# Questions tagged [real-analysis]

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

2,549 questions

**2**

votes

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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 ...