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0 votes
0 answers
241 views

Continuity of a function

Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$. Is the following function continuous for $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$: $$ F(z)=\bigg(\alpha-i\...
0 votes
1 answer
302 views

An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function: \begin{equation} g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k. \end{equation} Prove that $\exists C>0$ and $\phi(s)$ such ...
1 vote
1 answer
383 views

Solution of a PDE and its uniqueness

Hallo, consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE $\...
2 votes
0 answers
800 views

Controlling the Lipschitz norm of the limit of a sequence of functions

Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
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 $...
0 votes
1 answer
337 views

Integral inequality

Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...
3 votes
1 answer
2k views

A question about a formal power series manipulation

I want to find a function $f(x,y)$ which can satisfy the following equation, $\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = exp [ \sum _{n=1} ^\infty \frac{f(x^n,y^n)}{...
0 votes
0 answers
244 views

Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of ...
2 votes
2 answers
2k views

Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?

Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
2 votes
1 answer
190 views

Completeness for spaces of eventually bounded nets

Let $A$ be a directed set, and $\ell^\infty_A$ the (complex vector) space of all eventually bounded nets $A\to \mathbb{C}$. We can define the limit superior seminorm on $\ell^\infty_A$: $$ \vert\vert{...
2 votes
1 answer
469 views

If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?

If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
6 votes
3 answers
1k views

functional subrings

I should recall the notion of maximal subring of a commutative unitary ring $R$. Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
0 votes
0 answers
183 views

Continuity of the Shadow of a Nondecreasing Function

So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
2 votes
1 answer
403 views

The set of Upper semi-continuous functions as a ring.

I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set {$[a , b): a,b \in \mathbb{R} $} as it's base. If $X$ is a topological space, an upper ...
26 votes
3 answers
7k views

Dual of bounded uniformly continuous functions

Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$? I ...
5 votes
2 answers
774 views

Can we calculate the inner product of a semicontinous function with the Dirac delta function?

Dear all, It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
1 vote
1 answer
3k views

In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives

I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ...
0 votes
1 answer
721 views

Pointwise limit at Lebesgue's point

Dear MOs, I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by the Lebesgue ...
1 vote
1 answer
496 views

Convergence of Difference Quotients

Let $\gamma_{\varepsilon} \rightharpoonup \gamma$ in $W^{1,\infty}(0,1)$. Then for any fixed $s \in \mathbb (0,1)$ does the limit $\lim_{\varepsilon \rightarrow 0} \frac{\gamma_{\varepsilon}(s\...
4 votes
1 answer
471 views

Ask for theory about the weighted L^2(R^d) space.

Dear MOs, I am now considering the following norm: $$ ||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:. $$ where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
0 votes
1 answer
612 views

Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
0 votes
1 answer
238 views

A property of a quasiperiodic function

Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b. Does the limit of $\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$ Exist ...
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 ...
4 votes
1 answer
882 views

What is the domain of the "average operator"?

I can try to define an averaging operator for functions, namely let $$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$ by $$Af = \lim_{N\to\infty} \frac{1}{N}\int_0^N f(x)dx$$ whenever the limit ...
3 votes
0 answers
409 views

Continuous function sort

If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
7 votes
3 answers
4k views

Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous function. [Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable? If not, can one find a counter-example? Note that, for any $c$, ...
2 votes
0 answers
564 views

Young inequality in weighted spaces

Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$. Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$. Does ...
5 votes
1 answer
540 views

Cosets of groups of functions

Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$. The set $\mathcal ...
2 votes
3 answers
3k views

Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set

It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ...
6 votes
2 answers
4k views

Is there dual space of the distributions $\mathcal{D}'(R)$?

Dear MOs, Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
6 votes
0 answers
8k views

Dual space of continuous functions

Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
2 votes
1 answer
2k views

Modified Lebesgue differentiation theorem

Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
1 vote
1 answer
224 views

Can symmetrizing a contraction increase the speed of convergence?

Dear community, I have a problem which is very simple to state but seems to be hard to answer. Statement of the problem Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
2 votes
1 answer
942 views

A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$, $s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the singular values of a $2\times2$ matrix. Is it true that $$\left|s_{1}\...
6 votes
2 answers
2k views

Continuity of a convolution (Version 2)

Hello, This problem bothers me for some time. Suppose that $\mu$ is a non-negative Radon measure (or positive linear functional of the space of continuous functions with compact support); $\psi$ is ...
6 votes
2 answers
2k views

How to prove the Hahn-Banach constructively

I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space. Thanks in advance for any helpful answers.
4 votes
1 answer
1k views

Hausdorff dimension of graphs .

Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
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, ...
2 votes
2 answers
1k views

Characterization of Weakly measurable functions

I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong ...
1 vote
1 answer
275 views

Shift operator that generates separable orbit

Suppose, that $f$ is bounded measurable function, $T_h(f)(x) = f(x+h)$ is the shift operator. How to prove, that if the whole orbit $T_h(f):\, h\in\mathbb{R}$ has a dense, countable subset $T_{n_k}(f)$...
1 vote
0 answers
174 views

Eigenvalues of a Parametrized Family of Linear Functions

Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number. For each $\alpha$, it is given that $L(\alpha)$ is a ...
5 votes
1 answer
781 views

Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
2 votes
1 answer
465 views

What is the regularity of the argument of a complex function?

Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
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 \...
2 votes
3 answers
946 views

How can I measure the Morse index in infinite dimensions?

Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
2 votes
2 answers
317 views

Bibliography for topologies defined by a family of seminorms

Hello I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource. Thank you very much.
4 votes
2 answers
340 views

Embeddings of Weighted Banach Spaces

Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces $$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...

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