Questions tagged [functional-equations]

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33 views

Almost-differential functional equations

The ODE $y'(x)+P(x)y(x)=Q(x)$ has solution $$I(x)y(x)=\int I(x)Q(x)\,dx$$ where $I(x)=\exp\int P(x)\,dx$. Equivalently, $$Y(x)+P(x)\int_0^xY(t)\,dt=Q(x)\tag1$$ has solution $$Y(x)=\frac d{dx}\frac{\...
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1answer
80 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
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0answers
118 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
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2answers
122 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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3answers
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
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2answers
330 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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0answers
44 views

How do i get the $\Delta_a$ value that would maximize $(\Delta_d-\Delta_a)$

so this problem is derived from a real world problem i have so i'm unsure if it can be solved through maths or not. I'm currently "finding" the solution through tons of computer iteration ...
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0answers
92 views

What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?

Cross-post from MSE. The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...
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1answer
130 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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0answers
129 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
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1answer
127 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
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0answers
324 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
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1answer
51 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
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2answers
435 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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1answer
194 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
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1answer
123 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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0answers
94 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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1answer
104 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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0answers
56 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
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0answers
556 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
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1answer
225 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
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0answers
96 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
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1answer
173 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
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0answers
196 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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1answer
181 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
1answer
225 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
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1answer
352 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
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2answers
445 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
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0answers
37 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
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1answer
316 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
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1answer
302 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
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1answer
212 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
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0answers
44 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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0answers
90 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
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0answers
102 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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0answers
141 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
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2answers
352 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.
5
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0answers
156 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\...
3
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0answers
139 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 ...
2
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1answer
76 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:=\{...
3
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1answer
341 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 $x \in \mathbb{R}$...
5
votes
3answers
722 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]...
1
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1answer
304 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 ...
0
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1answer
96 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?
2
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0answers
187 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 ...
1
vote
1answer
410 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 ...
-1
votes
1answer
70 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\...
0
votes
1answer
153 views

A functional equation in real analysis

For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$, $\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$ ...
2
votes
1answer
173 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 ...
2
votes
0answers
45 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 ...