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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2answers
87 views

“Oddity” of a log-Bessel sequence happening at powers of $2$

Define the sequence $b_1=1$ and $$b_n=\sum_{k=1}^{n-1}\binom{n-1}k\binom{n-1}{k-1}b_kb_{n-k}.$$ By now, there is enough in the literature that $C_n$ is odd iff $n=2^k-1$ for some $k$ where $C_n$ are ...
3
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1answer
225 views

What are ways to compute polynomials that converge from above and below to a continuous and bounded function in $[0,1]$?

My interest is to take a coin of unknown bias $\lambda$ and use it to produce a coin of bias $f(\lambda)$. This is called the Bernoulli Factory problem, and only certain functions $f$ can be simulated ...
0
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1answer
143 views

Error function of multivariate Gaussian

I'm trying to prove that a sequence of functions $(k_N)_{N\in\mathbb{N}}$ $$k_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$ where $r>0$, $k>2$ and Edit: I have forgot to ...
7
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2answers
277 views

On a monotonicity property of Fourier coefficients of truncated power functions

Is it true that $$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$ is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$? This question is related to this previous one. Twice integrating by parts, one ...
18
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2answers
974 views

When are Fourier coefficients monotonic?

Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by $$ \hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\...
2
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3answers
304 views

how to numerically evaluate $\int_{0}^{\infty} \frac{1}{x!} dx$ [closed]

So I was graphing the equation $ y=\frac{1}{x!} $ for $ x \geq 0$ and tried the integral: $$\int_{0}^{\infty} \frac{1}{x!} dx$$ $$\int_{0}^{\infty} \frac{1}{\Gamma(x+1)} dx$$ $$\int_{0}^{\infty} \frac{...
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1answer
155 views

Integrability of $\int \log(f(x)) f(x) dx$ for a probability density function $f$

I am looking for weak conditions when a probability density function $f$ on $\mathbb{R}^d$ has a finite integral $$ \int_{\mathbb{R}^d} \log(f(x)) f(x) dx. $$ Any references would be appreciated.
4
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1answer
167 views

Is there a generalization of these q-series identities?

Denote $(q;q)_n=(1-q)(1-q^2)\cdots(1-q^n)$. The below three identities are known. \begin{align*} \sum_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\binom{n+1}2}}{(q;q)_n} &=1-\sum_{n\in\mathbb{Z}}(-1)^nq^{\...
7
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2answers
393 views

Bounding supremum norm of Lipschitz function by L1 norm

Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ? In ...
2
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1answer
121 views

How can a divergent nested radical be regularized (analogously to Cesaro sum regularization of a divergent series)?

The infinite series $1-1+1-1+\cdots$ diverges because the sequence of partial sums, $1,0,1,0,\ldots$ has no limit. However, it is well know that we can get around this problem in a number of ways; ...
2
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1answer
93 views

Approximating a limit of an integral

How can we prove the following asymptotic lower bound for the regularized Beta function when $n\rightarrow\infty$? $$\int_0^{1} I_{2 t - t^2}\left(\frac{n - 1}{2}, \frac{1}{2}\right) dt=\Omega\left(\...
2
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3answers
440 views

Ramanujan's Master Formula: A proof and relation to umbral calculus

The Ramanujan's master theorem states that: $$ \int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s} $$ I found a really strange proof recently on a personal blog: Define $...
9
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1answer
462 views

The infimum of a gradient over the whole $\mathbb{R}^d$

Let $\{f_k\}:\mathbb{R}^d\to\mathbb{R}$ be a sequence of $C^1$ functions which converges pointwise to 0. Is it true that $$\lim_{k\to+\infty}\inf_{x\in\mathbb{R}^d}|\nabla f_k(x)|=0?$$ If $d=1$ I ...
0
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0answers
85 views

Proof of Araki-Lieb-Thirring inequality

I would like to know a proof of Araki-Lieb-Thirring inequality. Let $H$ be a separable Hilbert space. Let $A$ and $B$ be positive and self-adjoint (not necessary bounded) operators on $H$. Let $BAB$ ...
-2
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1answer
131 views

Is a function piecewise continuous if it has a left-limit and a right-limit at every point in its interval domain and equals at least one of these? [closed]

Suppose a real-valued function f, whose domain is an interval, has the property that at every point in its domain it has a left-limit and a right-limit, and it equals at least one of these. Is it ...
1
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0answers
24 views

Hardy maximal function on the torus

A few years ago I asked a reference about the Hardy maximal function on the flat torus. Mateusz Kwaśnicki kindly answered in a comment, and confirmed my conviction that basically everything which is ...
12
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6answers
2k views

Alternative proofs sought after for a certain identity

Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so QUESTION. can you provide another verification for the problem below? Problem. Prove ...
0
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0answers
39 views

The function $u$ is in $H^1_a$?

