Questions tagged [functional-equations]

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$f' = e^{f^{-1}}$, 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 ...
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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 ...
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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 ...
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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: $$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$ $\...
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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 ...
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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(...
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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 ...
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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 ...
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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. ...
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  • 529
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0 answers
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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*} ...
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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+\...
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  • 2,858
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0 answers
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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 ...
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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 ...
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0 answers
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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^...
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  • 2,010
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 $...
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  • 6,286
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+...
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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 ...
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  • 403
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 ...
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  • 249
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 ...
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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 ...
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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 ...
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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 ...
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  • 235
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 $$\...
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  • 383
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 ...
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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. ...
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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 ...
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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 > ...
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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 $x$...
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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,...
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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 ...
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  • 516
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,\...
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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 $\...
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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 ...
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  • 6,286
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 ...
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  • 83
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 ...
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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)...
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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 ...
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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 ...
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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 ...
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  • 3,463
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 ...
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  • 1,643
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 ...
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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}...
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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 ...
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  • 3,463
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: $$...
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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 ...
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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 ...
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  • 383
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^{\...
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  • 3,686
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 $-\...
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  • 133
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.
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  • 2,997
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\...
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