All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
12
votes
1
answer
1k
views
Proof of Green's formula for rectifiable Jordan curves
$\newcommand{\Ga}{\Gamma}$
I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
- 128k
0
votes
1
answer
242
views
Harnack inequality for fractional laplacian
Let u be a positive solution of $s\in (0, 1) $
\begin{equation}
\left\{\begin{aligned}
(-\Delta )^{s} u &= 0 \text{ in } (-2T, 2T)\\
u &=g\quad\text{in}\quad \mathbb R\setminus(-2T, 2T).
\...
- 402
2
votes
1
answer
239
views
Projection of a ball in the ambient space to a manifold
Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
- 3,574
4
votes
1
answer
387
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
- 1,776
1
vote
0
answers
324
views
Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
- 3,799
5
votes
1
answer
4k
views
Rigorous multivariate differentiation of integral with moving boundaries (Leibniz integral rule)
The Leibniz integral rule, in its multivariate form, deals with differentiation of the following sort:
$$ \frac{\partial}{\partial t} \int_{D(t)} F({\bf x}, t) \, d{\bf x} \, , \qquad D(t)\in \mathbb{...
- 3,574
3
votes
0
answers
203
views
Identity for the product of two different associated Legendre polynomials
In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated:
$$
\small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...
- 16.5k
2
votes
1
answer
662
views
Clausen’s identity for associated Legendre polynomials
Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493)
$$P_n(\cos{\...
- 16.5k
6
votes
0
answers
227
views
Origins of the generalized shift operator exp(t*g(z)d/dz)
Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
- 10.5k
1
vote
1
answer
3k
views
Is there a standard proof that the L^1 norm > constant * sup norm for functions with derivative bounded above by K on the unit disk in R^n?
Suppose that you have a bounded function $f(x)$ on a compact domain in $\mathbb{R}^n$. It's easy to see from Holder's inequality that
$$
||f||_1 \leq \operatorname{Volume}(D) ||f||_\infty.
$$
There ...
- 747
9
votes
1
answer
749
views
property of convex functions
I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
- 1,211
24
votes
2
answers
2k
views
Reference for exponential Vandermonde determinant identity
I recently stumbled upon the following identity, valid for any real numbers $\alpha_1,\dots,\alpha_n$ and $\lambda_{n1} \leq \dots \leq \lambda_{nn}$:
$$ \mathrm{det}( e^{\alpha_i \lambda_{nj}} )_{1 \...
- 114k
7
votes
3
answers
2k
views
Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs
What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in
real analysis,
complex analysis,
functional analysis,
ODEs,
PDEs?
The ...
user60665
1
vote
1
answer
262
views
Relationship between $f(t,x)$ as $t \to \infty$ and $f(t/\epsilon, x/\epsilon^2)$ as $\epsilon \to 0$ (periodic functions)
Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous. ...
user89890
2
votes
2
answers
406
views
Solution to semilinear heat equation at $t=0$: $u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$
Consider the following Cauchy problem
$$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Suppose that $u \...
- 303
5
votes
2
answers
840
views
Decompostition of a Lipschitz domain
We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if:
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \...
- 291
2
votes
2
answers
261
views
Viscosity solutions for $u'(x) + \alpha u(x) - f(x) = 0$: supersolutions dominate subsolutions
Let $$u'(x) + \alpha u(x) - f(x) = 0,$$ with $x \in [0,\infty)$ and $\alpha \in \mathbb{R}$. Suppose $f \in C(\mathbb{R})$.
If
$u_1$ is a viscosity supersolution (or a viscosity solution, or a $C^...
user102027
5
votes
2
answers
1k
views
real analyticity, Fourier coefficients [duplicate]
Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.
- 43.2k
3
votes
0
answers
111
views
When does the constant term in the following expansion is nonzero?
