All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
3
votes
1
answer
529
views
Spectrum of self-adjoint operator
As a non functional analyst, I stumbled over the following question:
Given a self-adjoint Operator $T:D(T) \subset H \rightarrow H.$ Assume we know that $T$ has some eigenvalue $\lambda$ which is ...
- 173
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
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$...
- 1,649
3
votes
1
answer
657
views
Banach space of discontinuous functions(Killing continuous functions)
Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
$$\...
- 356
3
votes
1
answer
133
views
A recurrent sequence related to the Brouwer fixed-point theorem
Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u_0 \in K$, and define by recurrence $u_{n+1} = \frac{1}{n+1} \sum_{j=0}...
- 7,249
3
votes
1
answer
6k
views
Classical Derivative, Weak Derivative and Integration by Parts
Hello,
While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.
QUESTION
Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $...
- 895
3
votes
2
answers
949
views
Reference for proof that $C_b^* = rba$
The following theorem seems to have folk status:
The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, ...
- 459
3
votes
1
answer
187
views
Is this property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
3
votes
2
answers
429
views
Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
- 4,074
3
votes
1
answer
108
views
$L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality
For each $n \in \mathbb{N}$, the Hermite function $\psi_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by
\begin{equation}
\psi_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}...
- 3,477
3
votes
1
answer
110
views
Question on the existence/uniqueness of the fixed point
Let $E$ a Banach space ($E$ is the space of continuous functions on $[0,T]$ for my case). Let $F, G: E\times E\to E$ be contraction maps of contraction constant $\epsilon>0$. Given $b\in\mathbb R$, ...
- 1,334
3
votes
1
answer
762
views
Functions dense in $L^1[0,1]$ but not in $L^2[0,1]$
Is there a family of continuous functions $(f_n)_{n \in \mathbb{N}}$ on $[0,1]$ whose span is dense in $L^1[0,1]$ for the $L^1$-norm, but not dense in $L^2[0,1]$ for the $L^2$-norm?
Some preliminary ...
- 188
3
votes
1
answer
75
views
Analyticity of $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ in the complex plane?
Let I have the following function,
$f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$
Where, $x,y \in C$, $a,b\in R$ and $- m \le \Re (x),\Re (y),\Im (x),\Im (y)...
- 141
3
votes
1
answer
115
views
Approximation of vectors using self-adjoint operators
Let $T$ be an unbounded self-adjoint operator.
Does there exist, for any $\varphi$ normalized in the Hilbert space, a constant $k(\varphi)>0$ and a sequence of normalized $(\varphi_n)$ such that $$...
- 173
3
votes
1
answer
353
views
Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization)
I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also ...
- 1,467
3
votes
1
answer
316
views
Rademacher, maxima, convex hulls
Let $F\subset \mathbb{R}^n$ be a finite set and $\sigma$ be uniformly distributed over $\{-1,1\}^n$. The usual Rademacher average of $F$ (modulo normalizing factors) is
$$ R_n(F)=\mathbb{E}_\sigma \...
- 6,541
3
votes
1
answer
172
views
Infimum of an integral functional involving a symmetric matrix
I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int_{B_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du,
...
- 391
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
...
- 291
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
3
votes
1
answer
531
views
An argument in the proof of a compactness theorem
In the proof of a compactness theorem involving fractional derivatives in Temam's Navier-Stokes Equations, an argument as the following is made.
Suppose $X_0,X,X_1$ are Hilbert spaces such that
...
user14319
3
votes
1
answer
250
views
Characterization of a subset of [0,1] $II$
My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...
- 1,835
3
votes
1
answer
495
views
Inequality in the Sobolev space $H^1$
I've found the following inequality
$$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ^2\bigg)^{\frac{q}{2}-a}+\frac{c}{r^{2a}}\bigg(\int_{B_r}...
user45822
3
votes
2
answers
135
views
series representation of bivariate functions
Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...
- 53
3
votes
1
answer
352
views
Integral Equation with "convolution"
I've got the following problem I'm working on which is related to some of my research:
Solve:
$f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$
for f, given $G$ which has whatever smoothness ...
- 31
3
votes
1
answer
362
views
Cartesian product of test function spaces
Mini introduction
Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i \in \...
3
votes
1
answer
185
views
Uniformly closed ideals of smooth/real analytic functions
Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
- 513
3
votes
1
answer
153
views
Boundedness of an extension operator
Let $d \ge 2$ be a positive integer. For $x=(x_1,\dotsc,x_{d-1},x_d)$, we write $x'=(x_1,\dotsc,x_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x_d) \mid x_d>0\}$ denote the $d$-dimensional upper half-space.
