Uniform Norm of Polynomials Constrained with Many Zeros in Two Disjoint Intervals

Matthew He ()
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Matthew He: Nova University, Department of Mathematics

A chapter in Approximation, Probability, and Related Fields, 1994, pp 273-281 from Springer

Abstract: Abstract For nonnegative integers (s 1, s 2, m), consider the set of all real or complex polynomials of the form $$ P_n \left( x \right) = \prod\limits_{i = 1}^{s_1 } {\left( {x - x_i^{\left( {s_1 } \right)} } \right)\prod\limits_{j = 1}^{s_2 } {\left( {x - x_j^{\left( {s_2 } \right)} } \right)\sum\limits_{k = 0}^m {\alpha _k x^k } ,} } $$ where $${{s}_{1}} \geqslant {{\theta }_{1}}\left( {{{s}_{1}} + {{s}_{2}} + m} \right) = {{\theta }_{1}}n > 0$$ and $${{s}_{2}} \geqslant {{\theta }_{2}}\left( {{{s}_{1}} + {{s}_{2}} + m} \right) = {{\theta }_{2}}n > 0$$ , $${{\theta }_{1}} > 0,{{\theta }_{2}} > 0$$ and having s 1, s 2 zeros constrained so that $$x_{i}^{{\left( {{{s}_{1}}} \right)}} \in \left[ {d,1} \right],i = 1,2, \ldots ,{{s}_{1}}$$ , $$x_{j}^{{\left( {{{s}_{2}}} \right)}} \in \left[ { - 1,c} \right],j = 1,2, \ldots ,{{s}_{2}}$$ , respectively. For all $${{P}_{n}}{{\left( x \right)}^{\prime }}s$$ bounded on [-1,1], we found the smallest interval $$\left[ {a,b} \right] \subseteq \left[ { - 1,1} \right]$$ such that $$\parallel {{P}_{n}}{{\parallel }_{{\left[ { - 1,1} \right]}}} = \parallel {{P}_{n}}{{\parallel }_{{\left[ {a,b} \right]}}}.$$

Keywords: 30C15; 41A10; Uniform norm; incomplete polynomials; minimal set (search for similar items in EconPapers)
Date: 1994
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4615-2494-6_20

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DOI: 10.1007/978-1-4615-2494-6_20

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