Questions tagged [functional-equations]
The functional-equations tag has no usage guidance.
66
questions with no upvoted or accepted answers
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answers
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views
Zappa-Szép products of the group of integers with itself
Since my previous question didn't get much attention and I couldn't make any relevant progress on it, I thought it would be a good idea to "simplify" it by replacing monoids by groups. That is:
...
- 5,402
7
votes
0
answers
376
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General solution of the Multiplicative symmetry equation $f(xf(y))=f(f(x)y)$ in nonabelian groups
As we know, the functional equation $f(xf(y))=f(f(x)y)$ was completely solved in abelian groups (by J. G. Dhombres, Solution... $f(x\ast
f(y))=f(y\ast f(x))$, Aequationes Math. 15 (1977), 173--193, ...
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6
votes
0
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167
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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\...
- 61
6
votes
0
answers
136
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Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x
Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form
$$
g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...
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5
votes
0
answers
447
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modularity of Theta functions attached to Hecke characters
Let $K/\mathbb{Q}$ be a quadratic imaginary field, and let $\chi$ be a Hecke character on $K$. Using Poisson summation, one can show that the theta function
$$
\theta(z):=\sum_{I\subseteq \mathcal{O}...
- 3,232
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votes
0
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98
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Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,402
5
votes
0
answers
1k
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The functional equation $f(x) = qx + qxf(x) - f(x^2)$
A word (i.e., ordered string of letters) is bifix-free provided it has no proper initial string and terminal string that are identical. For example, the word $ingratiating$ has bifix $ing$, but the ...
- 399
4
votes
0
answers
77
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A functional equation: Functional families that are "weakly" closed under product?
Suppose that for any real number $a$, we have a function $f_a:\mathbb R \to \mathbb R$ or such that $f_a(x)$ is monotonically strictly increasing in $x$ and hence invertible on its image. We also ...
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4
votes
0
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119
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$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer
Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$
\begin{split}
f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\
f(...
- 733
4
votes
0
answers
100
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Associativity equation for topological rings and logarithms
Let $R$ be a topological ring of characteristic zero. Assume that $R$ is commutative. Let $G :R \times R \to R$ be a continuous function satisfying $G(G(x,y),z)=G(x,G(y,z))$ and $G(0,x)=x$, for all $x,...
- 433
4
votes
0
answers
109
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continuous linear recurrent relations
For a function $f:\mathbb{R}\rightarrow \mathbb{R}$ denote $f_0(x)=x$, $f_n(x)=f(f_{n-1}(x))$. Assume that $f$ satisfies a functional equation
$$f_n(x)+a_{n-1} f_{n-1}(x)+\dots+a_0 f_0(x)\equiv 0$$
...
- 103k
4
votes
0
answers
92
views
Archimedean $\varepsilon$-factors
Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined ...
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3
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226
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$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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3
votes
0
answers
191
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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 ...
3
votes
0
answers
362
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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 ...
- 2,817
3
votes
0
answers
241
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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)...
- 89
3
votes
0
answers
201
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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 ...
- 189
3
votes
0
answers
200
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Reflection Formulas for the $\Gamma$ Function
We have
$$\begin{align}
&\Gamma\Big(1~+~x\Big)~\cdot~\Gamma\Big(1-x\Big)~=~\frac{\pi x}{\sin\pi x}
\\\\
&\Gamma\Big(\tfrac12+x\Big)~\cdot~\Gamma\Big(\tfrac12-x\Big)~=~\frac\pi{\cos\pi x}
\\\...
- 655
3
votes
0
answers
74
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Solving a nonlinear integral equation for a distribution function
I have an equation of a probability distribution function $F$ on $[0,\infty)$, $$F(x)=e^{-\eta [1-\int_0^x F(x-y)g(y)\mathrm{d}y]},\quad x\in[0,\infty), $$
where $g$ is a probability density function, ...
- 31
3
votes
0
answers
308
views
Modified Jacobi’s theta function
Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$
Therefore $$\sum\limits_{k=1}^\infty ...
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2
votes
0
answers
67
views
Methods for holonomic recurrences
I wanted to ask if anyone knows of good texts/resources on methods for solving holonomic recurrence relations (if there are any general analytical approaches):
$$p_1(n)a(n)+p_2(n)a(n-1)+\dotsb+p_k(n)a(...
- 21
2
votes
0
answers
56
views
any ideas on how to solve matrix equation like this $X A_i Y = B_i$
the objective function is like
$$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix
I've tried gradient-descent, but as it turns out not well, I wonder if ...
- 21
2
votes
0
answers
148
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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 ...
- 759
2
votes
0
answers
126
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 ...
- 83
2
votes
1
answer
528
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 ...