...
- 721
3
votes
1
answer
292
views
Continuity of Legendre transform
Let $I \subset \mathbb{R}$ be an interval, and $f_n, f: I \rightarrow \mathbb{R}$ a convex function; then its Legendre transform is the function $f^{\ast}: I^{\ast} \rightarrow \mathbb{R}$ defined by
$...
- 1,043
3
votes
1
answer
354
views
Hölder inequality between different Orlicz spaces
If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$.
But if $g$ is a little bit more than $L^s$, say $L^s \...
- 363
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 $\tau \in \mathbb R$}.
$$
Does it follow that $\phi$ is a ...
- 4,143
3
votes
1
answer
302
views
Convergence of a level set when $f^n\to f$ in $C^1$ sense
Let $f^n$ be a family of $C^1$ functions and $f(x)=-|x|^2+1$ such that
$$f^n\to f$$
in $C^1$ sense as $\varepsilon\to 0$. I want to ask that does the level set $\{f^n=0\}$ converges to $\{f=0\}$ in ...
- 379
3
votes
1
answer
191
views
A convolution type singular integral operator with log
Define a convolution type operator $T_m$ by
$$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer.
Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
- 903
3
votes
2
answers
2k
views
A condition under which an Lp function is L-infinity [closed]
I am looking for a condition under which a function in $L_p(\Omega)$ is also in $L_\infty(\Omega)$. The condition may be on the function itself, or on $\Omega$.
In other words, is there anything that ...
- 93
3
votes
1
answer
203
views
Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$
Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
user139844
3
votes
2
answers
498
views
if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?
This question is related to solving $f(f(x))=g(x)$.
Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
user147204
3
votes
1
answer
149
views
Hardy-Littlewood-Sobolev for "componentwise product" of Riesz kernels
Let $d\in\mathbb N$ and $0<\alpha<d$. Define the Riesz kernel $K_\alpha(x):=|x|^{\alpha-d}$, and the associated convolution operator
$$K_\alpha f(x):=\int\frac{f(y)}{|x-y|^{d-\alpha}}~dy.$$
The ...
- 891
3
votes
1
answer
255
views
Closure of tensor product /tensor product semigroup
In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
user127612
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 ...
- 39
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.,
$$
u = \...
- 133
3
votes
1
answer
148
views
Prove existence of continuous function on $(0,1)$ with special properties [closed]
Consider the interval $I=(0,1)$ and let $f,g$ be two linearly independent continuous functions on $[0,1]$.
I am asking if there is a continuous function $h$ such that
$$\int_0^1 h(s) f(s) ds=0$$
$$...
- 501
3
votes
1
answer
144
views
Operator norm of almost mathieu operator
The almost Mathieu operator has become famous since it is the central object of the ten martini problem.
In this paper here a bound on the operator norm is given. Although the bound is of course ...
- 31
3
votes
1
answer
187
views
Free quantum evolution operator on Sobolev space
I am not a mathematician, but would like really like to get some confirmation on the things I am doing here.
Let $-\Delta: H^2(\mathbb{R}) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ then ...
- 95
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
3
votes
1
answer
133
views
Restrictions on spectral measure
Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\...
- 587
3
votes
1
answer
210
views
Using $H^2$ to find a cyclic vector in $\ell^2$
Let us consider $\ell^p(\mathbb{Z})$. We know that the vector $e_1=(\dots,0,0,1,0,0,\dots)$ is a cyclic vector in sense that given the right shift operator $S:(\dots,x_0,x_1,x_2,\dots)\mapsto (\dots,...
- 31
3
votes
1
answer
153
views
Separability of $R_+\times\mathcal{C}(R_+)$
Let $\mathcal{C}(R_+)$ be the space of continuous functions $f$ defined on $[0,+\infty)$ with $f(0)=0$. Denote by $\Omega$ the product of $R_+$ and $\mathcal{C}(R_+)$. Now endow $\Omega$ with the ...
- 1,835
3
votes
1
answer
171
views
Characterization of a set in $\mathbb{R}^d$
Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...
- 57
3
votes
1
answer
693
views
Equivalence of negative Sobolev norm of derivative to $L^2$-norm
Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the $L^2(S)...
- 2,270
3
votes
1
answer
643
views
Is a Cauchy principal value invariant under a "change of variables"?
Let $f \in C^{\gamma}_c(\mathbb{R}^n) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere ...
- 33
3
votes
1
answer
1k
views
Showing a singular integral operator takes Hölder continuous functions to Hölder continuous functions of the same order
I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as ...
- 31
3
votes
1
answer
220
views
What we know about the function in Fefferman's Theorem
In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
- 5,405