Hello! I would like to prove that the space of add functions:C^(\{infinity})_{odd} [0,1]):= \{ f: [0,1] -> R s.t f^(2k)(0)=0, for all k=0,1,2,3,... \} (where f^(2k) stands for the (2k)th derivative of f) is dense in the radial Sobolev space:H^(m)_{rad} (B^(d)(1)) := \{ u: (0,1) -> R s.t there exists a radial function v \in H^(d)(B^(d)(1)) with u(x)=v(|x|) and ||v||_(H^(d)(B^(d)(1))) < \infinity \}.Thank you so much in advance for your help.
Density in the radial Sobolev space
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The broken formatting (LaTeX not displaying, grouping symbols unmatched, etc) makes your meaning difficult to discern:
\(\displaystyle \mbox{I would like to prove that the space of additive functions:}\)
. . .\(\displaystyle C_{odd}^{\infty}\, [0,\, 1]\, =\, \left\{\, f\, :\, [0,\,1]\, \rightarrow\, \mathbb{R}\, \bigg| \, f^{2k}(0)\, =\, 0\, \forall\, k\, =\, 0,\, 1,\, 2,\, 3,\,...\,\right\}\)
\(\displaystyle \mbox{where }\, f^{2k}\, \mbox{ stands for the }\, 2k-\mbox{th derivative of }\, f,\, \mbox{ is dense in the radial Sobolev space:}\)
. . .\(\displaystyle H_{rad}^m\, (B^d(1))\, =\, \left\{\, u\, :\, (0,\, 1)\, \rightarrow\, \mathbb{R}\, \right\}\)
\(\displaystyle \mbox{such that there exists a radial function:}\)
. . .\(\displaystyle v\, \in\, H^{d}\left(B^d(1)\right)\)
\(\displaystyle \mbox{with:}\)
. . .\(\displaystyle u(x)\, =\, v\left(\vert x \vert \right)\, \mbox{ and }\, \Vert v \Vert_{H^d\left(B^d(1)\right)}\, <\, \infty\)
Please reply with corrections and clarifications. Also, please reply showing your thoughts and efforts so far. Thank you!
The following is my guess as to your meaning:
\(\displaystyle \mbox{I would like to prove that the space of additive functions:}\)
. . .\(\displaystyle C_{odd}^{\infty}\, [0,\, 1]\, =\, \left\{\, f\, :\, [0,\,1]\, \rightarrow\, \mathbb{R}\, \bigg| \, f^{2k}(0)\, =\, 0\, \forall\, k\, =\, 0,\, 1,\, 2,\, 3,\,...\,\right\}\)
\(\displaystyle \mbox{where }\, f^{2k}\, \mbox{ stands for the }\, 2k-\mbox{th derivative of }\, f,\, \mbox{ is dense in the radial Sobolev space:}\)
. . .\(\displaystyle H_{rad}^m\, (B^d(1))\, =\, \left\{\, u\, :\, (0,\, 1)\, \rightarrow\, \mathbb{R}\, \right\}\)
\(\displaystyle \mbox{such that there exists a radial function:}\)
. . .\(\displaystyle v\, \in\, H^{d}\left(B^d(1)\right)\)
\(\displaystyle \mbox{with:}\)
. . .\(\displaystyle u(x)\, =\, v\left(\vert x \vert \right)\, \mbox{ and }\, \Vert v \Vert_{H^d\left(B^d(1)\right)}\, <\, \infty\)
Please reply with corrections and clarifications. Also, please reply showing your thoughts and efforts so far. Thank you!