|WikiProject Mathematics||(Rated Start-class, Mid-priority)|
P or K?
I made a major expansion to this article, and changed the symbol for the kernel to K; this is in keeping with most other articles on WP where the letter K is used for kernels. By contrast, the letter P is usually reserved for projections, and since it is not uncommon to talk about projections and kernels at the same time, it would be unfortunate to call a kernel by the letter P. Is it really critical to break convention for this article? Alternately, is there some subject domain where P would be used instead of K? What subject domain would that be? I've seen this at best in control theory, and electrostatics, and am pretty sure P is never used in those domains. linas (talk) 22:34, 2 November 2008 (UTC)
- Let it be known that I really don't have strong opinions either way. However, in my own experience, the letter P is more common for the Poisson kernel than the letter K. Off the top of my head, green Rudin uses P, as does the book of Stein and Weiss (and its sequels). siℓℓy rabbit (talk) 22:45, 2 November 2008 (UTC)
- I haven't encountered K as the letter of the Poisson kernel. But my area of expertise is Harmonic analysis, and PDE, so it may be that in some other fields it gets denoted by K because of projections, but I am unaware of it. That is why I changed it back to P. Green Rudin is Rudin's real and complex analysis. Aside from Stien's books I can add Pinsky's book "Introduction of Fouirer Analysis and Wavelets". Also Javier Duoandikoetxea's book "Fourier analysis." I have some more in my office if needed. Out of curiousity who uses K? Thenub314 (talk) 07:35, 3 November 2008 (UTC)
You got me there; I don't believe I have any books that explicitly go out of their way to review the Poisson kernel in and of itself. I'm currently reviewing my shaky knowledge of Hardy spaces; it appears there, as a 'by-the-way' sort of topic, but as K. My insistence for K is simply that pretty much everything I see uses K for integral kernels (sometimes G for Green's functions, if its a physics book), without explicitly talking about some specific kernel or other. Hmm. The only book I have on specific cases is "Polynomial Expansions of Analytic Functions" by Ralph Boas and R. Creighton Buck which is a rather charming if olde-fashioned book that is filled with details of the regions of convergence for all sorts of 'obscure' series. Anyway, it uses K consistently for all the various kernels; it does not review the Poisson kernel. K is used pretty consistently in the vanishingly slim amount of reading I've done in Fredholm theory. I'm also browsing in spectral theory, operator theory, probability theory, dynamical systems, with K or G being used pretty consistently when it does show up. I've never seen P used there for anything other than projections (or probabilities). linas (talk) 02:43, 5 November 2008 (UTC)
- When I say 'K' of 'G' above, I mean "used for kernels in general", not the Poisson kernel in particular. The only book I have that actually talks about the Poisson kernel is a slim volume on linear operators, that, while good, wouldn't be authoritative. (although it does use K). linas (talk) 02:53, 5 November 2008 (UTC)
- I understood. Your right that when speaking of general kernels in general it is standard to use K, (and in PDE G is used often for greens functions). When the Poisson kernel is given some attention to itself, I usually see it as Pr or something similar (and Qr for the conjugate kernel). The two books we give has references also use P. I can't think of many places that this really causes trouble with other notation. You mentioned I should make changes of substance instead of style, next time I make a real edit, I will change it back to P. :) Thenub314 (talk) 06:19, 5 November 2008 (UTC)
Paragraph on complex analysis
I am taking out the following paragraph:
The Poisson kernel is important in complex analysis because its integral against a function defined on the unit circle — the Poisson integral — gives the extension of a function defined on the unit circle to a harmonic function on the unit disk. By definition, harmonic functions are solutions to Laplace's equation, and, in two dimensions, harmonic functions are equivalent to meromorphic functions. Thus, the two-dimensional Dirichlet problem is essentially the same problem as that of finding a meromorphic extension of a function defined on a boundary.
I don't know what is meant by "in two dimensions, harmonic functions are equivalent to meromorphic functions". The real part of a meromorphic function is harmonic, as is the imaginary part, but if one starts with an arbitrary function in Lp on the unit circle, one gets a harmonic function in the disk which is not meromorphic. If the function on the unit circle is real for instance, then the function in the disk will be real − which means it's not meromorphic. If someone wants to rewrite the paragraph in a way that is true, please do. I suppose what is meant is that the harmonic function is the sum of a holomorphic function and an antiholomorphic function. Eric Kvaalen (talk) 14:00, 12 December 2013 (UTC)