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  • 1
    ISSN: 1432-0940
    Keywords: 41A17 ; Markov- ; Bernstein- ; Nikolskii- ; Remez- ; Schur-type inequalities inL p for generalized nonnegative polynomials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We define generalized polynomials as products of polynomials raised to positive real powers. The generalized degree can be defined in a natural way. We prove Markov-, Bernstein-, and Remez-type inequalities inL p (0〈p〈∞) and Nikolskii-type inequalities for such generalized polynomials. Our results extend the corresponding inequalities for ordinary polynomials.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0940
    Keywords: Orthogonal polynomials ; Szegö's theory ; 42C05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let {ϕ n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofϕ n (dμ) when logμ'∈L 1. In what follows we will discuss the asymptotic behavior of the ratio φn(dμ 1)/φn(dμ 2) off the unit circle in casedμ 1 anddμ 2 are close in a sense (e.g.,dμ 2=g dμ 1 whereg≥0 is such thatQ(e it )g(t) andQ(e it )/g(t) are bounded for a suitable polynomialQ) and μ 1 ′ 〉0 almost everywhere or (a somewhat weaker requirement) lim n→∞Φ n (dμ 1,0)=0, for the monic polynomials Φ n . The consequences for orthogonal polynomials on the real line are also discussed.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-0940
    Keywords: 42C05 ; Orthogonal polynomials ; Szegö's Theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Consider a system {φ n } of polynomials orthonormal on the unit circle with respect to a measuredμ, withμ′〉0 almost everywhere. Denoting byk n the leading coefficient ofφ n , a simple new proof is given for E. A. Rakhmanov's important result that lim n→∞,k n /k n+1=1; this result plays a crucial role in extending Szegö's theory about polynomials orthogonal with respect to measuresdμ with logμ′∈L 1 to a wider class of orthogonal polynomials.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1432-0940
    Keywords: 42 C 05 ; Orthogonal polynomials ; Asymptotics ; Difference equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Asymptotic expansions are given for orthogonal polynomials when the coefficients in the three-term recursion formula generating the orthogonal polynomials form sequences of bounded variation.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Constructive approximation 2 (1986), S. 113-127 
    ISSN: 1432-0940
    Keywords: 41A17 ; 41A10 ; 26C05 ; Markov-Bernstein inequalities ; Polynomials ; Completeness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For the weights exp (−|x|λ), 0〈λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0〈λ〈1. The proof is based on the solution of the problem of how fast a polynomialP n can decrease on [−1,1] ifP n (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedL p spaces.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1432-0940
    Keywords: Orthogonal polynomials ; Szegö's theory ; 42C05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let {ø n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø n (dμ) when logμ′∈L 1. In what follows we will discuss the asymptotic behavior of the ratioø n (dμ 2)/ø n (dμ 1) on the unit circle whendμ 1 anddμ 2 are close in a sense (e.g.,dμ 2=gdμ 1, where g≥0 is such thatQ(e it )g(t) andQ(e it )/g(t) are bounded for a suitable polynomialQ) and μ 1 ′ 〉0 almost everywhere or (a somewhat weaker requirement) lim n→∞Φ n (dμ 1,0)=0 for the monic polynomial Φ n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.
    Type of Medium: Electronic Resource
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