# Questions tagged [functional-equations]

The functional-equations tag has no usage guidance.

206
questions

-2
votes

0
answers

56
views

### Imaginary delay

I'm looking at a particular delay differential equation:
Suppose we have $x'(t) = x(t - i \pi),$ with initial condition $\phi(t) = x_0(t),$ where $i \pi$ is imaginary.
Well, uh, how do we derive a ...

-1
votes

0
answers

77
views

### What are the process and fundamentals of solving delay differential equations?

I have a lot of confusion around delay/difference differential equations, I'm hoping someone can clear it up. This is a very niche, specialized subject, so I need to ask here among people who have ...

1
vote

0
answers

81
views

### Invariant polynomials under a non-standard group action

There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...

0
votes

2
answers

306
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.
...

1
vote

1
answer

162
views

### Is there a systematic procedure to Solve Abel's, Böttcher's, or Schröder's Equation

I've been interested greatly in the study of functional equations for some time now, I've learnt many different techniques for their solution. Currently I have been studying superfunctions and ...

1
vote

0
answers

25
views

### Hardy type inequality with singular weights

Recall that Hardy's inequality involving distance from the boundary of a convex set $\Omega \subsetneq \mathbb{R}^n ; n \geq 1$, asserts that
$$
\int_{\Omega}|\nabla u|^p \, d x \geq\left(\frac{p-1}{p}...

1
vote

0
answers

28
views

### Rätz orthogonality and involution

In the Rätz’s sens of orhtoganality, can we find an exemple of an involution u(different to -Id)such that x orthogonal to y then x orthogonal to u(y)

11
votes

2
answers

802
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 ...

4
votes

0
answers

110
views

### $f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer

Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$
\begin{split}
f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\
f(...

10
votes

2
answers

584
views

### Proving the simple form of a function from statistical mechanics

I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...

0
votes

1
answer

89
views

### Finding minimal $\gamma$ that satisfies the integral equation

I have a function $f(z) \in [0,\infty]$ that satisfies $-1 < f(z) < 0$.
I would like to find the minimal $\gamma$ that satisfies:
$$ \int_0^{\gamma} f(z)dz = \log(1+f(0)).$$
Clearly, I cannot ...

1
vote

0
answers

47
views

### Can I apply $q$-Lagrange Inversion formula?

Now I have equation $F(x) = x \sum_{k\ge 0} g_k F(x) F(qx) \cdots F(q^{k-1} x)$, I need to get the coefficient of $x^n$ in $F(x)$, can I apply $q$-Lagrange Inversion formula to this?
Moreover, I have ...

8
votes

1
answer

461
views

### $f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured.
$$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$
or variations of it like the weaker
$$f(f(f(f(z)))) = z ,\...

0
votes

0
answers

98
views

### Numerical approaches to functional equations

I'm interested in finding numerical approaches to solving functional equations such as
f(xy)=f(x)+f(y),
where the equations had no derivatives or integrals, and contains arguments involving x
and y
.
...

4
votes

2
answers

223
views

### Can every symmetric function be factorized through symmetric polynomials?

A symmetric function is a function $f:\mathbb R^n\to \mathbb R$ such that $f(x_1,\ldots,x_n)=f(\sigma(x_1,\ldots,x_n))$ for every permutation $\sigma\in S_n.$
The most commonly encountered symmetric ...

0
votes

0
answers

62
views

### How to solve a system of polynomial equations which are mostly factored?

If I have a system of $k$ equations
$$
\begin{cases}
F_1(x) = 0,\\
\quad\vdots \\
F_k(x) = 0.
\end{cases}
$$
with
$$
F_i(x) = c_i + \prod_{j=1}^n f^i_j(x)\quad i=1,\ldots,k
$$ where
$f^i_j: \mathbb{R}...

2
votes

0
answers

58
views

### Methods for holonomic recurrences

I wanted to ask if anyone knows of good texts/resources on methods for solving holonomic recurrence relations (if there are any general analytical approaches):
$$p_1(n)a(n)+p_2(n)a(n-1)+\dotsb+p_k(n)a(...

