# Questions tagged [fa.functional-analysis]

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

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### Inverse associated Legendre transform of products

Debnath and Harrell 1976 introduced the associated Legendre transform of a function $F(x)$ and its inverse, \begin{align} f(n,m) &= T_n^m[F(x)] = \int_{-1}^1 (1-x^2)^{-m/2} P_n^m(x) F(x) dx \\ F(x)...
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### Regarding the boundary of the range of a holomorphic map on the unit disc

Let $\Omega$ be a convex domain in $\mathbb{C}^n$. Let $f:\mathbb{D}\longrightarrow \bar\Omega$ be a holomorphic function. Let $z\in f(\mathbb{D})\cap \partial\Omega$. Let $\phi$ be a linear ...
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### Example when Kantorovich condition would not hold

Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator $$(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.$$ Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
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### Error bounds on the expansion of square root of matrix

I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...
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### Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)

Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...
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As we know the inhomogeneous Sobolev space (we only consider $s>0$) $${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ... 0answers 45 views ### Multiobjective Optimization with (too many?) functions Consider a multiobjective optimization problem$$\min\limits_{x\in \Omega} f(x),$$where f:\Bbb R^n \rightarrow \Bbb R^m and \Omega \subseteq \Bbb R^m. A point \bar{x} \in \Omega is said to be: ... 1answer 140 views ### For self-adjoint A and B, when is (A+iB)^* the closure of A-iB? Suppose that I have two self-adjoint operators A and B such that \mathcal{D}(A)\cap\mathcal{D}(B) is dense and B positive. Then A\pm iB (with domains \mathcal{D}(A)\cap\mathcal{D}(B)) are ... 0answers 105 views ### Conditions on the inequality with a gauge norm Let \Phi(x)=\int_0^x \phi(y)\,dy, x \in \mathbb{R}_+, be an N-function, and let u be locally inferable on \mathbb{R}_+. Consider the gauge norm$$ \rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$. Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...