کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4607384 | 1337853 | 2012 | 15 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: On the exact L2L2 Markov inequality on some unbounded domains in RdRd On the exact L2L2 Markov inequality on some unbounded domains in RdRd](/preview/png/4607384.png)
We consider the Markov problem of finding the so-called Markov factor M(U,K):=supu∈U‖Du‖‖u‖, of the set of differentiable functions UU, where Du:=|∂u|ℓ2Du:=|∂u|ℓ2 stands for the ℓ2ℓ2-norm of the gradient vector of uu, and ‖⋅‖‖⋅‖ is the weighted L2L2 norm on the set K⊂RdK⊂Rd. In the univariate case exact L2L2-Markov inequalities are known for algebraic polynomials on the real line, half line and intervals. We outline a variational approach to the above problem and show how this leads either to certain partial differential equations, or to a system of homogeneous linear equations. This method will be illustrated by using it to solve the L2L2 Markov problem for the cases of dd-dimensional spaces and dd-dimensional hyperquadrants. In the case of dd-spaces the solution is given for homogeneous polynomials, as well.
Journal: Journal of Approximation Theory - Volume 164, Issue 3, March 2012, Pages 391–405