{1+z^2}$ and $\zeta$ denotes the Riemann zeta function. Prove that if $\rho$ denotes the non trivial zeros of $\zeta(s)$ then, $$\sum_{|\alpha|&...
user292590
-1
votes
1
answer
1k
views
Kolodziej's acta paper "the complex monge-ampere equation"——a detailed ploblem [closed]
Recently,I am reading kolodziej's acta paper,there are some ditails that i do not know clearly.
In the top line of page 99,"it's no restriction to assume that for each s we have $\nu(\cup_{I\in{B_s}}...
- 247
-1
votes
1
answer
354
views
The grail of functional analysis?
Does $g\in C([0,1],[0,1])=A$ exist such that $\{g^n ,n\in\mathbb N\}$ is dense in $A$ provided with the uniform norm?
with $g^2=g \circ g $
If we can find $g$ then $F$ a closed of $A$, $id \in F$ ...
- 4,074
-1
votes
2
answers
409
views
$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]
$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is ...
- 263
-1
votes
2
answers
251
views
$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
- 1
-1
votes
2
answers
232
views
Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$
Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$.
I am wondering if it there is a constant $C > 0$ such that for all ...
- 420
-1
votes
2
answers
352
views
Image of a complex disc by this function? [closed]
I am working with the function $f(z) = \frac{z+1}{z-1}$, for a complex variable $z$. I understood that for $z$ in the unit disc, i.e $\lvert z\rvert \le 1$, $\mathrm{Re}(f(z)) \le 0$.
What if $z$ is ...
- 686
-1
votes
1
answer
113
views
Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
- 217
-1
votes
2
answers
510
views
inequivalent norms [closed]
I am thinking about the following question:
Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$.
When can I say that there exist inequivalent complete norms on $X$?
- 617
-1
votes
1
answer
512
views
Does $\int_{2}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$ converge on the real axis for $s>1/2$? [closed]
Consider the prime zeta function, defined for $\Re(s)>1$, by the infinite series
$$\sum_{p} p^{-s} = \sum_{m=1}^{\infty}\frac{\mu(m)}{m}\log \zeta(ms)$$ where $p$ denotes a prime, $\mu$ the Mobius ...
user146617
-1
votes
1
answer
227
views
Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]
We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
- 5,470
-1
votes
1
answer
149
views
Necessity of Lebegue's convergence criterion? [closed]
the well-known Lebesgue’s dominated convergence theorem states that pointwise convergence of a sequence of functions implies convergence of the sequence of integrals if an integrable function ...
-1
votes
1
answer
139
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
- 125
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votes
2
answers
407
views
Conditional expectation: commuting integration and supremum
Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
- 109
-1
votes
1
answer
200
views
Is the unordered sum of measurable functions measurable?
Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
- 167
-1
votes
1
answer
108
views
Are local diffeomorphisms Fredholm maps with index zero? [closed]
Does this statement correct? if it does how we can prove it. In Banach spaces
a map is local diffeomorphism if and only if it is a Fredholm map of index zero with no critical points?
- 71
-1
votes
2
answers
640
views
Invariance of spectrum under conjugation
Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
- 677
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votes
2
answers
466
views
What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]
How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy....
- 1,129
-1
votes
1
answer
122
views
On $\Re\frac{\zeta'}{\zeta}(s) \geq -A \log(|t|+4)$ for some $A>0$
On page 438 of "Multiplicative Number Theory I" by Montgomery-Vaughan one finds the following statement and its verification :
Assuming RH, there exists an absolute constant $A>0$ such ...
- 59
-1
votes
1
answer
109
views
Analyzing a Dirichlet series with log-oscillating terms via Fourier methods
I am investigating the series $S(z)$ defined as follows:
$$
S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)),
$$
where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$.
I want ...
- 375
-1
votes
1
answer
153
views
$\ell^q$ analog of square function
It is a classical result in harmonic analysis that
$$ \|\|P_kf\|_{\ell^2_k}\|_{L^p_x}\approx\|f\|_{L^p} $$
for $p\in(1,\infty)$, where $P_k$ is the Littlewood-Paley decomposition onto frquency $\...
- 5,169
-1
votes
1
answer
471
views
Infinite sum and product associated with the Weierstrass elliptic function [closed]
Can anyone help me figure out how the identity below was obtained?
$ \frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{...
