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I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise.

I define $$ g_2 (y) = \int_{-\infty}^{\infty} f(y+x) f(x) \mathrm{d} x $$ and recursively $$ g_n (y) = \int_{-\infty}^{\infty} g_{n-1}(y+x) f(x) \mathrm{d} x \, . $$

To make it easier and free myself from caring about integration limits, I expressed the solution as a Fourier integral

$$ \int_{-\infty}^{\infty} \mathrm{d} w \frac{1}{w^n} \left (e^{iwx} -1 \right )^n e^{-iwx} \, . $$

For $n = 1$, the resulting function should be a constant, and for $n = 2$ a triangle of the form $1-|x-1|$ (with some rescaling).

I would be grateful for any help. The Fourier approach might not be optimal.

Thanks a lot.

SSF
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1 Answers1

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One compact form for B-splines is $$ B_n(x)=\frac1{n!}\sum_{k=0}^{n+1}\binom{n+1}{k}(-1)^k(x-k)_+{}^n $$ where $x_+=\max(0,x)$.

Since $$ \sum_{k=0}^{n+1}\binom{n+1}{k}(-1)^k(x-k)^n $$ is the $(n+1)$-th iterated difference of a polynomial of degree $n$ the result is the zero function. From that the symmetry of the B-spline follows, especially that it is zero outside $[0,n+1]$.

$B_0$ with $0^0=0$ has a jump from 0 t0 1 at 0 and reverse at 1

$B_1=x_+-2(x-1)_++(x-2)_+$ has the correct kinks to be identical to $\max(0,1-|x-1|)$.

Lutz Lehmann
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