fa.functional analysis – better lower (and upper) bound for $i$’s moment of function of binomial random variable with $i = frac{1}{j}, j in mathbb{N}$

I want to derive a lower bound for $Eleft(left(frac{X}{k-X}right)^{i}right) $ with $X sim Bin_{(k-1),p}$ and $ k in mathbb{N} $. So far I could prove that
begin{equation}
Eleft(frac{X}{k-X}right) = frac{p}{1-p}-frac{p^k}{1-p}
end{equation}

by law of unconcious statistician and the identity $binom{k-1}{l}=frac{k-l}{l}binom{k-1}{l-1}$.
Now, I use as lower bound the fact that:
begin{equation}
frac{1}{k^{i}}Eleft(X^{i}right)leq Eleft(left(frac{X}{k-X}right)^{i}right) leq Eleft(X^{i}right)
end{equation}

Then:
begin{equation}
Eleft(X^{i}right) = sum_{l=0}^{k-1}l^{i}binom{k-1}{l}p^l(1-p)^{k-1-l} \= sum_{l=1}^{k-1}l^{i}binom{k-1}{l}p^l(1-p)^{k-1-l}geq sum_{l=1}^{k-1}binom{k-1}{l}p^l(1-p)^{k-1-l} = 1- (1-p)^{k-1}
end{equation}

So this is my lower bound. For the upper bound I use Jensen’s inequality on a concave function:
begin{equation}
Eleft(left(frac{X}{k-X}right)^{i}right)leq left(Eleft(frac{X}{k-X}right)right)^{i} = left(frac{p}{1-p}-frac{p^k}{1-p}right)^{i}
end{equation}

Thus, I have:
begin{equation} frac{1}{k^{i}}left(1- (1-p)^{k-1}right) < Eleft(left(frac{X}{k-X}right)^{i}right) < minBig{left(frac{p}{1-p}-frac{p^k}{1-p}right)^{i}, 1- (1-p)^{k-1}Big}
end{equation}

Do you have anything better than that?

fa.functional analysis – Condition on kernel convolution operator

I am studying a about O’Neil’s convolution inequality. It is stated that for $Phi_1$ and $Phi_2$ be $N$-functions, with $Phi_i(2t)approx Phi_i(t), quad i=1,2$ with $tgg 1$ and $k in M_+(R^n)$ is the kernel of a convolution operator.

The $rho$ is an r.i. norm on $M_+(R^n)$ given in terms of the r.i norm $bar rho$ on $M_+(R_+)$ by
$$
rho(f)=bar rho(f^*), quad f in M_+(R_+)
$$

Denote Orlicz gauge norms, $rho_{Phi}$, for which
$$
(bar rho_{Phi})_dapprox bar rho_{Phi}left(int_0^t h/tright).
$$

It is stated that
$$
rho_{Phi_1}(k+f)leq C rho_{Phi_2}(f)
$$

if
$$
(i) quad bar rho_{Phi_1}left(frac 1t int_0^t k^*(s)int_0^sf^*right)leq C bar rho_{Phi_2}(f^*)
$$

$$
(ii) quad bar rho_{Phi_1}left (frac 1tint_0^t f^*(s)int_0^sk^*right)leq C bar rho_{Phi_2}(f^*)
$$

$$
(iii) quad bar rho_{Phi_1}left(int_t^{infty}k^*f^*right)leq C bar rho_{Phi_2}(f^*).
$$

I cannot understand under which conditions on kernel those inequalities (i),(ii) and (iii) would hold.

fa.functional analysis – Topological constraints on a compact convex set admitting a strictly convex and subdifferentiable real function

It is a theorem of Hervé that

A compact convex set $K$ admits a strictly convex and continuous real function only if $K$ is metrizable. (The converse is also true.)

I’m wondering if any results of this sort are known if continuity is replaced with subdifferentiability. That is,

What topological constraints must a compact convex set $K$ meet in order to admit a strictly convex function that has a subgradient at every point in $K$? Does $K$ have to be first countable, for instance?

fa.functional analysis – Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus

In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim about the spectral mapping theorem’s possible proof. Let me attempt to bring the context here. I should mention there are some nice results in this paper that I wanted to use and generalize for my own research, I hope to accurately bring the context below.

