# van Cittert deconvolution method

In the early 1930s, van Cittert published a deconvolution method. Although his method was not perfect but it is the forefather of many improved spectral deconvolution methods. The basic idea is that if we know how the instrument distorts an input, we can rectify the experimentally observed spectrum. It is an approach for solving the convolution integral. I am trying (with some machine translation help) to read his original German paper, Burger, H. C., and P. H. Van Cittert. "Wahre und scheinbare Intensitätsverteilung in Spektrallinien". Zeitschrift für Physik 79, no. 11-12 (1932): 722-730, Zbl 0006.12203.

He basically shows that the apparent intensity distribution of the apparatus for a given (spectral) line is known to be determined by the integral...

Die scheinbare Intensitätsverteilung $$S(y),$$ die der Apparat für die betreffende Linie gibt, wird bekantlich bestimmt durch das Integral: $$S(y)=\int_{-\infty}^{+\infty} W(x) A(y-x)\,\mathrm{d}x$$

and afterwards with more text, he says,

Die Integralgleichung (1) ist durch einen Kunstgriff in eine Gleichung des Fredholmschen Typus umzuwandeln: $$S(y)=W(y)+\int_{-\infty}^{+\infty} W(x)\{A(y-x)-\Delta(y-x)\}\,\mathrm{d}x$$ $$\Delta(p)$$ ist eine Zackenfunktion, welche nur für $$p=0$$ von Null verschieden ist, und der Bedingung: $$\int_{-\infty}^{+\infty} \Delta(p)\,\mathrm{d}p=1$$ genügt.

Basically he is saying that the integral equation can be transformed into Fredholm type equation by a trick, using a Zackenfunction?

a) I wanted to know what trick is he referring to convert the convolution integral into a Fredholm type?

b) After searching the Google images for Zackenfunction, it seems it is a discontinuous function made from linear segments. I cannot find any English equivalent and the machine translation says "jagged function".

Thanks.

• I was not aware of this work and from the point of view of an R&D Electroncs Engineer, it is very interesting, so I added to the OP a few hyperlinks. However, feel free to revert my edits. Mar 12 at 18:44
• No, problem. I was also trying to do it but it was directly downloading the pdf. Mar 12 at 18:46

b) The German word Zackenfunktion ("spiky function") refers to the Dirac delta function, more commonly denoted with a lower case $$\delta$$.
a) The delta function has the property that $$\int_{-\infty}^\infty \delta(y-x)f(x)\,dx=f(y)$$, hence the identity used by Van Cittert.