Timeline for Can a sequence of polynomials in a translation invariant linear space pointwise converge to a polynomial which is not included in that space? In one variable this is impossible.
Current License: CC BY-SA 3.0
10 events
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| Oct 1, 2011 at 17:27 | comment | added | fedja | a) You can approximate $x$ by linear combinations of $x^2,x^3,\dots$ uniformly on arbitrarily large intervals, so, no, pointwise convergence does not imply the coefficientwise one. b) It is a nice problem, indeed, but I made it a rule not to answer to unknown(yahoo)'s. If the OP creates a unique username, I'll think of the question and vote it up. | |
| Jul 25, 2011 at 21:24 | comment | added | Pietro Majer | This question seems to me really interesting. I don't understand why it got one vote only. Would it meet with more success if the OP re-edited it and add some more details and motivation? | |
| Jul 21, 2011 at 3:04 | comment | added | user16456 | Coefficients are linear combinations of derivatives, derivatives are linear combinations of differences, differences are linear combinations of function values - hence point-wise convergence does imply coefficient-wise convergence. | |
| Jul 20, 2011 at 21:13 | comment | added | fedja | Erm, the pointwise convergence doesn't imply the coefficient-wise one either. | |
| Jul 19, 2011 at 3:27 | comment | added | user16456 | (Pietro Majer) Yeah, sorry. | |
| Jul 18, 2011 at 12:04 | comment | added | Pietro Majer | (precisely, point-wise convergence implies coefficient-wise convergence, but not the opposite, as the example $x^n$ shows). | |
| Jul 16, 2011 at 12:43 | comment | added | fedja | The first claim is true only if the degrees are uniformly bounded. | |
| Jul 16, 2011 at 6:37 | comment | added | user16456 | A sequence of polynomials converges point-wise to a polynomial if and only if it converges coefficient-wise to it. A linear space of polynomials is translation invariant if and only if it is invariant under partial differentiation. | |
| Jul 14, 2011 at 20:21 | history | edited | user16456 | CC BY-SA 3.0 |
added 28 characters in body
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| Jul 14, 2011 at 17:12 | history | asked | user16456 | CC BY-SA 3.0 |