All Questions
Tagged with fa.functional-analysis real-analysis
1,448 questions
4
votes
2
answers
126
views
Extension of a certain type of (very) smooth functions to a larger interval
Let $[a,b]\subset[-1,1]$ and $f\in C^{\infty}([a,b],\mathbb{R})$ a function which satisfies for all $n\in\mathbb{N}$ that $\sup_{x\in[a,b]}|f^{(n)}(x)|\leq n!$.
Does there exist a function $g\in C^{\...
- 5,343
5
votes
3
answers
668
views
Continuity and sequential continuity of a linear functional
Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...
- 60.6k
3
votes
1
answer
1k
views
Reference request: interpolation of Hölder spaces
On the Wikipedia page on interpolation space, it is written that the space $C^\theta([0, 1])$ is the (real) interpolation of $C^0([0, 1])$ and $C^1([0, 1])$, where $C^\theta([0, 1])$ denotes the space ...
- 2,670
5
votes
1
answer
795
views
How to define transfinite derivatives of a function?
There are all manners of theories generalizing the notion of derivative. Amongst them is the fractional calculus, a rich theory which gives a sense to the derivation and integration of non-integer (i....
10
votes
0
answers
845
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
CommunityBot
- 1
2
votes
1
answer
246
views
Does this sequence contain a nonnegative number?
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
- 109k
1
vote
1
answer
52
views
Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 5,169
4
votes
1
answer
184
views
Non-linear translation invariant functionals on $L^1$
I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...
- 4,361
1
vote
0
answers
126
views
identity involving spectral functions
Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true :
$$ A^* f(AA^*) = f(A^* A) A^*$$
- 2,118
6
votes
1
answer
213
views
A one-dimensional integral minimization problem
Let $\mathscr F$ be the collection of smooth functions $f \colon
\mathbb R \to \mathbb R$ such that
$f \in C^\infty_c(\mathbb R)$, with $\text{supp } f \subset [-1,1]$;
$\int_0^1 x f(x) dx ...
- 128k
7
votes
2
answers
998
views
Uniform continuity of heat semigroup
I would like to illustrate my question with an example:
It is well-known that $\Delta$ is the generator of a strongly continuous semigroup $(T(t))$ on $L^2(\mathbb R^n),$ i.e. the heat-semigroup.
It ...
- 16.2k
7
votes
1
answer
1k
views
A variational problem - some guidance
This is a problem I'm thinking about, to learn some more advanced calculus of variations on my own. I would appreciate some help, or a solution, just to have a sample to compare in the future.
Let
$\...
CommunityBot
- 1
3
votes
1
answer
876
views
Is Quantum Mechanics (norm)-consistent?
I edited a few small comments to the question in order to make it perhaps more comprehensible.
Today I came across the following question in quantum mechanics.
In Quantum mechanics it is common to ...
- 24.2k
2
votes
1
answer
93
views
Lipschitz bound on semigroups
Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator.
Then one can study the unitary groups $R_T(t):=e^{itT}$ and $R_S(t):=e^{itS}.$
Now if you think about ...
- 13k
1
vote
0
answers
45
views
Shifting Sobolev norms in a hyperbolic estimate
Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate:
$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
- 63.9k
3
votes
1
answer
172
views
Infimum of an integral functional involving a symmetric matrix
I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int_{B_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du,
...
- 128k
2
votes
1
answer
151
views
Rademacher‘s Theorem and Baire category
Is there a variant of Rademacher‘s Theorem where the smallness of the points of non-differentiability is measured in terms of Baire category instead of measure?
More precisely, let X be a separable ...
- 42.8k
4
votes
1
answer
203
views
If $T_1T_2 = T_2T_1$, why $r(T_1 + T_2) \leq r(T_1) + r(T_2)$?
Let $T_1$ and $T_2$ be two bounded linear operators in a complex banach space $X$.
If $T_1T_2 = T_2T_1$, I want to know how to show that
$$
r(T_1+T_2) \leq r(T_1) + r(T_2),
$$
where $r(A)$ ...
- 17.2k
3
votes
1
answer
431
views
Can I approximate a function of bounded variation with orthogonal polynomial?
Let function $u\in BV(\Omega)$ be a function of bounded variation and $\Omega\subset \mathbb R^2$ be a smooth domain. I know it is possible to approximate function $u$ with polynomials, i.e.,
$$
u = \...
- 28k
5
votes
1
answer
211
views
Pointwise convergence in functional calculus
Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$
$$\left\lVert A_n x-Ax \right\rVert \...
- 4,729
1
vote
1
answer
131
views
Convergence of $L^p$ of approximation
Let $f \in L^p(\mathbb R^n)$ be given. Consider a partition of rectangles $I_{ij}:=[x_i,x_{i+1}]\times [x_j,x_{j+1}]$ of $\mathbb R^2.$
Then, we may define the coefficients
$$\alpha_{ij}= \frac{1}{\...
- 128k
1
vote
0
answers
237
views
On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
-2
votes
1
answer
99
views
A question on the zeros involving the equation containing exponential factor [closed]
I recently encounter a puzzle that: how to show that for any constant $c_1,c_2,c_3,c_4 \in \mathbb{R}$ the equation
$$c_1 e^t+c_2e^{-t}+c_3 e^{\alpha t}+c_4 e^{-\alpha t}=0$$
has at most only one ...
