`============================== A math story: =============================` ``` When I was little, growing up in Denmark, [before kindergarten; only there wasn't any then...],-- so before I heard of 'The tinder box', or 'The ugly Duckling' from the books of Hans Christian Andersen, my father told me about low-pass and high-pass filters. He was a telephone engineer and worked on the filters used in signals transmitted over long cables, just after the war, WWII. The 'high' and 'low' part of the story refers to frequency bands of the sound signals. Not that this meant much to me at the time. Rather, I was fascinated by the pictures in the EE journals that were stacked up on the floor next to me, and I spent hours looking at them [--that was all there was, there on the floor!], so these pictures of filter design, some in color, occupied me on long Sundays while my dad was building instruments in the living room. Nothing else for me to do! Then, after going to school, I forgot all about my dad's explanation of quadrature mirror filters; no wonder(!!), and they were out of mind for a very long time. I never had any particular reason to think much about them at all, I mean the low-pass frequency bands and all that, but I am sure they in some strange way created a lasting visual impression for me. If I heard them mentioned later, I might have been slightly amused, but no more than that. Perhaps not before wavelet math, or rather my interests in wavelets in the late 1980ties, did all of that stuff about frequency bands gradually resurface, from out of a mist, recalled from the back reaches of my mind. So it was only after I grew up and matured myself, that I found that these subband filters define operators in Hilbert space which satisfy all kinds of abstract relations, now known as Cuntz- and Cuntz-Krieger relations, and thought in some circles to have been invented in 1977, by Cuntz, or in 1965 by Dixmier. These are tools from math that I had gotten involved with in the late 1970ties for completely different reasons. My impression is that the operator relations that are called the Cuntz relations in math go way back, probably way back to before I was born; and they are and have been used every day, and twice on Sundays, ever since, by signal processing engineers, and others I probably don't even know about. In addition to their extensive use in several areas of math! Our matrix functions from math are actually called poly-phase matrices by engineers, and they are scattering matrices in other circles, and quantum gates in physics. In fact a lot of the things we do in operator theory are known and used in other fields, but known under different names, and known in different ways. And important for different reasons! In any case, they *are* really important, and for all kinds of good reasons, not least of which is their rediscovery in operator algebras, and in wavelets. ----------------------------------------------- Palle Jorgensen; Math University of Iowa Iowa City, IA 52242 USA http://www.math.uiowa.edu/~jorgen/```