- 作者：Szilá ; rd Gy. Ré ; vé ; sz
- 关键词：Bernstein– ; Markov inequalities ; Turá ; n's lower estimate of derivative norm ; Logarithmic derivative ; Chebyshev constant ; Convex domains ; Width of a convex domain
- 刊名：Journal of Approximation Theory
- 出版年：2006
- 期刊代码：71_00219045
- 类别：mt
- 出版时间：August 2006
- 卷：141
- 期：2
- 页码：162-173
- 文件大小：197 K
The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán, who showed the lower bounds ve&_udi=B6WH7-4JYKP8N-3&_mathId=mml5&_user=10&_cdi=6843&_rdoc=7&_handle=V-WA-A-W-AW-MsSAYZW-UUW-U-AAZCBUUECU-AAZWYYADCU-VBBUZDUA-AW-U&_acct=C000050221&_version=1&_userid=10&md5=f51096c14ca70bc411a9bbbc6717f75b" title="Click to view the MathML source">p′(n/2)p for the unit disk D and for the unit interval ve&_udi=B6WH7-4JYKP8N-3&_mathId=mml7&_user=10&_cdi=6843&_rdoc=7&_handle=V-WA-A-W-AW-MsSAYZW-UUW-U-AAZCBUUECU-AAZWYYADCU-VBBUZDUA-AW-U&_acct=C000050221&_version=1&_userid=10&md5=d0461613f99690442332e74067dd80cf" title="Click to view the MathML source">I[-1,1]. Although partial results provided general lower estimates of order , as well as certain classes of domains with lower bounds of order n, it was not clear what order of magnitude the general convex domains may admit here. Here we show that for all bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K ve&_udi=B6WH7-4JYKP8N-3&_mathId=mml9&_user=10&_cdi=6843&_rdoc=7&_handle=V-WA-A-W-AW-MsSAYZW-UUW-U-AAZCBUUECU-AAZWYYADCU-VBBUZDUA-AW-U&_acct=C000050221&_version=1&_userid=10&md5=4e1414c9b421a93d01dcee016d26be00" title="Click to view the MathML source">p′c(K)np holds true, while ve&_udi=B6WH7-4JYKP8N-3&_mathId=mml10&_user=10&_cdi=6843&_rdoc=7&_handle=V-WA-A-W-AW-MsSAYZW-UUW-U-AAZCBUUECU-AAZWYYADCU-VBBUZDUA-AW-U&_acct=C000050221&_version=1&_userid=10&md5=01fbc399ce04cf8201598cee4446dffe" title="Click to view the MathML source">p′C(K)np occurs for any K. Actually, we determine ve&_udi=B6WH7-4JYKP8N-3&_mathId=mml11&_user=10&_cdi=6843&_rdoc=7&_handle=V-WA-A-W-AW-MsSAYZW-UUW-U-AAZCBUUECU-AAZWYYADCU-VBBUZDUA-AW-U&_acct=C000050221&_version=1&_userid=10&md5=2fd39b39f62d7dbebd55a7ed1f8ee8b4" title="Click to view the MathML source">c(K) and ve&_udi=B6WH7-4JYKP8N-3&_mathId=mml12&_user=10&_cdi=6843&_rdoc=7&_handle=V-WA-A-W-AW-MsSAYZW-UUW-U-AAZCBUUECU-AAZWYYADCU-VBBUZDUA-AW-U&_acct=C000050221&_version=1&_userid=10&md5=514b677351b3f50f58b6e43e1ee416e9" title="Click to view the MathML source">C(K) within a factor of absolute numerical constant.