Dyson's Theorem
The constant term in the expansion of
$$\prod_{1\leq i\neq j\leq n}\left(1-\frac{x_i}{x_j}\right)^{a_i}$$
is the multinomial coefficient
$$\frac{(a_1+\cdots+a_n)!}{a_1!\cdots a_n!},$$
...
- 1,997
3
votes
2
answers
306
views
Asymptotics for the number of digits of the ratio of binomial coefficients
Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. ...
- 128k
11
votes
2
answers
1k
views
Two divergent series conspiring?
Consider the sequence $a_n=2^{2n}\binom{2n}n^{-1}$. Stirling's approximation shows that $a_n\sim \sqrt{\pi n}$, thus
$$\sum_{n\geq0}\frac{\pi}{2a_n}\qquad \text{and} \qquad
\sum_{n\geq0}\frac{a_n}{2n+...
- 43.2k
10
votes
1
answer
580
views
About certain infinite products with the property $f(a)=f(1/a)$
In the paper Transformations of infinite series, Bryden Cais gives the following transformations of infinite products
Theorem 4. If
$$
f(t) = \frac{\cosh(\pi t)-1}{\sinh(\pi t)}\frac{\cosh(2\pi t)+1}{...
- 5,624
1
vote
1
answer
380
views
Infinite compositions of holomorphic functions, is there literature on the subject?
I've recently become very intrigued by infinite compositions. To get at what I mean by the term, I'll be as explanatory as possible.
Consider a sequence of holomorphic functions $\{\phi_j\}_{j=0}^\...
user78249
5
votes
0
answers
329
views
Could there be something like a Grzegorczyk hierarchy in Analysis?
My most prevalent interest in mathematics has always been hyper-operators. I first learned about them when I was in highschool, and quite frankly, they amazed and dazzled me. For those who've yet to ...
user78249
3
votes
1
answer
202
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 884
43
votes
3
answers
7k
views
Could the Riemann zeta function be a solution for a known differential equation?
Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...
5
votes
0
answers
136
views
Solving the difference equation in exotic scenarios
The difference equation, as referenced in the title, is a very specific object I'm referring to. If you have a holomorphic function $\phi$ on a domain $G$, then a solution $F$ to the difference ...
user78249
4
votes
0
answers
672
views
Proofs of the second fundamental theorem of calculus
I am referring to the following version of the theorem, in the setting of the Lebesgue integral.
Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
- 18.7k
1
vote
0
answers
176
views
Coefficient perturbations of polynomials with real roots only
Let
\begin{align}
P(x) &= x^n+a_{n-1} x^{n-1} +\ldots+a_0 = \prod_{i=1}^n (x-p_i)\\
Q(x) &= x^n + b_{n-1} x^{n-1} + \ldots +b_0 = \prod_{i=1}^n (x-q_i)\\
p_i, q_i& \in \mathbb{R},\ 0<...
- 175
10
votes
1
answer
833
views
This is not a dyadic cosine-product
The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$
into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
- 43.2k
1
vote
2
answers
539
views
Generating function for products of complex Hermite polynomials
By making use of the generating function
$$\sum_{m=0}^\infty \frac{H_m(x)}{m!} t^m=e^{-t^2 + 2xt} $$ for the real Hermite polynomials $H_m$, we get easily that
$$(*)\quad \sum_{m,n=0}^\infty \frac{u^...
- 650
1
vote
1
answer
990
views
Is the existence of $\lim_{n\to\infty}\cos(n!\pi x)$ for given arbitrary irrational $x$ an open problem?
Motivated by a recent MSE question about the sequence of function $\cos(n!\pi x)$, I have read related several related questions:
On the behaviour of $\sin(n!\pi x)$ when $x$ is irrational.
Is there ...
user14319
3
votes
0
answers
113
views
the topological equivalence of linear autonomous system
N. Ladis and Kuiper gave the classification of the topological equivalence of linear autonomous system. More precisely, they proved if two linear autonomous system $\dot{X}= AX, \dot{X}= BX$ are ...
