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Tuesday, October 22, 2002

Tuesday October 22, 2002

Filed under: General,Geometry — m759 @ 1:16 AM

Introduction to
Harmonic Analysis

From Dr. Mac’s Cultural Calendar for Oct. 22:

  • The French actress Catherine Deneuve was born on this day in Paris in 1943….
  • The Beach Boys released the single “Good Vibrations” on this day in 1966.

“I hear the sound of a
   gentle word

On the wind that lifts
   her perfume
   through the air.”

— The Beach Boys

 
In honor of Deneuve and of George W. Mackey, author of the classic 156-page essay, “Harmonic analysis* as the exploitation of symmetry† — A historical survey” (Bulletin of the American Mathematical Society (New Series), Vol. 3, No. 1, Part 1 (July 1980), pp. 543-698), this site’s music is, for the time being, “Good Vibrations.”
 
For more on harmonic analysis, see “Group Representations and Harmonic Analysis from Euler to Langlands,” by Anthony W. Knapp, Part I and Part II.
 
* For “the simplest non-trivial model for harmonic analysis,” the Walsh functions, see F. Schipp et. al., Walsh Series: An Introduction to Dyadic Harmonic Analysis, Hilger, 1990. For Mackey’s “exploitation of symmetry” in this context, see my note Symmetry of Walsh Functions, and also the footnote below.
 
† “Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey… to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries.”
The importance of mathematical conceptualisation,
by David Corfield, Department of History and Philosophy of Science, University of Cambridge

5 Comments

  1. I want to comment, but you’ve got it all locked up opaque.

    Comment by HomerTheBrave — Tuesday, October 22, 2002 @ 2:25 AM

  2. Sorry… I was working on this entry repeatedly after first posting it… Something I often do, and I’ve wondered if it screws up possible readers. Apparently so. Or it might have been another problem… The non-Xanga site where I store images and midis is down, so my Xanga site is blind and silent for the time being.

    Comment by m759 — Tuesday, October 22, 2002 @ 3:59 AM

  3. That wasn’t the problem at all. I meant it’s all circular and internally-consistent. There’s not much to add.

    Comment by HomerTheBrave — Tuesday, October 22, 2002 @ 4:08 AM

  4. “Circular and internally consistent” is a good phrase; it describes the circular functions, on which harmonic analysis is based, and the analysis itself.

    Of course, we mathematicians try for something more, something we call “elegance”… for which, see Deneuve.

    Comment by m759 — Tuesday, October 22, 2002 @ 4:16 AM

  5. Unfortunately, both of you brainiacs have me rolling …

    My eyes. 

    Harmonic Analysis is something for which I look to Big Sur parties and beach bon fire gatherings.  Typically the guy with the guitar has one.  And we analyze it.  Then we play more music.

    Catherine Deneuve caught my eye in “The Hunger” … a very ‘elegant’ vampire movie starring the vixen, David Bowie and Susan Sarandon.  It was glorious.  And I wear Coco … by Chanel, but there IS that number five again.

    Comment by oOMisfitOo — Wednesday, October 23, 2002 @ 2:36 AM

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