Talk:Multiplier (Fourier analysis)

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In mathematics, a multiplier is a kind of operator, or transformation of functions.

I wonder how one might rephrase this with a qualifier, saying that that's one way of using the word multiplier in mathematics. There are also such things as Lagrange multipliers and the "exponential multipliers" used in differential equations. Michael Hardy 21:51, 7 Jan 2005 (UTC)

References?[edit]

Excellent survey: clear, concise… but I miss some bibliographic references: In the article there is mention to proofs and papers, but not too their location in specific books or journals. —Preceding unsigned comment added by 65.25.219.172 (talkcontribs)

Solid references definitely needed here...[edit]

I'd argue that the missing references problem is particularly problematic here. In particular, personally, I would define an L^p Fourier multiplier to be the symbol itself of an operator that is necessarily L^p bounded (see, for example, Loukas Grafakos (2008) Classical Fourier Analysis (Second Edition). Springer, ISBN 9780387094311). I would call the operator itself a Fourier multiplier operator. However, this is at odds with the definition given here, which describes the "multiplier" as the operator itself and doesn't require any boundedness. It's conceivable that some mathematicians would use this definition, but without a solid reference it just feels contradictory to those of us who don't at the moment. In fact, I discovered this as I came to cross-reference to this article, only to find that the definition given didn't match the one I was trying to convey, requiring me to reword the article I was writing. Tcnuk (talk) 15:45, 21 August 2009 (UTC)

(Just to clarify, the boundedness bit doesn't concern me too much, as I am familiar with texts that just talk about "Fourier multiplier operators" without requiring any particular boundedness (e.g. Torchinsky's book on Harmonic analysis). What I haven't seen anywhere other than here is the use of "multiplier" to refer to the actual operator. Tcnuk (talk) 12:15, 28 August 2009 (UTC))
I tried to find references that backed up the definition given, but couldn't. For lack of anything else forthcoming, I've adjusted the article to correspond to the viewpoint of the references which I've now been able to provide. I've also moved the article to a title which is now more appropriate. Tcnuk (talk) 18:09, 1 September 2009 (UTC)

I agree with your changes that m is the multiplier and Tm the multiplier operator. I think we are in agreement also that boundedness is not really essential. One is usually interested in Lp multipliers, but also multipliers between other Sobolev spaces are sometimes considered in the literature, and they need not be bounded. Sławomir Biały (talk) 17:10, 4 September 2009 (UTC)

Yeah, absolutely in agreement. I got a little carried away in my first comment as in informal discussion I tend to contract "L^p Fourier multiplier" to "Fourier multiplier", and talk about establishing that something "is a Fourier multiplier", by which I mean "is an L^p Fourier multiplier" for some particular p. When I thought about what I'd written further, I consulted Torchinsky and added the italicised comments. Anyway, I'm glad to have somebody else's opinion on this. :-) Tcnuk (talk) 17:41, 4 September 2009 (UTC)
I disagree with the terminology. These operators it is far from the usual terminology. I am certainly familiar with the use of the term multiplier to apply to the operator. For the moment I restored only the title, but will check the references before I change the definition. Thenub314 (talk) 18:00, 4 September 2009 (UTC)

Hmm, well I agree that it seems likely that some people do use "multiplier" to refer to the operator, but I think we should try and reach some kind of consensus before everything is changed right back. My original hope was that somebody could find a reference to support this definition and add the comment to the current article along the lines of "also sometimes known as...". To support your viewpoint "it is far from the usual terminology", I think you'd need to find a significant number of references that support your proposed definition, since the ones I've cited do support the one that's currently in place. Since you have raised the issue now, if I get time on Monday, I will attempt to consult a few more references to see if I can establish more strength for either side of the argument, since I'm more than happy to accept either terminology. For the timebeing, however, based on the citations, the evidence supports the article as it currently is.