They bring up the continuos functional calculus $phi: C(sigma(A)) rightarrow L(H)$ for a bounded, self-adjoint operator on a Hilbert space A. This is an algebraic *-homomorphism from the continuous functions on the spectrum of $A$ to the bounded operators on $H$. The paper’s spectral mapping theorem basically says in this context $$ sigma(phi(f)) =f(sigma(A)) $$ and the paper says something nice about this. It does not actually give a proof but it says there is a nice way to prove it using both inclusions with the inclusion $ f(sigma(A)) subseteq sigma(phi(f)) $ sketched in the following way: the author supposes $ lambda in f(sigma(A)) $ and says “it is very obvious” that there exists a vector $h in H$ with $|h|=1$ such that $|phi(f)-lambda)h|$ is arbitrarily small which shows $lambda in sigma(phi(f))$ which shows the desired inclusion.

The author says that it is “very obvious” to show this but I am a bit stumped. The way I would construct the continuous functional calculus is to start with polynomials and then generalize to $ C(sigma(A)) $ based on the Weierstrass approximation theorem on the real compact set $sigma(A)$ and the BLT theorem. The inclusion $sigma(phi(f)) subseteq f(sigma(A))$ is, I think, quite obvious but the other one in the above context has me stumped. Since I am already working on generalizing some results, I would really love to know how the author proves the inclusion with the method of showing the mentioned vector exists. Maybe use approximation in some way, but even though I suspect it is simple, I still do not see the author’s proposed proof. Can someone here please help me recover it? I thank all interested persons.

fa.functional analysis – A density question for the Hilbert transform

Let $mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $mathbb R$. Are the set of functions
$$ {(f+mathscr Hf)_{|_{(0,1)}},:, f in C^{infty}(mathbb R)quad text{and}quad textrm{supp} f Subset (0,infty)}$$
dense in $L^2((0,1))$?

fa.functional analysis – Dimensional scaling implementation

I’m not familiar with the concept of dimensional scaling at all. Anyone please help me with this problem.

I need to understand how they obtained eq(16) from eq(15) in the paper here so I can do the same when I change Q0 from a constant to a variable function in time. Summary as below:

Scaling:

$ξ=x/l,ζ=z/h_* ,τ=t/t_* ,γ=l/l_* ,λ=h/h_* ,Π=p/p_* ,Ψ=q/q_* ,bar Ψ=bar q ̅/q_* ,Ω=w/w_* ,bar Ω = bar w /w_* (15)$

“The six characteristic quantities that have been introduced are identified by setting to unity, six of the dimensionless groups that emerge from the governing equations when they are expressed in terms of the dimensionless variables. The remaining two groups are numbers that control the problem. After some algebraic manipulation, we obtain the following expressions for the characteristic quantities:”

$ l_*=frac{πH^{4}∆σ^{4}}{4E^{‘3} μQ_0 } $, $ h_*=H $, $ t_*=frac{π^{2} H^{6} ∆σ^{5}}{4μE^{‘4}Q_0^{2} } $ , $ p_*=∆σ $, $ q=frac{Q_0}{2H} $, $w=frac{πH∆σ}{2E^{‘}}=M_0∆σ (16)$

Thanks a lot.

fa.functional analysis – Is the restriction map $C^1ni fmapstoleft.fright|_K$ a continuous map?

Let $E$ be a $mathbb R$-Banach space, $Thetasubseteq C^{0,:1}(E,E)$ be a $mathbb R$-Banach space and $iota$ be a continuous embedding of $Theta$ into $C^1(E,E)$.

I would like to show that, given a compact $Ksubseteq E$, there is a $cge0$ with $$sup_{xin K}left|{rm D}(iota f)(x)right|_{mathfrak L(E)}le cleft|fright|_{Theta};;;text{for all }finTheta.tag1$$

If I’m not missing something, $$C^1(K,E):=left{left.gright|_K: gin C^1(U,E)text{ for some open neighborhood }Utext{ of }Kright}$$ equipped with $$left|gright|_{C^1(K,:E)}:=maxleft(sup_{xin K}left|g(x)right|_E,sup_{xin K}left|{rm D}g(x)right|_{mathfrak L(E)}right);;;text{for }gin C^1(K,E)$$ should be a $mathbb R$-Banach space. If that’s true, we may be able to show $$Thetani fmapstoleft.(iota f)right|_Ktag2$$ is a continuous embedding of $Theta$ into $C^1(K,E)$, from which the desired claim would follow.

