# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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### Infinite partial fraction expansions to compute fractional iterations and recurrences

Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...
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### Convergence of fractional laplacian

Let $s\in(0,1)$, let: $$\mathcal{S}_s=\biggl\{ f\in C^\infty(\mathbb{R}^n): \sup_{x\in\mathbb{R}^n}(1+|x|^{n+2s})|\partial^\alpha(x)|<\infty,\,\forall \alpha\in\mathbb{N}_0^n\biggr\}.$$ The linear ...
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### A lower-bound on matrix-function with vector product

I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...
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### A question about Fourier transform of a function defined by an integral

I have the function: $$G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta},$$ for all $x\in\mathbb{R}^n$ and $k>0$....
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### Expressing the measure of a set in terms of the characteristic function of the measure

Let $\mu$ be a discrete, finitely supported probability measure in $\mathbb{R}^d$ and denote by $\phi$ be the characteristic function of $\mu$, i.e. $\phi(t)=\mathbb{E}e^{i<t,X>}$, where $X$ is ...
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### Decomposition of a probability measure into positively and negatively supported ones

Consider the class $\mathcal{M}$ of probability measures $\mu$ on $\mathbb{R}$ whose moment generating function $\int e^{tx} d\mu(x)$ is finite for $t$ in a neighborhood of zero. Let $\mathcal{M}_{+}$ ...
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### Interpolation theory and $C^k$-spaces

Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets. Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}... 1answer 99 views ### “Approximating” linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients In a lecture I once attended, I remember the speaker using a result of the following nature:$$Let$\{A_n\}_{n=1}^\infty \subset \mathbb R$be a sequence satisfying a recursion of the form $$P(n) ... 1answer 75 views ### Kolmogorov entropy of a subset of L^1 How can we estimate the Kolmogorov \epsilon-entropy$$H_\epsilon (A,L^1(\mathbb R))$$where A = \{f:\mathbb R \to [0,K] \text{ s.t. f \in L^1 and has total variation TV(f) \le M}\} ? 0answers 86 views ### Lower bound on iterated matrix application Let n \in \mathbb Z^2 such that the non self-adjoint weighted Laplacian is$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$the adjoint operator is then$$(\Delta^* u)(n)=... 1answer 152 views ### Optimal constant in Sobolev embedding It is well-known that the Sobolev space$H^1(0,s)$embeds continuously in the space of continuous functions$C[0,s]$; in fact, Marti has found in 1983 that the optimal embedding constant is$\sqrt{\...
Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and \$\...