Questions tagged [functional-equations]
The functional-equations tag has no usage guidance.
181
questions
-1
votes
0
answers
246
views
$f' = e^{f^{-1}}$, a third time
I am of the impression the differential equation $f' = e^{f^{-1}}$ was considered on mathoverflow for the first time here: How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?
It was found ...
0
votes
0
answers
62
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 ...
7
votes
2
answers
531
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
345
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
802
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
210
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
265
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
139
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
181
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.
...
0
votes
0
answers
34
views
Solving a difference functional equation by using Laplace transform
Consider the operator $T:L^{2}(0,r+1)\longrightarrow L^{2}(0,r+1)$:
\begin{equation*}
Tu(x)=:u(x)+a\mathbf{1}_{(0,1)}(x)u(x+r)+b\mathbf{1}_{(1,1+r)}(x)u(x-1),%
\text{ }x\in (0,r+1),
\end{equation*}
...
6
votes
1
answer
556
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
$$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
0
votes
0
answers
46
views
a functional equation in signal processing
I would like to get help in solving the following functional equation born in a signal processing problem.
Let $\phi(t)$ be well-behaved smooth function and $\tau$ and arbitrary delay; define the ...
3
votes
0
answers
174
views
Fermat's Last Theorem (FLT) in standard model space corresponding to an infinite Blaschke product
Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the ...
0
votes
0
answers
99
views
What are the functions such that $ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^p$?
Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that
$$ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^...
0
votes
0
answers
32
views
Sufficient conditions for domination of derivatives by involutions
Let $h:(0,1) \to (-\infty,0)$ be a $C^1$ function, with $h'>0$.
I am looking for sufficient conditions on $h$ that imply the existence of a $C^1$ decreasing function $g:(0,1) \to (0,1)$ such that $...
0
votes
1
answer
94
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
122
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
141
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
3k
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
368
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
131
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
143
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
141
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
344
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 ...
1
vote
1
answer
120
views
General solution of a linear functional equation
As we know, general solution of the linear functional equation $f(x+1)-f(x)=g(x)$ ($g$ is a known function) is $f=f_0+\phi$, where $f_0$ is an its special solution and $\phi$ any 1-periodic function.
...
3
votes
2
answers
440
views
What does this paper have to do with Hilbert's fifth question?
Apparently, there is a paper
M. Sablik, Final part of the answer to a Hilbert's question. Functional Equations - Results and Advances. Edited by Z. Daróczy and Zs. Páles, Kluwer Academic Publishers ...
5
votes
1
answer
201
views
A functional equation in two complex variables
Let $X$ be a compact metric space, or just $X=\mathbb T$, the unit circle, if it helps. We consider only continuous, complex-valued functions on $X$.
Let $\varepsilon >0$. Is there $\delta > ...
7
votes
1
answer
134
views
On the functional equation $f(xf(y))=\frac{f(f(x))}y$ on arbitrary groups
In this answer, it was shown that there is no function $f\colon\mathbb Q_{+}^{*}\to\mathbb Q_{+}^{*}$ such that
\begin{equation}
f(xf(y))=\frac{f(f(x))}y \label{1}\tag{1}
\end{equation}
for all $x$...
4
votes
0
answers
97
views
Associativity equation for topological rings and logarithms
Let $R$ be a topological ring of characteristic zero. Assume that $R$ is commutative. Let $G :R \times R \to R$ be a continuous function satisfying $G(G(x,y),z)=G(x,G(y,z))$ and $G(0,x)=x$, for all $x,...
0
votes
1
answer
118
views
Functional equation for $\eta(s)$ following Riemann's $2^{nd}$ method
I'm crossposting.
Being
\begin{equation*}
\eta(s)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots
\end{equation*}
and following ...
0
votes
0
answers
60
views
Differential equation
Consider $u = u(\phi,\psi)$ where $\phi = \phi(x)$ and $\psi =\psi(x)$ are both analytic function. The following equation
$$\partial_x u - u\partial_x (\phi-\psi)=0$$
has a trivial solution $u(\phi,\...
5
votes
0
answers
692
views
Twisted functional equations in Goldfeld's book
I am confused about the contents of D. Goldfeld's book "Automorphic forms and L-functions for the group $\mathrm{GL}(n,\mathbb{R})$". In the process of deriving the converse theorem on $\...
3
votes
1
answer
242
views
When is the optimum of an optimization problem affine in the constraint parameter?
While working on a variational problem I have reached to the following question:
Let $f:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $f(1)=0$. Suppose that $f(x)$ is strictly increasing ...