4
votes

2
answers

306
views

### Solving the functional equation $2f(x)=f(x+a_n)+f(x-a_n)$

Let $a_n$ be a sequence of strictly positive real numbers such that $\lim_{n \to \infty}a_n=0$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ that admit primitives (i.e. there exists a function $F:...

9
votes

4
answers

2k
views

### How may I find all continuous and bounded functions g with the following property?

Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.

2
votes

2
answers

242
views

### Non-exponential functions $f(x)$ satisfying $f(x+c)=\gamma(c)f(x)$

Question:
what can be said about the existence of functions
\begin{align}
f:x\mapsto f(x)&\implies x+c\mapsto \gamma(c)f(x)\\ f(x)\ne f(y)&\iff\frac{f(x)}{\gamma(x)}\ne\frac{f(y)}{\gamma(y)}
\...

2
votes

0
answers

56
views

### any ideas on how to solve matrix equation like this $X A_i Y = B_i$

the objective function is like
$$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix
I've tried gradient-descent, but as it turns out not well, I wonder if ...

0
votes

0
answers

68
views

### How are eigenvalues of two psd kernels related?

Suppose $K(x,x')$ and $R(y,y')$ are two positive semi-definite kernels on $(x,x')\in \mathbb
{X}\times\mathbb{X}$ and $(y,y')\in\mathbb{Y}\times\mathbb{Y}$, respectively, and satisfying the following ...

0
votes

1
answer

46
views

### Symmetric and nearly additive bounded functions

Let $(y_n)_{n\ge 1}$ be a sequence with values in $(0,1)$ such that $\lim_n y_n=1$. Let also $f: [0,1]\to \mathbb{R}$ be a bounded function such that $f(0)=0$ and satisfies
$$
\forall n\ge 1, \forall ...

3
votes

1
answer

228
views

### A pexiderization of the sine addition law on semigroups

Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?

2
votes

2
answers

249
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 ...

4
votes

2
answers

368
views

### Is there a recurrence for the coefficients of the Laurent series expansion of $\frac{1}{1-e^{e^x - 1}}$?

So there's an elementary (but in my opinion quite interesting!) result which is that the Laurent series expansion of
$$\frac{1}{1-e^x} = -\frac{1}{x} + \frac{1}{2} - \frac{1}{12}x - \frac{1}{720}x^3 \...

0
votes

1
answer

35
views

### A set of equations for two (cumulative distribution) functions and their inverses

I am trying to characterize a set of distributions that satisfy two conditions. It is easy to characterize distributions fitting each of those conditions separately, but I am unable to make progress ...

1
vote

0
answers

51
views

### Functional approximation with derivatives

I am trying to solve a functional approximation problem.
Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...

4
votes

2
answers

245
views

### A functional inequality which calculates the limitation of human eyes

Find all pair of function $f^-,f^+:[0,1]\rightarrow[0,1]$ such that:
(1)$f^-(x)\leq x\leq f^+(x)$.
(2)$f^-(x)+f^+(1-x)=1$.
(3)$f^-(x)f^-(y)\leq f^-(xy)\leq f^-(x)f^+(y)$.
(4)$f^+(x)f^-(y)\leq f^+(xy)\...

1
vote

1
answer

107
views

### Spectral family associated with the Laplacian operator in $L^{2}(\mathbb{R}^{n})$ [closed]

Let $\lambda>0$ be given. Define
$$G_{\lambda}(\xi) = \chi_{_{\lbrace |\xi|^{2} \leq \lambda \rbrace }}.
$$
and
$$
E_{0}(\lambda)f = \mathcal{F}^{-1}[G_{\lambda}(|\xi|^{2})\mathcal{F}(f)], \ \ f \...

0
votes

0
answers

72
views

### Solving the integral equation $y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$

Could you please help me to solve the following integral equation?
$$y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$$
I tried to turn the exponentiential term into its trigonometric ...