2
votes
0
answers
57
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 ...
- 545
2
votes
0
answers
44
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 ...
- 545
2
votes
0
answers
87
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&...
- 1,121
2
votes
0
answers
131
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Analytic continuation of $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s}$. Possibly related to Riemann Zeta function $\zeta(s)$?
I'm trying to find the analytic continuation for
$\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s} ,$
with $a\in \mathbb{N^+}$ and $s<1$. I need most likely only the values for $s=\frac{1}{2}-m$...
- 129
2
votes
0
answers
120
views
Which functions $f: \mathbb{R} \to \mathbb{R}$ is injective over some subinterval of $(x,y)$ whenever $x<y$ and $f(x) \ne f(y)$?
Under what conditions on a function $f: \mathbb{R} \to \mathbb{R}$ can we say that given any real numbers $x,y$ with $x<y$ if $f(x) \ne f(y)$ then there is a sub-interval $S_{(x,y)}$ of $(x,y)$ ...
- 812
2
votes
0
answers
112
views
Family of (Cumulative Distribution) Functions
I'm looking for a 2 (or more)-parameter family of functions $F$ with the following properties:
For each $f \in F$, $f(0)=0$, $f(1)=1$, and $f$ is (weakly) increasing.
$F$ is closed under products.
$F$...
- 341
2
votes
0
answers
187
views
Solve this functional equation with respect to $f$
Let $v\not= 1$ be a real number. Let $f(s)$ be real analytic on an open interval containing $v$ and $1$, with a zero of order $m\ge 1$ at $s=1$.
My question is: Can we solve this functional equation ...
- 403
2
votes
0
answers
140
views
A Convolution Integral Equation
Is there any close-form solution for a function $f(t)$ satisfied the below equation:
$f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known ...
- 23
2
votes
0
answers
560
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The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is well-...
- 19.5k
2
votes
0
answers
314
views
elementary Abel function of a polynomial
Is there an elementary real function $F$ such that
$F(1+F^{-1}(x))$ is a polynomial of degree at least 2 without real fixpoints.
- 737
2
votes
0
answers
259
views
Algebraic Dirichlet series and beyond
I wonder what the "right" notion of "algebraic Dirichlet series" might be. Here I'm thinking of formal Dirichlet series $D(s)=\sum_{n\geq 1} a_n/n^s$, say with $a_n$ being rational numbers.
I'm ...
- 5,563
1
vote
0
answers
34
views
automorphisms and mellin transforms
If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...
- 101
1
vote
0
answers
82
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Maximizing the integral of a transformation that depends on a neighborhood of values of the original function
I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. Also, any suggestion on changes that might make the problem better are welcome.
...
- 11
1
vote
1
answer
63
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Finding closed form roots for pseudo-trinomial
I have the below function:
$$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) ...
- 11
1
vote
0
answers
54
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...
- 131
1
vote
0
answers
117
views
Invariant polynomials under a non-standard group action
There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...
1
vote
0
answers
34
views
Hardy type inequality with singular weights
Recall that Hardy's inequality involving distance from the boundary of a convex set $\Omega \subsetneq \mathbb{R}^n ; n \geq 1$, asserts that
$$
\int_{\Omega}|\nabla u|^p \, d x \geq\left(\frac{p-1}{p}...
- 419
1
vote
0
answers
29
views
Rätz orthogonality and involution
In the Rätz’s sens of orhtoganality, can we find an exemple of an involution u(different to -Id)such that x orthogonal to y then x orthogonal to u(y)
1
vote
0
answers
51
views
Can I apply $q$-Lagrange Inversion formula?
Now I have equation $F(x) = x \sum_{k\ge 0} g_k F(x) F(qx) \cdots F(q^{k-1} x)$, I need to get the coefficient of $x^n$ in $F(x)$, can I apply $q$-Lagrange Inversion formula to this?
Moreover, I have ...
- 11
1
vote
0
answers
56
views
Functional approximation with derivatives
I am trying to solve a functional approximation problem.
Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...
1
vote
0
answers
42
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 ...
- 31
1
vote
0
answers
104
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 ...
- 161
1
vote
0
answers
139
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
$$Z(X_0,t) = \pm q^{\...
- 4,124
1
vote
0
answers
154
views
Solving an equation of function
How to solve, or at least how to proceed to solve, the following equation for $g(u)$
$$\int_0^{\infty} \{1-\cos(2\pi uh)\} g(u)du = (1+h^{\alpha})^{\beta/\alpha} -1?$$
Here $0<\alpha\leq2$ and $-\...
- 133
1
vote
0
answers
93
views
Fredholm integral equation of third kind
Let us consider the following integral equation
$$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 ...
- 545