# Questions tagged [functional-equations]

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153
questions

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40 views

### A hard function with 2 solutions [closed]

Recently my juniors came across an exam. It turned out to be the hardest problem
Determine all the functions $f:\mathbb R\rightarrow\mathbb{R}$ such that the equation satisfies
$f(2rs+f(r+s))=rf(s)+...

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78 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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**1**answer

158 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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73 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**answer

177 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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**1**answer

207 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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317 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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255 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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36 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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**1**answer

309 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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**1**answer

209 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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211 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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40 views

### Infimum of odd path functional

Let $c>0$, $A_c\triangleq \left\{ f \in C_0([0,1]:\mathbb{R}^d): \|f_t\| >c \mbox{and} \dot f \mbox{ exists-a.e.}\, \mbox{ for some } t \in [0,1]
\right\}$, and set $g(x)=\left(\max\{x_i,0\} \...

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39 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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70 views

### Can we solve this integral equation?

Let $(E,\mathcal E,\lambda)$ be a measure space, $p,q_i$ be positive probability densities on $(E,\mathcal E,\lambda)$ for $i=1,2$, $\mu:=p\lambda$, $\sigma_{ij}:E^2\to[0,\infty)$ be $\mathcal E^{\...

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85 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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94 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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140 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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**2**answers

345 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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152 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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106 views

### The uniqueness of a solution of a differential equation on a unit circle

Trying to solve one problem in the geometry of 2-dimensional Banach spaces, I arrive to the problem of uniqueness of the following differential equation on a function $r:\mathbb T\to(0,1]$ defined on ...

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127 views

### Criteria for $f(f(x))=g(x)$

I 'm searching about the solvability of the functional equation $f(f(x))=g(x)$. I have three questions about it:
Let's be $g$ an arbitrary function and the functional equation $f(f(x))=g(x)$. Are ...

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**1**answer

72 views

### Sets closed by sum and solutions to the Cauchy functional equation

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a solution to the Cauchy functional equation
$$f(a+b)=f(a)+f(b),\quad\forall a,b\in\mathbb{R}.$$
Observe that
$$A:=\{a\in\mathbb{R}:f(a)\geq 0\},\quad B:=\{...

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**1**answer

290 views

### Existence of function satisfying $f(f'(x))=x$ almost everywhere

My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$...

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703 views

### Does there exist another form of the derivative for polynomials?

Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]...

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251 views

### Another functional inequality

Is there some general solution to the functional inequality:
$$ f(xy) \leq y f(x) + x f(y)$$
Where $x,y\in[0,1]$?
I can find many particular solutions but I just wonder if there is a more general ...

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votes

**1**answer

94 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?

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175 views

### Find real function $f(x)$ such that $f(f(x))=f'(x)$ [duplicate]

Absolutely there is a trivial solution $f(x)=0$. Actually, assuming $f(x)$ being smooth and expanding $f(x)$ into power series one can get $f(0)=0\to f(x)=0$. Also, in the complex field there are ...

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368 views

### Formal group law and Koenigs function conjecture?

Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). \tag{1}$$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$
This equation has many ...

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**1**answer

67 views

### On probabilistic extension for Bernstein polynomials

Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...

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148 views

### A functional equation in real analysis

For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$,
$\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$
...

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170 views

### Functional equation $\int_z^{2z} [f(x)-f(z)] dx = 0$

Suppose a continuous function $f:[0,1] \to \mathbb{R}$ satisfies the following equation for all $z \in \left(0,\frac{1}{2}\right)$,
$$\int_z^{2z} [f(x)-f(z)] dx = 0.$$
It is clear that a constant ...

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**0**answers

43 views

### Integral equation with kernel defined in a rectangle

Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$
Observe that the kernel is not defined on a square.
My question: Can I apply the classical ...

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**0**answers

51 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 ...

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40 views

### A special integral equation of Volterra type

Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation:
$$
f(t)\int_0^t a(s)\,ds + \int_0^t a(t - s) f(s) \, ds = 0
$$
My question is : under what condition ...

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**1**answer

296 views

### A variant of Cauchy-type functional equation conjecture

Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that
$$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$
Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$
The answer is ...

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230 views

### Background on the functional equation $F(x+1)+F(x)=f(x)$ [closed]

In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-...

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388 views

### Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$
I think the following system of equations ...

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103 views

### A generalized Cauchy type functional equation

Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...

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157 views

### How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?

Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...

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**1**answer

124 views

### Solvability of a certain functional equation in simple $C^*$ algebras

For which simple unital $C^*$ algebras does the following functional equation have a solution:
$$ d^2=0,\;{(d+d^*)}^2=1$$
The Calkin algebra and $M_{2n}(\mathbb{C})$ are some examples. It is not ...

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**1**answer

89 views

### 1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation
$$
\begin{split}
\mathbb ...

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**1**answer

74 views

### Equation for a geometrical half surface from folding a flat curved in one direction surface in half [closed]

I cannot not formulate the problem as i do not know how to model this idea. This is an open question not an mathematical exercise so if anybody has a good proposition i'm happy to use it.
Sorry in ...

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**2**answers

81 views

### Find the general solution to the Forsyth/Abel functional equation

Let $R(m,n)$ be defined on all the integers such that $R(m,0)=m, R(0,n)=n, R(m,n)=R(n,m)$ and $R(R(m,n),p)=R(m,R(n,p))$ for all integers $p$. Thus $R$ satisfies the Abel associativity equation. Let $...

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67 views

### Characterisation of functions for which the Fourier transform commutes with a particular operator

Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...

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**2**answers

274 views

### Seeking proof to an asymptotics of a recursion or functional equation

My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion
$$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\...

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76 views

### Functional equation involving integrals and exponential

Can we find on $\mathbb{R}^+$ a real positive function $f(x)$ (in $C^{\infty}$) such that:
$$\int_0^{\infty} f(x) e^{\lambda \int_1^{x} f(t)^2 dt} dx=0$$
where $\lambda$ is a complex number (with $0&...

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**2**answers

85 views

### A functional equation with a quadratic solution

I have the following problem. I have a function $v(x, \theta)$ that can be expressed in two ways, for all $x, \theta \in \Re$:
$v(x, \theta) = u(x - \theta)$, where $u$ is strictly concave and ...

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**0**answers

53 views

### Existence of a couple of functions solution of a differential equation (with additional constraint)

I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$):
$$...

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**2**answers

377 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? ...