I've added the "consensus needed" tag the article in the hope of promoting discussion. :-) Tcnuk (talk) 09:16, 5 September 2009 (UTC)

While I agree with Charles below that the terminological issue doesn't seem very critical, I do think that the operator should be called the multiplier operator and m the multiplier. Stein seems to prefer this convention in his books. While it is standard practice to conflate the two in practice (Duoandikotxea does this somewhat apologetically), I think the article marginally benefits from having a clear delineation between the two things. As for the title of the article, I don't have a strong preference, but I think that Multiplier (Fourier analysis) is marginally better because the article covers both the multipliers and the operators. Sławomir Biały (talk) 12:26, 5 September 2009 (UTC)
I meant to delete the sentence I struck through, and accidentally left a segment of it which doesn't really make any sense. Let me explain that I mostly disagreed with the term "Fourier multiplier" and "Fourier multiplier operator". It was the use of the term Fourier that I felt was not standard in the literature. I can certainly agree that using the terms "multiplier operator" for the operator, and "multiplier" for the symbol in this article. Though it would be nice to have a sentence that explains that the terms "multiplier operator" is sometimes shortened to multiplier. (For example this most often happens in titles of papers, but is also done occasionally in the references given.) Thenub314 (talk) 09:22, 6 September 2009 (UTC)

Ah, I understand now. Well, my own estimate would probably be that the word "Fourier" is most commonly dropped in research papers and other expert sources (I'd guess initially for brevity, later dropping into mainstream usage), but that introductory texts usually leave it in. This being said, I don't have my references here to double-check what they each do. I also feel that it gives more clarity to the article leaving the term "Fourier" in, as the subject matter is more apparent at a glance to somebody who is new to the concept. I don't have any strong feelings, though.

I guess it's obvious that I feel that the other title is more appropriate as the article's main name (else I wouldn't have moved it!), but I don't feel strongly about this either.

Finally, as I indicated above, I think a sentence explaining that "multiplier operator" is sometimes shortened to "multiplier" would be great if it can be backed up by an in-line reference. If you have a good one, I'd go ahead and put it in. :-) Tcnuk (talk) 11:13, 6 September 2009 (UTC)

Like you I am away from my references, but maybe tomorrow. It sounds from the conversation like Douandikotxea does it with calls the operators multipliers with comment, and Stein does it without at some point, either way that should be enough for a reference. In defense of the title, Grafakos, Stien, Torchinsky etc. title the section in their books "multipliers" as opposed to something else, but I will check some more basic references as well. Thenub314 (talk) 08:23, 7 September 2009 (UTC)
Fair point about the titles. I didn't completely understand your comment about Duoandikoetxea. Whatever, I haven't located the desired reference for the "sometimes known as" and I don't have time or inclination to look for it now.
I've now glanced at (Little) Stein, Grafakos and Torchinsky (I don't have copies of Duoandikoetxea or Katznelson to hand at the moment) and here's what I've found out (obviously, from what I've said before about finding references, when giving formal definitions, none of them use "multiplier" for the operator). Torchinsky, in section V.4 ("Multipliers") does not seem to insert the word "Fourier" at all, just using "multiplier" and "multiplier operator" throughout. Grafakos (N.B. second edition), in section 2.5.5 ("The Space of Fourier Multipliers \mathcal{M}_p(\mathbb{R}^n)") introduces only Lp Fourier multipliers, saying "Elements of the space \mathcal{M}_p are called Lp multipliers or Lp Fourier multipliers". Later in section 3.6 on the toral case ("Multipliers, Transference, and Almost Everywhere Convergence"), he starts out using the word "Fourier", but drops it almost immediately (as soon as section 3.6.1 starts). Finally, Little Stein uses the term "multiplier transformation" in the guide to notation at the start. However, in Section IV.3, he doesn't seem to give a particular name to the operators themselves.
So, make of that what you will! You've convinced me that "multiplier operator" is probably more common than "Fourier multiplier operator". However, I still feel that the latter is better, at least near the beginning of the article, in a general reference like Wikipedia as its subject matter is clearer at a glance and there is less room for people to get confused with Lagrange multipliers etc. (though I realise there is already a comment to this effect). This being said, I'm not going to lose any sleep if the word "Fourier" is dropped! :-) Tcnuk (talk) 16:57, 7 September 2009 (UTC)

