On the Boundedness and Unboundedness of Polynomials
نویسنده
چکیده
Throughout this paper P,(x) will denote a polymonial of degree n ; c, c 1 c 2 ," . will denote positive absolute constants not necessarily the same if they occur at different places . Tn other words : for no choice of the points X 1 ,"',Xn can we deduce from the boundedness of I Pn _ 1(x) I, 1 <_ 15 n the boundedness of I P,,,(x) I in the whole interval ( 1, + 1) . Bernstein [1] asked himself the question whether one can deduce the boundedness of I P,(x) I in ( 1, + 1) if we know that I P„(x) I < 1 for in > (1 + c)n values of x. His answer was affirmative . In fact he showed that if I P„(x("'~) I < 1 for all roots of the in h Tchebicheff polynomial T,,(x) where m > n(1 + c), then
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