8
votes

2
answers

642
views

### Solving functional equation $f(xy)=f(x+y)$ and Diophantine equations

It is well known that the only solution is $f$ a constant function. However, by putting some restrictions on the functional equation, we might get other solutions, with potential implications to ...

8
votes

1
answer

414
views

### Functional equation with Fourier transform and $\frac{1}{x} f(\frac{1}{x}) $

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation:
$$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$
$\...

10
votes

3
answers

822
views

### Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses

The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...

3
votes

0
answers

224
views

### $f(x)>0$ and $f(y)>0$ implies $f(x+y)>0$, then there must exist an linear function $g$ such that $g(x)>0$ iff $f(x)>0$?

Background: Let $x,y\in\mathbb (0,+\infty)^n$. $f$ is a continuous function on $\mathbb R^n_+=(0,+\infty)^n$. Consider the following condition (1), the sign of $f(x+y)$ is dependent on the sign of $f(...

5
votes

2
answers

286
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
...

3
votes

1
answer

156
views

### A question about decompositions of rational functions

Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...

2
votes

1
answer

199
views

### Largest asymptotic growth for $2f(x)-f(2x)$

I am trying to find smooth functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that the quantity $$\Delta_f(x) := 2f(x)-f(2x)$$
is positive for $x$ large enough and has the greatest asymptotic growth.
...

6
votes

1
answer

565
views

### Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?

Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via
$$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...

3
votes

0
answers

185
views

### Fermat's Last Theorem (FLT) in standard model space corresponding to an infinite Blaschke product

Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the ...

0
votes

0
answers

102
views

### What are the functions such that $ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^p$?

Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that
$$ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^...

0
votes

1
answer

169
views

### Is it possible to numericaly solve functional equation

Given a functional equation of form $f(f(x))=T(x)$ is there any good ways to solve it numerically? If not then at least approximate in some small region $x\in(-a;a)$.
E.g. with the equation $f(f(x))=x+...

2
votes

0
answers

142
views

### Square root of a function on a finite set

Let $S$ be a finite set and $f \colon S \to S$ be an arbitrary function. How can we find all functions $g \colon S \to S$ with $f = g \circ g$?
If $f$ and $g$ are both required to be invertible, the ...

2
votes

2
answers

183
views

### Functional equations and normal distribution

Let $\alpha \neq 1.$
If $X,Y$ are two independent random variable such that $U=X+Y$ and $V=X+\alpha Y$ are independent, then $X$ and $Y$ are normally distributed.
In term of characteristic functions ...

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 ...

10
votes

2
answers

452
views

### A functional equation involving the inverse function

$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...

0
votes

1
answer

140
views

### Solve $(x-a)^{\alpha +1} - \lambda*(b-x)^{\alpha + 1} = C(\frac{a+b}2 - x)^{\alpha}$ over $\mathbb R$ [closed]

I have been having trouble solving the following equation. Any help would be appreciated.
Let a,b,C,$\alpha$,$\lambda$ be real numbers with $C < 0$, $0 < \alpha < 1$, $\lambda > 1$. We ...

0
votes

0
answers

192
views

### Functions that satisfy a reverse triangle inequality: do they have a name?

Let $f : \mathbb{R}^n \to \mathbb{R}_+$ satisfy
$$f(a) - f(b) \le C f(a - b)$$
$\forall a, b \in \mathbb{R}^n$ for some $C \ge 0$. Is there a name for such functions? (I would be happy to have a name ...

2
votes

1
answer

164
views

### Existence of a distinguished continuous version of the logarithm of a continuous function

Let $E$ be a $\mathbb R$-Banach space and $\varphi\in C^0(E,\mathbb C\setminus\{0\})$ with $\varphi(0)=1$.
I want to show that there is an unique $\psi\in C^0(E,\mathbb C)$ with $\psi(0)=0$ and $$\...

3
votes

0
answers

358
views

### Is the study of additive functions dead?

I am learning more about the theory of additive functions ($f(nm)=f(m)+f(n)$) and I am struck by how powerful the theorems given are. For example, we have a complete characterization of not only what ...