State of the article[edit]

I've done a little work on the lead, re-ordering and throwing in an informal comment or two. It doesn't seem too bad to me. The terminological issue doesn't appear really critical: if, for the purist, the "multiplier" (symbol) itself doesn't appear until the second topic sentence, I wonder who if anybody will actually be misled. The functional analysis slant (operator over shaping function) seems justified here, to keep a clean topic definition in the first sentence, and reduce the "so what" business.

I would suggest some work on the rest of the article, starting with an example or two before the definition for general G. For myself I find examples with Fourier series easiest to comprehend. The scope of the concept might be well illustrated by easy examples, one on Fourier series and one on Fourier transforms on the line. Charles Matthews (talk) 09:55, 5 September 2009 (UTC)

If we include examples, I would say we first include differentiation. It is simple to see why this is a multiplier. Then, perhaps, we could follow this up with the Hilbert transform which is certainly less clear but may illustrate some of the importance of studying multipliers. Thenub314 (talk) 09:28, 6 September 2009 (UTC)
Fine, differentiation of Fourier series, Hilbert transform on the line: these seem representative. Charles Matthews (talk) 09:37, 6 September 2009 (UTC)

Agreed. I think these would be excellent examples. Tcnuk (talk) 11:13, 6 September 2009 (UTC)

Thanks for the edits, Thenub314, which I guess closes the "consensus" issue, so I removed the tag that I placed there before. I made a few small corrections for typos, grammar etc. Tcnuk (talk) 16:20, 8 September 2009 (UTC)

Having read through the whole article a bit more carefully I would like to get peoples opinion about making some other not to minor changes. In particular, I would like to try to make the definition section more concrete sticking with \mathbb{R}^n and the circle, and mention later that this can be extended into any setting in which the Fourier transform can be defined. (About this: We require Amenability for some reason. Follow Katznelson we should be able to define the Fourier transform on a locally compact Abelian group. Amenability seems out of place here. Am I missing some point?)

Also the General considerations section needs some work. The ordering of it seems to say that because of the properties of C* algebra's multipliers commute, instead of this following directly from the definition. Also sentences like: "Since every translation-invariant operator is a multiplier operator, we conclude that multiplier operators are translation-invariant." I agree with the conclusion, but not with the reason we give for it, which again follows form the definition. I will try my hand at things when I have some time.Thenub314 (talk) 10:36, 9 September 2009 (UTC)

I agree that the approach article should emphasize Rn, the circle, and the torus. The case of locally compact abelian groups should, I think, be marginalized as much as possible to a separate section. The article should start from the specific and build to the general, not the other way around.
The sentence that you object to is true, though poorly expressed. Of course, the operation of translation is itself a multiplier, and so commutes with every other multiplier. Thus multipliers are translation invariant.
I don't know why amenability is there either. I am unable to find any source that requires this for multipliers. However, various authors use various different related definitions. One can define multipliers, for instance, on compact non-abelian groups, or multipliers for operators on functions with values in a Hilbert space. Presumably there are special applications that require amenability. The larger problem is that there presently no sources in the article for multipliers on groups, and so we are having difficulty verifying the material. Sławomir Biały (talk) 13:33, 9 September 2009 (UTC)
I'd definitely prefer the article to be written from the Euclidean perspective, so I'd certainly support that. I'm no expert in abstract harmonic analysis, but I've never seen the amenability condition anywhere but here, either.
I don't have issue with the C* algebra section. The commutativity can be seen as following from the homomorphism property, not the C* algebra properties (i.e. the homomorphism gives that things commute on the operator side because they commute on the multiplier side). Feel free to reword if you want to make it clearer, though. Tcnuk (talk) 09:28, 10 September 2009 (UTC)