Timeline for Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel

Current License: CC BY-SA 4.0

10 events

when toggle format	what		by	license	comment
Dec 25, 2018 at 22:18	history	edited	Minkov		edited tags
Dec 24, 2018 at 2:13	history	edited	Minkov	CC BY-SA 4.0	deleted 2 characters in body; edited tags
Dec 23, 2018 at 12:08	comment	added	Alex Gavrilov		Even this is not so ``natural'' a choice. What is the big reason to take all of this terms with weight one? (It is not invariant with respect to scaling, for example.) I admit that in this case the question actually has an answer, but I also admit that, personally, I totally would not care about it.
Dec 23, 2018 at 10:12	history	edited	Minkov	CC BY-SA 4.0	edited title
Dec 23, 2018 at 9:58	history	edited	Minkov	CC BY-SA 4.0	edited title
Dec 23, 2018 at 9:43	comment	added	Minkov		@AlexGavrilov I totally agree with you on that the norm and inner product are not unique. We are interested in the "natural" choice of Sobolev norm and inner product $<u, v> = \sum_{i=1}^k \langle D^i u, D^i v \rangle$, especially how to construct the corresponding basis, for example, the eigenvector of the reproducing kernel. Also, for $0 \leq s_1, s_2 < d/2$, is it possible to align the orthonormal bases of $H^{s_1}(\Omega)$ and $H^{s_2}(\Omega)$ so that they only differ up to a rescaling of magnitudes?
Dec 23, 2018 at 9:34	history	edited	Minkov	CC BY-SA 4.0	added 16 characters in body
Dec 23, 2018 at 9:32	comment	added	Alex Gavrilov		I believe that this is not a meaningful question. You see, even though $H^s(\Omega)$ is called a Hilbert space it is technically not, in the sense that the actual Hilbert norm on it is not fixed. It is a sort of infinite dimensional analog of a linear space of finite dimension: you are free to chose any of the (equivalent) scalar products. Naturally, an orthonormal basis would very much depend on the choice.
Dec 23, 2018 at 9:17	history	edited	Minkov	CC BY-SA 4.0	added 10 characters in body
Dec 23, 2018 at 9:08	history	asked	Minkov	CC BY-SA 4.0