Can we show this?

fa.functional analysis – Analytic functions where all derivatives vanish at infinity and which are bounded

Yes.

Let $phi$ be any smooth function with compact support on the interval $(-1,1)$.

Set $f$ to be the inverse Fourier transform of $phi$.

Since $phi$ is in Schwartz class, so is $f$, and all of its derivatives decay to zero as one approach $pminfty$.

You can estimate

$$ |f^{(k)}(x) | lesssim | |xi|^k phi(xi) |_{L^1} leq 2 |phi|_{L^infty} =: C$$

$f$ is analytic by Paley-Wiener.

fa.functional analysis – Solution space of first order PDE

We consider the following first order PDE

$$(partial_x + ipartial_y) u(x,y)+ begin{pmatrix} 0 & cos(2x+y) \ cos(2x-y) & 0 end{pmatrix}u(x,y) =0,$$

where $u in mathbb C^2$ is vector-valued and periodic on $(0,2pi)^2.$

I ask: What is the dimension of the solution space to that equation?-My conjecture is that it is two-dimensional.

The reason is that if
the equation was

$$(partial_x + ipartial_y) u(x,y)+ begin{pmatrix} 0 & cos(2x+y) \ cos(2x+y) & 0 end{pmatrix}u(x,y) =0,$$

then we could explicitly state the two-dimensional solution space

$$u(x,y)=operatorname{exp}left( frac{i}{2i-1} begin{pmatrix} 0 & sin(2x+y) \ sin(2x+y) & 0 end{pmatrix}right)x_0$$

for any $x_0 in mathbb{C}^2.$

fa.functional analysis – Definition question: asymptotic-$ell_{p}$ versus coordinate-free asymptotic-$ell_{p}$

Let $(e_{j})_{j=1}^{infty}$ be a basis for the Banach space $X$. If there exist constants $zeta_{1},zeta_{2}>0$ such that for all $Ninmathbb{N}$,
begin{equation*} zeta_{1}left(sum_{i=1}^{N}|x_{i}|^{p}right)^{frac{1}{p}}leqleftVertsum_{i=1}^{N}x_{i}rightVertleqzeta_{2}left(sum_{i=1}^{N}|x_{i}|^{p}right)^{frac{1}{p}} end{equation*}
for all block sequences $(x_{i})_{i=1}^{N}$ that satisfy $M=M_{N}leqmintext{supp}(x_{1})$, then $X$ is said to be (stabilized) asymptotic-$ell_{p}$ with respect to $(e_{j})_{j=1}^{infty}$.

There is also coordinate-free generalization of a Banach space being asymptotic-$ell_{p}$ without reference to a basis. In this situation, there exist $zeta_{1},zeta_{2}>0$ such that for all $Ninmathbb{N}$, there are subspaces $Y_{1},ldots,Y_{N}$ of finite-codimension such that
begin{equation*} zeta_{1}left(sum_{i=1}^{N}|y_{i}|^{p}right)^{frac{1}{p}}leqleftVertsum_{i=1}^{N}y_{i}rightVertleqzeta_{2}left(sum_{i=1}^{N}|y_{i}|^{p}right)^{frac{1}{p}} end{equation*}
for all $y_{i}in Y_{i}$. My question is the following: why do we want the subspaces $Y_{i}$ to have finite codimension? In particular, the block vectors $x_{i}$ in the first definition are members of finite-dimensional subspaces of $X$ (not finite co-dimensional subspaces) and I am wondering why a coordinate-free generalization of the first definition wouldn’t take the following form:

$X$ is coordinate-free asymptotic-$ell_{p}$ if there exist $zeta_{1},zeta_{2}>0$ such that for all $Ninmathbb{N}$ there exist pairwise disjoint finite dimensional subspaces $Y_{1},ldots,Y_{N}$ of $X$ such that
begin{equation*} zeta_{1}left(sum_{i=1}^{N}|y_{i}|^{p}right)^{frac{1}{p}}leqleftVertsum_{i=1}^{N}y_{i}rightVertleqzeta_{2}left(sum_{i=1}^{N}|y_{i}|^{p}right)^{frac{1}{p}} end{equation*}
for all $y_{i}in Y_{i}$.