- 799
2
votes
3
answers
233
views
Difference equation and formal series
For a given formal series $g(x)=\sum_{k=0}^\infty g_k x^k$ I would like to find a formal series $f(x)=\sum_{k=0}^\infty f_k x^k$ such that they satisfy the difference equation
$$
f(x+1)-f(x)=g(x).
$$
...
- 1,343
2
votes
1
answer
231
views
Entire composite square roots of functions of finite order
A composite square root of a function $g$ is a function $f$ such that $f(f(z)) = g(z)$. Not surprisingly, for arbitrary $g$ a function like this is hard to find. Specifically I am looking at functions ...
user78249
0
votes
1
answer
125
views
A slight generalization of triconfluent Heun equation: what is known?
I have recently come across an ODE of the form
$$y''+(a+b x^2)y'+(c+dx+h/x^2)y=0 \hspace{30mm} (*)$$
where $y=y(x)$ and $a,b,c,d,h$ are arbitrary constants.
As far as I understand (please correct ...
- 312
8
votes
1
answer
338
views
A representation of the Bernoulli numbers
Let
\begin{equation}
\ell_{m,p}:=\sum_{j=1}^m\gamma_{m,j}\sigma_{p,j},
\end{equation}
where
\begin{equation}
\gamma_{m,j}:=\frac{2 (-1)^{j-1} }{j }\binom{2 m}{m+j}\Big/\binom{2 m}{m},\quad
\...
- 128k
1
vote
0
answers
341
views
Reference for PDE problem book
What I need is a source of solved exercises, problems in Partial Differential Equations; to be hard enough (olympiad style) and in areas like Calderon-Zygmund theory and applications, Paley-Littlewood ...
- 85
2
votes
3
answers
471
views
Existence of solution to SDE with perscribed initial and terminal conditions
The SDEs \begin{equation}
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
\end{equation} with prescribed initial conditions are well studied. My question came up in my research and I have not found much on ...
- 5,405
4
votes
2
answers
371
views
Literature on ZS-AKNS systems with independent potentials
For those with some familiarity with integrable systems, I'll summarize my question as such:
Where can I find literature on ZS-AKNS systems, and their solution via the inverse scattering transform, ...
- 319
3
votes
2
answers
807
views
Cubic splines convergence?
I am looking for a basic, classical, result on approximating a smooth function using cubic and linear splines. Is there a reference on the convergence, in some sense, of the splines to the function of ...
- 31
5
votes
3
answers
478
views
Nonlinear ODE: $y'=(1+axy)/(1+bxy)$
Consider the first order nonlinear ODE problem:
$$
y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0
$$
where $a, b>0$ are some constants. I would like to know if these kind of equations were ...
- 3,946
3
votes
2
answers
780
views
An integral identity evaluating the gamma function
While reading a number theory paper I encountered the identity
$$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$
...
- 7,347
2
votes
1
answer
455
views
Smooth dependence on the initial condition of the integral of an ODE
I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$.
I assume that my ODE ...
- 562
7
votes
5
answers
1k
views
Generalizations of the Euler–Maclaurin Summation Formula
I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in Apostol - An Elementary View of Euler's Summation Formula does not give me ...
- 3,574
1
vote
0
answers
617
views
History of Cauchy-Euler Equations
As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -
$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\cdots ...
- 3,574
9
votes
1
answer
630
views
$C^{k,\alpha}$ diffeomorphisms and vector fields
This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
- 14.4k
3
votes
1
answer
2k
views
What are the best known bounds on the Hermite polynomials?
The best I could find on the net is this paper,
http://arxiv.org/pdf/math/0401310.pdf
Has this been improved?
- 2,246
3
votes
2
answers
153
views
An English version Borok's work on finite-infinite systems of ordinary differential equations
I am looking for the English translation of the paper by V. M. Borok (originally in Russian)
The Cauchy problem for finite-infinite systems of linear differential equations. This work is about the ...
1
vote
0
answers
120
views
Interpolation functional for BV spaces?
Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
- 121