- 188k
0
votes
1
answer
115
views
Verifying that a map to $L^2_{\text{loc}}$ is continuous
Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
- 1,913
1
vote
0
answers
50
views
Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
2
votes
1
answer
311
views
Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
- 536
1
vote
1
answer
165
views
Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?
Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$
Moreover, we know ...
- 128k
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
1
vote
1
answer
337
views
Bochner measurable; continuous operator
It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $...
- 66
2
votes
1
answer
224
views
Strongly continuous semigroup: continuous or continuous componentwise?
Let $T(t)_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$
Then, one can consider the function
$f(t_1,t_2):= T(t_1)S T(t_2)x$ where $x$ is a fixed element of the Hilbert space ...
- 5,005
1
vote
0
answers
170
views
non-analytic functions with arbitrary large derivatives [closed]
This may be a trivial question but I can't see it immediately.
Suppose $\{a_k\}$ is an increasing sequence of positive reals. Does there exist a smooth function $f \in C^{\infty}([0,1])$ such that $\...
- 4,115
1
vote
2
answers
435
views
Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix [closed]
I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$
is it sufficient to prove the norm of ...
- 159
5
votes
1
answer
2k
views
Commuting with self-adjoint operator
Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a ...
- 42.8k
0
votes
0
answers
60
views
Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
2
votes
2
answers
857
views
Hölder functions dense in space of bounded continuous functions (for non-compact manifolds)
Let $M$ be a non-compact manifold and denote by $C_b(M)$ the space of bounded continuous functions on $M$. Is it true that the space of Hölder functions is dense in $C_b(M)$ (in the $C^0$ norm: $||f||=...
- 60.6k
25
votes
2
answers
2k
views
$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$
Let $f$ be a real function with domain R.
If $f^2$ and $f^3$ are both infinitely differentiable on R,
how to prove $f$ is infinitely differentiable on R?
I have been thinking about this problem for a ...
- 91.8k
7
votes
1
answer
337
views
Flows in Hilbert spaces
Let $\varphi: [0,T] \rightarrow H$ be a Hilbert space valued $C^1$-function. Let $H = X \oplus X^{\perp}$ such that $\varphi(0) \in X$ and the implication $\varphi(t) \in X \Rightarrow \varphi'(t) \in ...
- 25.3k
1
vote
0
answers
101
views
Non standard Lipschitz extension
Consider a ball B and let $f(x) \in L^1(B)$ such that $\int_B f(x) dx = 0$. Furtheremore, there exists a closed set $E \subset B$ such that $f|_E$ is Lipschitz. The standard Lipschitz extension ...
- 455
5
votes
0
answers
195
views
What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?
Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
- 13.8k
3
votes
1
answer
209
views
A particular measure of noncompactness?
I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
...
- 128k
2
votes
0
answers
125
views
Constant periodic Sobolev embedding
Dear mathoverflowers,
I would like to have a reference regarding the optimal constant in the Sobolev embedding
$$
\|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s},
$$
($H^s$ denotes the standard L^2 ...
- 28k
1
vote
1
answer
642
views
Interchange of integration order (of a not absolutely convergent integral with sinus)
Can we interchange the integral order of this integral to start integration on $x$ ? (Taking $g$ and $f$ two functions of rapid decrease which are $o(x^2)$ near zero)
$$A=\int_{0}^\infty \int_0^{\...
- 128k
1
vote
2
answers
333
views
What gives a "Parseval like" equation mixing cosine and sine Fourier transforms ?
Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by:
$$\mathcal{F}^c [f (x)]=\int_0^{\infty} f(t) \cos(xt) dt $$
$$\mathcal{F}^s [f (x)]=\...
- 1,199
4
votes
0
answers
212
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
2
votes
1
answer
144
views
Convergence of sequence of images of Schur multipliers
Let $\eta$ be a continuous bounded function on $(0, \infty)^{2}$ so that $\eta(0,0)=1$. Let $A$ be a bounded operator on $\ell^{2}(\mathbb{Z}_{\geq 0})=\ell^{2}$ (by bounded operator I will always ...
- 31
11
votes
2
answers
2k
views
Operator that commutes with projections
We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$
Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
- 128k
2
votes
1
answer
964
views
Is the Delta distribution a continuous functional on $H^1(\mathbb{R})$? [closed]
While it is easy to see that $H^1(\mathbb{R})$ are Hölder $1/2$-continuous, I started wondering whether this implies that $\delta_x(\varphi)=\varphi(x)$ is continuous as a functional
$$\delta_x:H^1(\...
- 28k
2
votes
1
answer
149
views
The infinite set of $SBV$ function?
Let $u\in SBV(\Omega)$ where by $SBV$ we denote the special bounded variation function and $\Omega\subset \mathbb R^N$ is open bounded.
Let's identify $u$ by its approximation representative (see ...
- 512
5
votes
0
answers
349
views
Tietze extension theorem for lower semi continuous functions
On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
- 29.8k
3
votes
1
answer
262
views
Can the $L^{\infty}\to L^{\infty}$ norm be bounded by the trace norm?
Let $k\in C(\mathbb{R}^2; \mathbb{R})$ be a continuous function. Suppose that the operator $K\colon L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ defined by the formula
$$(Kf)(x)=\int_{\mathbb{R}} k(x,...
- 128k