- 45
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votes
2
answers
407
views
Almost isometric subspaces of $\ell_p$
1) Given $p\in (1,\infty)$.
2) Let us fix two, non-isometric subspaces $X,Y\subseteq \ell_p$ isomorphic to $\ell_p$.
3) Are there an $\varepsilon\in (0,1)$ and an isomorphism $S\colon X\to Y$ such ...
- 537
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
-1
votes
1
answer
168
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
-1
votes
1
answer
168
views
A question in functional analysis about selfadjoint operator [closed]
In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$.
I ...
- 1
-1
votes
1
answer
246
views
Density of normal elements in a C*- algebra [closed]
Let $A$ be a unital C*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
- 149
-1
votes
1
answer
102
views
Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]
Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
- 131
-1
votes
2
answers
129
views
Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]
Under which condition can it form a Hilbert space? Or what space can it form?
You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
- 11
-1
votes
1
answer
215
views
Dense linear span implies closed convex hull has non-empty interior
Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
- 5,405
-1
votes
1
answer
122
views
Approximation of function in general measure space
Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
- 411
-1
votes
1
answer
135
views
Prove that Cartesian composition $c_0 \times c_0$ is not isometric isomorphic [closed]
Prove that Cartesian composition $c_0 \times c_0$ with rate $ \Vert (x_1 ,x_2)\Vert = \Vert x_1 \Vert_{c_0} + \Vert x_2 \Vert_{c_0} $ is not isometric isomorphic to space $c_0$.
- 21
-1
votes
1
answer
346
views
Riesz representation theorem for Hilbert-to-Hilbert mappings [closed]
Assume $\phi:\mathbb{H}_1\rightarrow \mathbb{H}_2$ is a continuous linear mapping between two real Hilbert spaces $\mathbb{H}_1$ and $\mathbb{H}_2$. If $\mathbb{H}_2=\mathbb{R}$, then the Riesz ...
- 313
-1
votes
1
answer
237
views
Theorem with an example [closed]
i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?
- 277
-1
votes
1
answer
280
views
Showing there is a unique spectral measure
All the books I have seen have proved that, for a normal bounded operator $T$, there is a unique spectral measure $E$ such that $\int_{\sigma(T)}^{}\lambda\,dE=T$ by first proving in it for a general ...
- 127
-1
votes
1
answer
208
views
smooth holomorphic functions are CR on the boundary? [closed]
Is this true that any holomorphic functions in a domain with smooth boundary, and which is smooth on the boundary is a CR function ?
- 19
-1
votes
1
answer
116
views
Riemann xi function strictly increasing along a half-plane
Matiyasevich, Saidak, Zvengrowsk proved the following result:
Let $σ_0$ be greater than or equal to the real part of any zero of ξ. Then $|ξ(s)|$ is strictly increasing in the half-plane $σ > σ_0$.
...
- 59
-1
votes
1
answer
250
views
Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$
Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...
user485483
-1
votes
1
answer
77
views
Applications and motivations of resolvent for elliptic operator
Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2
\...
- 1,219
-1
votes
1
answer
83
views
"Large" compact sets in separable normed space
Let $(X, \lVert \cdot \rVert)$ be a separable normed space. Can we always guarantee that there is a nonempty compact set $K \subseteq B_X$, where $B_X$ is a closed unit ball in $X$ such that:
$$\...
- 820
-1
votes
1
answer
116
views
Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse
Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $
\delta_n := \{x \in [0,1]^n:\,\Sigma_k x_n =1, (\forall i)\,x_i>0\}
$ are homeomorphic. What I'm ...
- 5,405
-1
votes
1
answer
120
views
Definition of a $\psi$-Banach space [closed]
Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...
- 291
-1
votes
1
answer
176
views
How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]
How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ?
Is the pressure limited or can it be any amount?
- 11
-1
votes
1
answer
70
views
Is this kind of interpolation correct?
Let $f=\sum f_j$ be a finite sum. Assume that
$$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$
$$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$
Then can we conclude that for $2<p<\infty$
$$\|f\|_p\le C^{1-\...
- 1
-1
votes
1
answer
243
views
On a certain representation of the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
- 105
-1
votes
1
answer
328
views
About the critical points of quasi-convex functions
What do we know about the structure of critical points of quasi-convex functions?
I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...
- 2,246
-1
votes
1
answer
346
views
An infinite set in a compact space
Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
-1
votes
1
answer
360
views
Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]
Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
- 7
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
