Questions tagged [functional-equations]

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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)+...
2
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0answers
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 ...
4
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1answer
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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0answers
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)...
1
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1answer
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 ...
3
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1answer
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 ...
5
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1answer
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 ...
1
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2answers
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 ...
1
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0answers
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 ...
2
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1answer
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}...
5
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1answer
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 ...
2
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1answer
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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0answers
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\} \...
2
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0answers
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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0answers
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^{\...
1
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0answers
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 ...
1
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0answers
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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0answers
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 $-\...
4
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2answers
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.
5
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0answers
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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0answers
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 ...
3
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0answers
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 ...
2
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1answer
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:=\{...
3
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1answer
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 $x \in \mathbb{R}$...
4
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3answers
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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1answer
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 ...
0
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1answer
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?
2
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0answers
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 ...
2
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1answer
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 ...
-1
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1answer
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\...
0
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1answer
148 views

A functional equation in real analysis

For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$, $\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$ ...
2
votes
1answer
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 ...
2
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0answers
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 ...
1
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0answers
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 ...
2
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0answers
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 ...
7
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1answer
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 ...
0
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1answer
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)-...
5
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2answers
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 ...
1
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0answers
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)...
10
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2answers
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 ...
3
votes
1answer
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 ...
0
votes
1answer
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 ...
1
vote
1answer
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 ...
0
votes
2answers
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 $...
1
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0answers
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 ...
3
votes
2answers
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\...
2
votes
0answers
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&...
1
vote
2answers
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 ...
0
votes
0answers
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, $0<Re(\alpha)<1$): $$...
2
votes
2answers
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? ...