Talk:Mercer's theorem

message: Two pages link to here: kernel trick and James Mercer.  Yet, when I click on "what links here" I am told that no pages link to this one.  Michael Hardy 22:22, 24 Aug 2003 (UTC)
replies: text: Kernel trick is showing up for me -- perhaps the changes just take a while to propagate? -- Tlotoxl 22:33, 24 Aug 2003 (UTC)
thread_title: [Untitled]

message: Here's the proof:  If K is a p.d. continuos kernel on a compact space, then there is a Hilbert space H and a measurable function phi:X -> H such that
K(x,y) = <phi(x) , phi(y)>
Proof: Let H = l^2 and let 
phi(x) = sum_i sqrt{lambda_i} e_i(x)
By the trace property of the associated operator, this thing is well defined.
I think this is the right statement of the Kernel Trick (I don't think you can get by with L^2 kernels, because you don't get fast enough convergence.CSTAR 04:36, 13 May 2004 (UTC)
thread_title:  The Theorem you need? 

message: In fact, the kernel trick mentioned above has really nothing to do with Mercer's theorem.  It results from a general procedure for constructing Hilbert spaces out of  positive definite functions on arbitrary sets: If K is positive definite function, there is a Hilbert space H and
a function phi :  X --> H
K(x,y) = <phi(x) , phi(y)>
This is an old trick.  Define a (possibly degenerate) inner product on functions of finite support on X using the kernel K as foolws
< a, b > = sum_x,y K(x,y) a(x) b(y)
Divide out by vectors of norm zero ancd complete.  This is the oldest trick in the book of Hilbert space constructions. So you can forget about continuity., Mercer's theorem and everything else.
CSTAR 05:23, 13 May 2004 (UTC)
thread_title:  Further clarification 

message: A more precise reference of where to find the proofs
of the theorems in the article would be nice.
130.238.5.7 18:47, 26 January 2006 (UTC)
replies: text:  This follows from the structure theorem for Hilbert-Schmidt operators. Note that it is required that K be a square-integrable kernel. I'm sure this is wel-known, but I'll have to dig up a reference.--CSTAR 14:02, 7 December 2006 (UTC)
replies: text:  Thanks for the quick reply, the reference would be very useful. 158.143.77.11 14:42, 7 December 2006 (UTC)
thread_title:  More precise reference 

message: There is a separate article on Mercer's condition. Should that be incorporated here?
Kjetil Halvorsen 01:21, 25 March 2011 (UTC)
replies: text: Yes. Suggested the merge. Qwertyus (talk) 09:43, 2 July 2012 (UTC)
thread_title:  Article on Mercer's condition 

message: The section Generalization has noun-verb disagreement and run-on sentences that make it difficult to figure out what is going on. Is there someone qualified who can clear it up? Thanks.
128.171.78.188 (talk) 09:52, 16 April 2012 (UTC)
thread_title:  Confusing grammar 

message: Shouldn't it say "If K is a symmetric non-negative definite L2 kernel ..." to be precise?
134.106.106.150 (talk) 10:39, 22 October 2014 (UTC)
thread_title:  Last generalization in the article 

message: Hello,
I believe that Mercer's Theorem and Mercer's Condition should be combined, as they are closely related and of the same works and time period of Mercer's life. 
Further, I think that these two articles should be merged with the barren page of James Mercer, the mathematician that propelled these outcomes and has their namesake.  — Preceding unsigned comment added by 2601:280:4A00:CF61:199A:2745:F0E:3CA8 (talk) 19:31, 17 May 2016 (UTC)
replies: text: Don't merge: Mercer's theorem and Mercer's condition seem to be sufficiently distinct to warrant their own pages, and given the complexity of the topic for amateur readers, I feel that they are best kept separate. Klbrain (talk) 22:14, 18 June 2016 (UTC)
replies: text: Reading the earlier correspondence, I have been inclined to changed my mind given the comments of subjcet experts ... suggesting merge. Klbrain (talk) 11:20, 20 June 2016 (UTC)
thread_title:  Combining Mercer's Theorem and Mercer's Condition 

message: The article seems to imply that Mercer's theorem is a special case of what is now called the spectral theorem. Can this be tightened up? Does Mercer's theorem ever say anything more than what the spectral theorem might say? Or is it a pedagogical, historical curiosity that has been supplanted by more modern language, tools and techniques? Likewise, this article talks of "kernels", which, as far as I can tell, is identical to a Fredholm kernel. Is there any sense in which the so-called "kernel" in this article is not (a special case of) a Fredholm kernel? 84.15.182.62 (talk) 08:04, 28 June 2023 (UTC)
thread_title:  Spectral Theorem?