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      Questions tagged [real-analysis]

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

      2
      votes
      2answers
      34 views

      Rate of convergence for eigendecomposition

      Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta_D$ on a set of cardinaltity 4 is the matrix $$\Delta_D := \left( \begin{matrix} 2 &...
      0
      votes
      0answers
      23 views

      Approximation of functions by tensor products

      Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...
      5
      votes
      0answers
      47 views

      Minimum intersection of two $\ell_p$-norm balls separated by a fixed $\ell_1$-distance

      Given two $\ell_p$ norm balls in $\mathbb{R}^d$, $$B_a(x) = \{ z\in \mathbb{R}^d: \|x - z\|_p \leq a \}$$ and $$B_b(x + \delta) = \{ z \in \mathbb{R}^d: \|x + \delta - z\|_p \leq b \}\hbox{ with }x \...
      0
      votes
      1answer
      113 views

      Is the function $f(x=\sum a_n/2^n)=\sum na_n/2^n$ continuous and nowhere differentiable? [on hold]

      Let $x=\sum_{1<=n<\inf}a_n/2^n$ be the binary expansion of a real number in [0,1]. Assume that infinitely many of a_n are 0 so that the expansion is unique. Define $f(x)=\sum_{1<=n<\inf}...
      2
      votes
      0answers
      43 views

      Approximation of functions in $L^p(R^d;L^\infty)$

      Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that $$...
      3
      votes
      1answer
      256 views

      Where to find the proof of this property?

      I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...
      0
      votes
      0answers
      51 views

      A hard to answer Digamma question [on hold]

      I just learning online obout Polygamma function, and I want to know what x equal to (and how to get it) when ψ(x)=1 and x>1.
      -2
      votes
      0answers
      47 views

      What is the relationship (mapping) from a reciprocal function 1/r to a exponential function exp(-r)? [on hold]

      The mathematical problem: Consider the mapping from $r$ to $u$. for large $r$ the theory suggests a formulation like $u=a_1 e^{-a_2 r}$, which means that the function decay exponentially. for ...
      1
      vote
      0answers
      55 views

      How to see the divergence of a series is not faster than some order? [on hold]

      $$ \sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p} $$ For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.
      5
      votes
      0answers
      77 views

      Square of Jacobian

      Given a smooth vector-valued function $u:B_1(0)\subset \mathbb{R}^n\to \mathbb{R}^m$, one can look at the matrix \begin{equation} Q_{ij}= \sum_{k=1}^m \nabla_i u^k \nabla_j u^k \end{equation} This is ...
      -3
      votes
      0answers
      35 views

      How to find the volume using triple integral spherical coordinates? [on hold]

      A hemispherical bowl of radius $5 cm$ is filled with water to within $3cm$ of the top. Find the volume of water in the bowl? How can I find the volume using spherical coordinates? writing it like ...
      -1
      votes
      0answers
      70 views

      Counter-example of Subsequence Criterion? [migrated]

      The last argument shows that if $X_n\to X_\infty$ a.s. and $N(n)\to\infty$ a.s., then $X_{N(n)}\to X_\infty$. We have written this out with care because the analogous result for convergence in ...
      -2
      votes
      0answers
      42 views

      Closed union of all connected subsets that contain x [on hold]

      If Cx(S) is the union of all connected subsets of S which contain x, it is connected. I understand that, but what I don’t understand is that if S is closed, then Cx(S) is closed. Isn’t that like ...
      3
      votes
      3answers
      138 views

      Existence of solution to linear fractional equation

      We consider the equation $$?\sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$ where $\lambda_j>0$ and $x_j$ are real distinct numbers. I want to show that if $\lambda_k$ is small compared to the ...
      4
      votes
      0answers
      96 views

      Approximation of a compactly supported function by Gaussians

      Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...

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