# Questions tagged [ca.classical-analysis-and-odes]

Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

2,136 questions

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38 views

### Time-varying perturbations of continuous-time hyperbolic orbits

My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then ...

**1**

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58 views

### Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...

**2**

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**1**answer

77 views

### The blow-up rate of a nonlinear oscillator

(Related to this Math.SE question.)
For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$
...

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37 views

### symmetry of solutions

Consider the problem $$-u''+u= u^p \text{ on }\mathbb I; \quad u=0 \text{ on } \partial \mathbb I$$ where $\mathbb I $ is symmetric bounded interval. If $u$ is a least energy solution, then it is ...

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122 views

### $L^1$-continuity estimate for ODE solutions in terms of $L^1$ distance of vector fields (only one of them being Lipschitz)

Consider the following ODE initial value problems
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in ...

**2**

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**1**answer

92 views

### Optimal control theory of PDEs

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...

**-2**

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**0**answers

51 views

### Estimate on the difference between the measure of the sublevels of two functions in terms of their $L^1$ distance

Fix $R\gg 1$. How can I estimate the difference between the Lebesgue measures $$\mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): f(x)>0\} - \mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): g(x)>0\...

**2**

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**1**answer

130 views

### Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...

**6**

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**1**answer

173 views

### Asymptotic Expansion of Bessel Function Integral

I have an integral:
$$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$
and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically ...

**6**

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**1**answer

328 views

### Interesting behaviour of binomial coefficients

Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...

**3**

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**1**answer

215 views

### Prescribing a gradient direction

Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...

**4**

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**1**answer

99 views

### Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...

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**1**answer

99 views

### Delay equations

In an effort to solve a delay partial differential equation
$$\partial_t f(t,x)= \Phi(x) f(t,x)+\Psi(x) f(t,x-\alpha),$$
with
$$f(0,x)=1,\hspace{0.3cm} f(t,0)=1$$
Where $\alpha$ is the delay ( a real ...

**1**

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**0**answers

58 views

### Best constant for Hölder inequality in Lorentz spaces

It's well known (and proved by R. O'neil) that there is a version of Hölder's inequality for Lorentz spaces, namely
$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...

**32**

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**2**answers

2k views

### When can a function be made positive by averaging?

Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\...

**-2**

votes

**2**answers

112 views

### What is the limit of this integral as $n$ approaches infinity for integer $k\geq 0$ and real $m\geq 1$? [closed]

$\int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du$

**-5**

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**1**answer

145 views

### How do you prove the validity of this formula for $H(n)$? [closed]

I'm looking for a proof of the identity
$$\sum_{k=1}^{n}\frac{1}{k}=\frac{1}{2n}+\pi\int_{0}^{1} (1-u)\cot{\pi u}\left(1-\cos{2\pi n u}\right)\,du.$$
There is a generalization of this formula for $...

**12**

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**5**answers

2k views

### Reference request: Oldest calculus, real analysis books with exercises?

Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there.
Edit. Unsolved exercises ...

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46 views

### Convex functions and infinite convex combinations

Let $f: D\rightarrow \mathbb R$ be a convex function on a convex subset in $\mathbb R^n$. Let $t_i>0$ with $\sum_{i=1}^\infty t_i=1$ and $x_\in D$ be such that the series $\sum_{i=1}^\infty t_i x_i ...

**1**

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**1**answer

92 views

### Are bilinear sparse bounds for local operators trivial?

I'm thinking about a recent result by M. Lacey which says that a dyadic spherical maximal function satisfies a sparse bilinear bound. To be precise, define the unit scale dyadic spherical maximal ...

**0**

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**0**answers

21 views

### Convergence of the asymptotic expansion solution of homogeneous linear ODE of order 2

Consider ODE $w''+pw'+qw=0$, $p$ and $q$ are functions of $z$. Denote $w_1$ and $w_2$ the ODE's two linear independent solutions. As $z\to0$, in which situation:
Do both $w_1$ and $w_2$'s ...

**4**

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**2**answers

176 views

### Integral equality of 1st intrinsic volume of spheroid

Computations suggest that
$$\int_{0}^{\infty}\int_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$
The question ...

**5**

votes

**1**answer

207 views

### Finding an asymptotic solution for a first order ODE

Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...

**3**

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**2**answers

284 views

### how to calculate the following integral related to Chebyshev polynomials

Chebyshev polynomials of the second kind $V_n(x)$ can be defined as
$$V_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$
or through the recurrence relation
$$V_{n+1}=xV_n-V_{n-1}, V_0=1, V_1=...

