Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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Nonseparable counterexamples in analysis

When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
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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. ...
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Smoothness of distance function to a compact set

Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$. Suppose that: $K$ is regular : ...
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What is the distribution of the following limit?

Assume $x \in \mathbb{R}$. We already know that $$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$ Here $\delta_x$ denotes the Dirac distribution. If we ...
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Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$V(x) = V_0 \mathbf 1_{[-a,a]}(x),$$ where $\mathbf 1_{[-a,a]}$ denotes the ...
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Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions

Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
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Reference request for type of specific integral equation in two variable:

Consider the following integral equation: $$\int_0^\infty K(t,y)\phi(t,x)dt=0$$ Here, $K(t,y)$ is a trigonometric kernel and $\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$). I want to find the ...
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Another question about asymptotic models in Banach spaces

The array $(x_{i}^{k})_{i=1,k\in\mathbb{N}}^{\infty}$ of normalized $M$-basic sequences in a Banach space $X$ is itself called $M$-basic if, for every $k\leq i_{1}<i_{2}<\ldots$, the diagonal ...
280 views

Existence of a translation-invariant basis of $\ell^2$

This question is heavily inspired by this other one, but is meant to be a hopefully more accessible variant of it (and I think slightly more natural). I give four equivalent formulations of the same ...
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Reflexive subalgebras of $B(X)$

Let $X$ be a reflexive Banach space, and let $B(X)$ denote the set of all bounded linear operators $X\to X$. Does there exist a subalgebra $A\subseteq B(X)$ with the following properties? A is unital....
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How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
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Embedding theorems for fractional Sobolev spaces $W^{s,p}(\Gamma)$ where $\Gamma$ is closed piecewise $C^1$ curve in $\Bbb R^2$

I am interested in embedding theorems for the fractional Sobolev space $W^{s,p}(\Gamma)$ where $\Gamma$ is closed piecewise $C^1$ curve in $\Bbb R^2$ such as the boundary of a triangle or rectangle. ...
665 views

Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
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Green function of symmetric stable process in dimension 1 and 2

Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
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Reference request for semilinear PDEs in dimension 2

I am interested in the study of the (semi-linear, I suppose) equation $$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\ u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$ ...
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Let $X$ be a compact, Hausdorff, topological space and let $B$ be a unital C*-algebra containing $C(X)$, and sharing units with the latter. In other words $$1_B\in C(X)\subseteq B.$$ For each $... 2answers 229 views Are equicontinuous function dominated by a continuous function? Let$f_n: [0, 1] \to \mathbb R$be an equicontinuous sequence of functions. Does there exist a continuous function$f$that dominates$f_n $in the following sense? We say$f$dominates the sequence$...
The picture below is taken from this paper: http://real.mtak.hu/22877/. The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...