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I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$. Given any $x,y \in M$, $$d(x,y) = ||x-y||_{L^2} + |V_0^1(x) - V_0^1(y)|$$

Fourier series is not convergent in this metric! But if I use this (not proven yet!) concept, which is a slightly modified form of Fourier expansion, the resulting series converges in the above metric.

My question is, is this metric been studied anywhere in the literature, if so, what are the interesting properties/aspects of it. My hope is that, if this metric is interesting, then it would serve as a good motivation for validating this concept.

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  • $\begingroup$ Since the variation of a constant is zero, the variation itself is not a norm. But you get a norm by adding something to take care of the one extra dimension. You have chosen $\|f\|_2$, I have not seen it done this way. I have seen $\|f\|_1$, I have seen $\|f\|_\infty$, and I have seen $|f(0)|$. But of course all of these norms are "equivalent" (differ by at most a constant factor from each other). Note bounded variation implies square-integrable. $\endgroup$
    – Gerald Edgar
    Commented Nov 17, 2015 at 16:04
  • $\begingroup$ @GeraldEdgar : What I have described is a metric, not a norm! My intention is to define a metric, not a norm. you might have slipped into that assumption after seeing the first term, if i got it correctly. $\endgroup$
    – Rajesh D
    Commented Nov 17, 2015 at 16:39
  • $\begingroup$ agree on BV implies square integrability. $\endgroup$
    – Rajesh D
    Commented Nov 17, 2015 at 16:40
  • $\begingroup$ $d(x,y)$ has the form $\|x-y\|$ for a norm $\|f\| = \|f\|_2+V_0^1(f)$.. $\endgroup$
    – Gerald Edgar
    Commented Nov 17, 2015 at 16:44
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    $\begingroup$ I believe what you're interested in, but for $||x-y||_{L^1}$ instead of $||x-y||_{L^2},$ can be found near the bottom of p. 422 of The space of functions of bounded variation and certain general spaces by Clarence Raymond Adams [Trans. AMS 40 (1936), 421-438]. See also Abstract #1 on p. 19 here (1937) and Abstract #1 on p. 27 here (1940). $\endgroup$
    – Dave L Renfro
    Commented Nov 17, 2015 at 22:01

1 Answer 1

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This is closely related to the so-called metric of strict convergence which is $$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$ where $TV(u)$ denoted the total variation of $u$. This is indeed a metric on the space $BV(\Omega)$ (also for $\Omega\subset\mathbb{R}^n$). Hence, strict convergence of $u_n$ to $u$ is nothing else than saying $$ u_n\to u\quad\text{in}\quad L^1,\quad\text{and}\quad TV(u_n)\to\ TV(u). $$

Moreover, it has the nice property of meterizing the so-called weak-$*$-convergence in BV, which is defined as $$ u_n\to u\quad\text{in}\quad L^1,\quad\text{and}\quad \int u_n\, \mathrm{div}\phi \to \int u\,\mathrm{div}\phi\quad\forall \phi\in C^1_0(\Omega). $$ This is described, for example, in

Ambrosio, Fusco, Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000

and googling for something that contains "strict convergence" and something like "bounded variation" or "total variation" gives a lot of hits.

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  • $\begingroup$ Hi Dirk, thanks for the answer and reference. This was pioneered by C Raymond Adams in his 1936 paper as mentioned by DaveLRenfro in the comments. I request you to add this to the answer. CA Raymonds also wrote a series of papers on this, where there is some general form for a linear functional on space $E$. $\endgroup$
    – Rajesh D
    Commented Nov 19, 2015 at 13:59
  • $\begingroup$ ams.org/journals/tran/1940-048-01/S0002-9947-1940-0002017-4/… $\endgroup$
    – Rajesh D
    Commented Nov 19, 2015 at 14:07
  • $\begingroup$ This is 1940 paper which came later. $\endgroup$
    – Rajesh D
    Commented Nov 19, 2015 at 14:08
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    $\begingroup$ If the one-dimensional case is enough for you, you may well go with the Adams et al papers from the 30s. Nowadays there is well developed theory of BV spaces in any dimensions I personally like to read to newer more polished literature on the topic. You can, however, edit the post (which will then go the review cue). (On a different matter: I usually do not follow links that go directly to some pdf but would prefer a link to a site which states title/abstract and to on so that I can decide whether I not I would like to read the article.) $\endgroup$
    – Dirk
    Commented Nov 19, 2015 at 14:11
  • $\begingroup$ $$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$ What is more interesting is that the space BV(0,1), under this metric, is separable, uniformly bounded, compact and complete! $\endgroup$
    – Rajesh D
    Commented Dec 1, 2015 at 12:40

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