Let $a\in (-1,1)$, let $H^1_a$ be the completion of $C^\infty_c(\mathbb{R}^n\times(0,\infty))$ under the norm: $$\|v\|_a=\sqrt{\int_{\mathbb{R}^n\times(0,\infty)}y^a|u(x,y)|^2\,dx\,dy+\int_{\mathbb{R}^...
0
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0answers
38 views

definition of functions that “weakly vanishes as $y\to\infty$” and find a proof of Theorem 9 in the article

I'm reading "Extension problem and fractional operators: semigroups and wave equations" by P.R. Stinga, i have two questions: in Theorem 7 the author use the state "weakly vanishes as $...
2
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0answers
34 views

A question for regularity of solutions to wave equation

let $T>0$ and suppose $\Omega$ is a bounded domain with smooth boundary. Let $g \in L^\infty(0,T;H^1(\partial \Omega))$ and consider the wave equation \begin{equation}\label{pf0} \begin{aligned} \...
1
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1answer
79 views

Realizing a set as the image of a smooth map

Consider the following subset of $\mathbb{R}^2$: $S = \{ (x, y) \in \mathbb{R}^2 : |y| \leq |x|^{3/2} \}$ (See here for a plot on Wolfram Alpha.) The origin $(0, 0)$ is a kind of singular point of $S$....
5
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1answer
224 views

Find a combination of convex function so that it is positive

A student in my class asked me the following question, I did know what tools will be needed to attack it. But I found it is an interesting question. Let $f_1,f_2$ be two convex functions on $[0,1]$ ...
0
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1answer
106 views

“Arc” length parametrization for surfaces

If we have a function $\phi:\Omega\subseteq\mathbb{R}^2\to\mathbb{R}$, $\phi\in C^2(\Omega)$ is there a way to find a function $d:\Omega\to\mathbb{R}, d\in C^2(\Omega)$ so that: $|\nabla d(x,y)|=1,\ \...
3
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2answers
146 views

Asymptotic bound for $\sum_{x=0}^\infty \sum_{y=0}^\infty (x+y)^m e^{-\frac{x^2}{2i} - \frac{y^2}{2j}}$ for $i$ and $j$ large

Note: This question relates to two previous questions on math.stackexchange (1 and 2), neither of which had satisfactory answers after posting bounties. Whilst trying to count certain types of ...
3
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1answer
154 views

Euler-Lagrange equation for a functional

What does it mean that the equation: $$ \text{div}_{x,y}(y^a\nabla_{x,y}u)=0,\quad \text{in }\mathbb{R}^n\times(0,\infty),$$ is the Euler-Lagrange equation for the functional: $$ J(u)=\int_{\mathbb{R}^...
2
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0answers
74 views

A problem of uniqueness

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem: $$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$ $$ u(x,0)=f(x),\quad\forall x\...
0
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1answer
102 views

Some multivariate Taylor series and corresponding smoothness balls

Suppose I have a multivariate function $f$ from $\mathbb{C}^d$ to $\mathbb{C}$ that accepts a Taylor expension of the form $$f(\mathbf x) = \sum\limits_{\mathbf k \in \mathbb N^d} a_{\mathbf k} \...
1
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1answer
65 views

Properties of Magnus expansion

The Magnus expansion $\Omega(t, t_0) = \sum^\infty_k \Omega_k(t,t_0)$ is so that the solution $$ Y(t) = e^{\Omega(t,t_0)}\,Y_0, $$ solves an ODE $$ Y'(t) = A(t)\,Y(t), \qquad Y(t_0) = Y_0. $$ ...
0
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0answers
34 views

How to approximate the function $u$ in the weighted $L ^ 2$ norm?

Let $s\in(0,1)$, $f\in \mathcal{S}(\mathbb{R}^n)$, define the function: $$ u(x,y)=\int_{\mathbb{R}^n} \frac{y^{2s}}{(|x-\xi|^2+y^2)^{(n+2s)/2}}f(\xi)\,d\xi,\quad\forall(x,y)\in\mathbb{R}^n\times(0,\...
3
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0answers
113 views

Opposite of the curl operator and Biot-Savart kernel

Note: I just realized that using $\omega$ and $w$ might not have been the smartest choice of notation -- Sorry about that. Let $\renewcommand{\div}{\operatorname{\div}}Q_0, Q_1$ be two real numbers, $...
2
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0answers
57 views

wave equation with L^2 boundary data via spectral decomposition

It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation \begin{equation}\label{pf2} \begin{aligned} \begin{cases} \partial^2_{t}u- \Delta u=0\,\...
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2answers
169 views

An elementary-looking integral inequality

This might seem a bit easy but I still like to ask it for pedagogical reasons. QUESTION. Is this inequality true for non-negative integers $n$? $$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
1
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1answer
95 views

Existence of uniform approximator that also approximates derivative

Let $S$ be a subset of $C^1([0, 1], \mathbb{R})$. It is a well-known fact that given a function $f\in C^1([0, 1], \mathbb{R})$ and a sequence $\{f_n\}\subset C^1([0,1], \mathbb{R})$ such that $f_n\to ...
6
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2answers
231 views

Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?

Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma>0$. Is there a way to reconstruct the ...
0
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1answer
46 views

Existence of integral kernel

I know the following statement ture. Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$. Then, $T$ has the integral kernel $...
1
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0answers
50 views

Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?

$\DeclareMathOperator\rad{rad}$ Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ be compactly embedded in $L^{\sigma}(\mathbb{R}^n)$? In $H_{\rad}^1(\mathbb{R}^3)$, by Struass estimate $|f(x)| \lesssim |x|^{-1} ...
1
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1answer
70 views

Sufficient conditions for the convexity of the discrete Fourier transforms

Let $f : [0,2\pi] \to \mathbb{R}$ be some function. Then the discrete Fourier transform of $f$ when sampled at $2\pi i/N$ is then given by $$ X_n := \sum_{i=0}^{N-1}\cos\left(\frac{2\pi n i}{N}\right)...
2
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0answers
81 views

Eigenvalues of the operator $A = -v'' + B(x) v$

How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that $$ \left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
3
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1answer
102 views

Variational formulation of an elliptic pde

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, what is the variational formulation of the following problem: $$ \text{div}(y^a\nabla_{x,y}V)=0,\quad\text{on }\mathbb{R}^n\times(0,\infty),$$ $$ V(x,...
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1answer
65 views

A concave function as supremum of upper semi continuous is upper semi continuous

We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
1
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1answer
91 views

Convexity of discrete Fourier transform

Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
0
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1answer
59 views

Slowly-varying functions near zero

I am cross-posting the question below, which I asked in Mathematics StackExchange a week ago and did not receive answers there. Thank you for your help! It is well known that, if $x\mapsto f(x)$ is a ...
0
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1answer
85 views

Connectedness of the set having a fixed distance from a closed set 2

This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $F$ is a closed and connected set in $\mathbb{R}^n$ ($n>1$). Suppose the complement ...
4
votes
1answer
280 views

“Interlacing property” of certain polynomials

I posted this question on MO which was quickly and decidedly answered by Noam D. Elkies. Once more referring to the same set of polynomials $$u_n(x) = {2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1) -{2\,n-1\...
7
votes
1answer
708 views

Real-rooted polynomials

I proposed this question at MO which was resolved neatly by Gerald Edgar in the form $$ u_n(x) = {2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1) -{2\,n-1\choose n-1}\prod _{k=0}^{n-1}(x+k).$$ Now that we ...
4
votes
2answers
364 views

Periodic eigenfunctions for 2D Dirac operator

Consider the 2D Dirac operator $$H = \begin{pmatrix} 0 & \partial_{\bar z} \\ \partial_z & 0 \end{pmatrix}$$ where $\partial_z = \partial_x - i \partial_y$ and $\partial_{\bar z} = \partial_x +...
1
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0answers
150 views

Justify $\int_0^\infty e^{-ax^2}\ \mathrm{d}x$ for complex $a$ and zero real part [closed]

(Reposted from math stack exchange) I have searched and failed to find a rigorous proof showing that $$\int_{0}^\infty e^{-ax^2}\ \mathrm{d}x = \frac{\sqrt{\pi}}{2\sqrt{a}}$$ is true for $\Re(a)=0$ ...
5
votes
2answers
141 views

Upper bounds for Bessel functions

Cosider the K-Bessel function $$K_\nu(x):= \frac\pi 2 \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu\pi)}.$$ See also Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press, ...
4
votes
0answers
100 views

Harmonic functions in upper half plane

Let $\mathbb H^+$ denote the upper half plane in $\mathbb R^2$. Consider the following equation \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \Delta u=0\,\quad &\text{on $\mathbb H^+$},...
0
votes
0answers
21 views

Minimization of a palindromic-like sequence and asymptotics

Suppose that I have a sequence of $n$ numbers $x_1, \ldots, x_n$ all taken from the real interval $[0,1]$. I am interested in minimizing the infinity norm of the vector $$ v = \left( \frac{x_{1}}{x_2},...

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