Consider the vector space of continuous real valued functions on a finite interval and the inner product defined by the integral over the interval . If we have a family of orthogonal polynomials such that their span is dense then each polynomial has exactly n distinct roots . I was wondering if these roots might be dense in the interval because i tried to think of these polynomials as interpolation polynomials.
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$\begingroup$ Are you assuming these polynomials form a dense basis? Why are you assuming the roots are distinct? $\endgroup$– Owen SizemoreFeb 14, 2017 at 17:59
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$\begingroup$ Related: math.stackexchange.com/questions/12160/… $\endgroup$– Jack D'AurizioFeb 14, 2017 at 18:10
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$\begingroup$ Sorry i forgot to mention that my bad , yes i am assuming the span of this set is dense ( the term in french is "Famille totale" i can't find the english equivalent ) $\endgroup$– Mac SatFeb 14, 2017 at 19:28
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1 Answer
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The answer appears to be "Yes". I have found a proof but it is too long for me to write here . The proof is an admission exams to the french "École Polytechnique" here is a link to the exam X-ENS PSI 2006 .I will try to write the proof in english in my spare time and add it here as soon as possible .