Linked Questions

I would like to formulate a bump function (link) $f:\Bbb R \to\Bbb R$ with the following properties on the reals: $$ f(x) = \begin{cases} 0, & \mbox{if } x \le -1 \\ 1, & \mbox{if } x = 0 \\...

The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions: a functional–integral equation$\require{action} \require{enclose}{^{\texttip{\dagger}{a ...

Is possible to construct infinitely differentiable functions that interpolate through arbitrary points, the way polynomial splines do? If so, do they have a name and is there an algorithm for ...

Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) \...

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...

Suppose I have a sequence defined via its first term and a recurrence relation involving summation over all previous values with some coefficients. Here is the sequence I am interested in right now (...

Calculate the infinite product $f_q(x):=\prod_{n=0}^\infty\frac{\sin(q^n x)}{q^n x}$, where $x$ is real and $0<q<1$. In other words, $f_q$ must satisfy the functional equation $f_q(x)=f_q(qx)\...

Consider the function $$f(x)=\prod_{n=0}^\infty\operatorname{sinc}\left(\frac{\pi \, x}{2^n}\right),\tag1$$ where $\operatorname{sinc}(z)$ denotes the sinc function. It arises as a Fourier transform ...

We want to find an elementary function $f(x)$ that is smooth and strictly increasing on some interval $x\in(0,\epsilon)$, satisfying $\lim\limits_{\,x\to0^+}f(x)=0$, whose asymptotic for $x\to0^+$ is $...

Yesterday I learned about the strange Fabius function $f$ in this question. Given my interest in neural networks and the fact that this function has a distinct sigmoid shape, I became curious about ...

Statement Consider an advanced functional differential equation $$ Lf(x) = f(2x+\pi)+f(2x-\pi),\quad L\equiv\frac{d^2}{dx^2}+1. \tag{1} $$ Let's construct a solution of Eq. $(1)$ with finite ...

Looking for closed-form examples of Delay Differential Equations I found that $$u'(x)=2u(2x+1)-2u(2x-1)$$ could had as solution a modification of the Fabius function $F(x)$, which "is an example ...

I am trying to build a function $q(x)$ from another function $f(x)$ as: $g(x)=f(x+1)-f(x-1)$ $z(x)=\int\limits_{-2}^{x} g(u)\ du$ $q(x)=z(2x)$ from the function I took from here: $$f(x) = \begin{...