**-2**

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**3**answers

169 views

### A Specific Linear Homogeneous System of Differential Equations with Variable Coefficients

Is there an analytical solution satisfying these 3 equations with non-constant z?
$$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$
$$\frac{dy}{dt}=z\cdot\sin(\omega t)$$
$$\frac{dz}{dt}=x\cdot\cos(\omega t) - ...

**9**

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**1**answer

195 views

### The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]...

**24**

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**4**answers

986 views

### show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$

I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version....

**3**

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**3**answers

200 views

### ODE with Bessel decay

This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the ...

**6**

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**0**answers

277 views

### Is this proof of Basel identity known?

Today, to divert myself, I tried to find a new proof of Basel identity $\boxed{\sum_{j=1}^\infty\frac{1}{j^2}=\frac{\pi^2}{6}}$. I came up with the following, which essentially interprets the identity ...

**12**

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**0**answers

414 views

### Is it possible that the following integral is $0$?

Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint.
It is not difficult to see that
$$\int_{1<|z|&...

**1**

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**0**answers

91 views

### How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...

**2**

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**0**answers

85 views

### Analytic continuation of an NLS soliton

The attractive nonlinear Schroedinger equation $i \partial_t \psi = - \frac {1} {2} \Delta \psi - |\psi|^{p-1} \psi$ in the $H^1$-subcritical case $1 < p$, $\frac {d} {2} + \frac {2} {p-1} < 1$ ...

**0**

votes

**1**answer

45 views

### Strict positive type function on hypersurface also of positive type in neighborhood?

Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...

**0**

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**1**answer

81 views

### Chebyshev interpolation [closed]

Let's define the n-th degree Chebyshev polynomials by
$$ T_{n} (x)=\cos(n\arccos(x)).$$
Find a polynomial $P$ such that
$$\mid y- P (x) \mid$$
is minimal, using the first three Chebyshev ...

**3**

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**0**answers

215 views

### Periodic orbit for certain Hamiltonian on the tangent bundle

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point.
Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...

**0**

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**0**answers

43 views

### Function approximation via an orthonormal basis (with singular weight)

If you don't mind, please consider the eigenvalue problem
$$ (1-x^2)u''+ \lambda u=0 \ \ \ \forall x\in (-1,1), $$
$$ u(\pm 1) = 0. $$
Observe that for suitable values of $\lambda$, the ODE resembles ...

**1**

vote

**1**answer

61 views

### additive discrepancy under a multiplicative constraint

Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints:
(1). $\sum_{i=1}^K a_i, \sum_{i=1}^K b_i, \sum_{i=1}^K c_i \ge 1/2 + \epsilon$;
(...

**0**

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**0**answers

29 views

### Existence of solutions to NLS: Local existence and boundedness

I was wondering when the following argument is valid:
Consider a nonlinear Schrödinger equation
$$i \partial_t \varphi = -\Delta \varphi+ N(\varphi)$$
where $N$ is a nonlinearity.
Often it is ...

**-2**

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**1**answer

166 views

### Strong estimates for the zeta function on natural numbers

Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
be the Riemann zeta function (here we just consider real $s$).
We do have a description given by
$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...

**2**

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**1**answer

88 views

### Quotient with positive second derivative in the limit?

I am studying the quotient of
$$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$
and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$
for some $\...

**16**

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**0**answers

323 views

### An integral in Gradshteyn and Ryzhik

Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...

**4**

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**0**answers

91 views

### Injectivity of product functions on natural number sequences

Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.
We now define for each $k \geq 2$ ...

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**0**answers

88 views

### Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices

Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...

**3**

votes

**1**answer

132 views

### Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...

**2**

votes

**1**answer

398 views

### Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?
This seems to be a well-known result, but I can ...

**-2**

votes

**1**answer

120 views

### Relationship between “Radial” Fourier transform and Fourier transform, especially at infinity

Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support.
What is the relationship between
$$
\widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...

**27**

votes

**1**answer

2k views

### A remarkable almost-identity

OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$.
Mikhail Kurkov noticed that it ...

**3**

votes

**0**answers

34 views

### Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...

**3**

votes

**2**answers

122 views

### Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$

I am working in data science and I have to deal with the following problem for which I would like to find a simplification:
We call a function almost positive if $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,...

**4**

votes

**1**answer

125 views

### Method of characteristics beyond the Lipschitz setting

I have come across the following easy-looking problem that is driving me mad.
I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...