All Questions
Tagged with ca.classical-analysis-and-odes functional-equations
28 questions
8
votes
1
answer
436
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
1
vote
1
answer
120
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...
- 141
1
vote
1
answer
134
views
Functions for which $\lambda f(x)=f(\alpha_\lambda x + \beta_\lambda)$
How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ containing an interval containing $1$ such that for any $\lambda\in S$ there exist $\...
- 3,065
3
votes
2
answers
620
views
Are there any functions $f$ beyond trivial examples where $\int f(x +f(x + f(x +\dotsb)))\,dx$ $= F(x+F(x+F(x +\dotsb))) + C$ for some function $F$?
Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...
- 149
0
votes
2
answers
369
views
How can I derive functional properties of (the solutions of) this simple functional differential equation?
I've not yet finished a course in functional analysis so I'm unsure how to go about this, but I've always been fascinated by a simple functional differential equation I concocted for almost no reason.
...
11
votes
2
answers
852
views
What are the iterates of $x \mapsto 1 - \sqrt{1-x^2}$?
Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
- 32.5k
4
votes
2
answers
267
views
One question about a specific first-order differential equation
Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below:
$$f'(x)=f(x+c),$$
where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$.
I found two families of ...
- 696
5
votes
2
answers
309
views
A functional equation for a family of functions indexed by the symmetric group $S_3$
$\newcommand{\C}{\mathbb C}$A question asked recently was as follows:
For the symmetric group $G:=S_3$, is it possible to construct functions $t_g\colon\C\to\C$ that satisfy the convolution identity
...
- 128k
18
votes
3
answers
4k
views
Is there a general solution for the differential equation $f''(x) = f(f(x))$?
I'm currently an undergraduate studying differential equations and I've been fixated on the differential equation $f''(x) = f(f(x))$ for the past 2 days. I can't seem to crack it but it feels like it ...
- 307
3
votes
0
answers
247
views
Solutions of the differential equation $f'=(f^{-1})^{[n]}$
For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation.
After reading these two posts (here and here)...
- 89
5
votes
1
answer
761
views
Is it possible to express the functional square root of the sine as an infinite product?
Cross-post from MSE.
It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that ...
- 4,829
0
votes
1
answer
101
views
How to create a function whose harmonic is a sine wave [closed]
How do I solve the following equation for $f(\cdot)$?
$f(x)+\frac{1}{n}f(nx)=\sin(x)$
That is, how do I create a function which, when combined with its nth harmonic, will be a sine wave?
- 103
1
vote
0
answers
97
views
Fredholm integral equation of third kind
Let us consider the following integral equation
$$a(x)u(x) + \int\limits_0^2 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p \in [1,\infty]$ and let $K \in L^q((0,2) \times (0,1))$. Assume ...
- 617
2
votes
2
answers
416
views
What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?
What is the solution, $f(n)$, of the following functional equation:
$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$
where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...
- 153
10
votes
1
answer
752
views
A conjecture about certain values of the Fabius function
The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations
$$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$
and
$$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\...
- 6,819
31
votes
1
answer
2k
views
$f'=e^{f^{-1}}$, again
This question is a spin-off of this one, in which the OP asks whether there is a solution $f:\mathbb R\to\mathbb R$ of the functional equation (not exactly an ODE) $f'=e^{f^{-1}}$, where $f^{-1}$ is ...
- 5,169
3
votes
0
answers
316
views
Modified Jacobi’s theta function
Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$
Therefore $$\sum\limits_{k=1}^\infty ...
- 293
4
votes
0
answers
110
views
continuous linear recurrent relations
For a function $f:\mathbb{R}\rightarrow \mathbb{R}$ denote $f_0(x)=x$, $f_n(x)=f(f_{n-1}(x))$. Assume that $f$ satisfies a functional equation
$$f_n(x)+a_{n-1} f_{n-1}(x)+\dots+a_0 f_0(x)\equiv 0$$
...
- 109k
6
votes
0
answers
138
views
Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x
Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form
$$
g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...
- 679
3
votes
2
answers
369
views
Consistent price index
This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem.
Let $Y=({\...
- 16.4k
5
votes
1
answer
667
views
solution of functional equation $f^{\circ k}(x) = x$
The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]:
...
- 183
12
votes
4
answers
831
views
Relating the roots of polynomials to the solution sets of certain functional equations
Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in \mathbb{R},\;\text{and}\;f^0=\text{...
- 453
2
votes
0
answers
563
views
The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is well-...
- 19.6k
13
votes
6
answers
4k
views
Finding f such that f(f(x))=g(x) given g
Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...
- 15.4k
0
votes
1
answer
580
views
Approach to solving a differential-functional equation
What could be an approach to solving such equations?
$$f'(x)=C \prod_{k=0}^x f(k)$$
$$\frac{g'(x)}{g(x)}=C+ \sum_{k=0}^{x-1} g(k)$$
Here the product and the sum are understood as indefinite sum and ...
- 10.1k
0
votes
1
answer
321
views
Does any iterative equation of n-th order have exactly n independent solutions?
Does any iterative equation of n-th order which does not inclute derivatives of order higher than 1 have exactly n independent solutions?
Let's designate n-th iterate of a function $y(x)$ as $y^{[n]}(...
- 10.1k
7
votes
4
answers
827
views
$n$th root of $(a,b) \mapsto (\operatorname{gm}, \operatorname{am})$
Suppose $0 < a < b$, and let GM and AM be respectively the geometric and arithmetic means of $a$ and $b$. Does the mapping $(a,b) \mapsto (\operatorname{GM}, \operatorname{AM})$ have a well-...
5
votes
2
answers
410
views
How to find a solution to a particular Bottcher equation
Functional equations of the form $f(g(x))=(f(x))^p$, where $g(x)$ is known, is called Bottcher equation. Generally, we have only asymptotic formula for the solution $f(x)$ under certain conditions. In ...
- 1,858