2
votes
0
answers
105
views
Analytic properties of motivic L-functions twisted by Dirichlet characters
Let $M$ be a pure motive over $\mathbb{Q}$ and consider the (completed) $L$-function $\Lambda(M, s)$ attached to its $\ell$-adic realization. Let us assume that this $L$-function admits analytic ...
4
votes
1
answer
181
views
Functional inequalities involving the condition $\left(\int_0^t f(x)dx\right)^2 \ge \int_0^t f(x)^3dx$
I was reading the solution to a functional inequality in an article when the author made the following remark without giving any proof: let $f(x): [0, \infty]\to[0, \infty]$ be locally integrable and ...
3
votes
0
answers
207
views
Solutions of the differential equation $f'=(f^{-1})^{[n]}$
For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation.
After reading these two posts (here and here)...
1
vote
1
answer
186
views
The functional equation $f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$
Incidentally, I came across the following functional equation
$$f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$$
that is to hold for all $x,y\in \mathbb R$. Is there a neat way to find all ...
3
votes
1
answer
255
views
Polynomial satisfying a functional equation [closed]
I am currently stuck with the following question:
Let $q$ be a polynomial of degree $n+1$ with distinct positive zeros $x_0, ... , x_n$. Find a polynomial $p \in P_n$ that satisfies the functional ...
6
votes
1
answer
386
views
What is known about the functional square root of the Riemann zeta function?
Let us consider the Riemann zeta function $\zeta(s)$, where $s$ can take on values on the domain $\mathbb{R}_{>1}$:
$$\zeta(s) := \sum_{n=1}^{\infty} \frac{1}{n^{s}} .$$
I wonder what is known ...
1
vote
2
answers
798
views
Simple bound on $\log(x)/x$
I would like to pick $x$ as small as possible while guaranteeing that $\log(x)/x \leq \epsilon$ where $\epsilon \ll 1$. Clearly $x$ should (roughly) be of the order $1/\epsilon$; I would like simple ...
1
vote
0
answers
39
views
Nonlinear fixed-point equation with linear solutions?
Let $S$ be an $N\times N$ row-stochastic matrix and let $w'$ be the left Perron eigenvector of $S$ (i.e., $w$ is the stationary distribution of the Markov chain represented by $S$). Let $T$ be the ...
3
votes
1
answer
328
views
On functional equation $f\circ \exp=\exp \circ Df$ on a Riemannian manifold or a Lie Group
Let $M$ be a Riemannian manifold or a Lie group whose corresponding exp map (in corresponding context) is denoted by "exp" which is a map $\exp:TM\to M$
We search for the set $\mathcal{H}...
5
votes
1
answer
343
views
Is it possible to express the functional square root of the sine as an infinite product?
Cross-post from MSE.
It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that ...
2
votes
1
answer
213
views
Given functions $A(x), B(x)$ find $f(x)$ s.t. $A\big(f(x)\big)=f\big(B(x)\big)$
Currently, I am facing this problem:
Given two real functions $A( \vec x )$ and $B( \vec x ):\Bbb R^N\to \Bbb R$, I want to find a third real, monotonic function $f(x):\Bbb R\to\Bbb R$ such that:
$$...
2
votes
0
answers
48
views
book recommendation about iterative functional equations [closed]
I would like to learn about iterative functional equations.
I read this book, but it doesn't include such functional equations. I tried this, but it was too general for my purpose. Finally I read ...
1
vote
0
answers
93
views
Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant
Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...
1
vote
0
answers
113
views
What is behind the constant in the functional equation for the Hasse-Weil zeta function?
Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ of dimension $n$. The Weil conjectures assert that the zeta function $Z(X_0,t)$ satisfies the functional equation
$$Z(X_0,t) = \pm q^{\...
1
vote
0
answers
147
views
Solving an equation of function
How to solve, or at least how to proceed to solve, the following equation for $g(u)$
$$\int_0^{\infty} \{1-\cos(2\pi uh)\} g(u)du = (1+h^{\alpha})^{\beta/\alpha} -1?$$
Here $0<\alpha\leq2$ and $-\...
4
votes
2
answers
354
views
Equality in $\mathbb F_q\left(\left(\frac1T\right)\right)$
Can one characterize the $a\in\mathbb F_q\left(\left(\frac1T\right)\right)$ such that $a(T+1)=a(T)$? Although this seems elementary, I did not manage to find a answer.
Thanks in advance for any help.
6
votes
0
answers
164
views
Boolean functional equations
My current approach to investigating reversible quantum gates requires the solution of Boolean functional equations. For example,
$$f(x,y,z) = f(x,y \oplus f(x,y,z), z \oplus f(x, y, z)),$$
where $f\...