Extremal problems for Lorentz classes of nonnegative polynomials in 𝐿² metric with Jacobi weight

Milovanović, Gradimir V.; Petković, Miodrag S.

doi:10.1090/S0002-9939-1988-0920987-2

Extremal problems for Lorentz classes of nonnegative polynomials in $L^2$ metric with Jacobi weight
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by Gradimir V. Milovanović and Miodrag S. Petković

Proc. Amer. Math. Soc. 102 (1988), 283-289

DOI: https://doi.org/10.1090/S0002-9939-1988-0920987-2

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Abstract:

Let ${L_n}$ be the Lorentz class of nonnegative polynomials on $[-1,1]$. Extremal problems of Markov type, in ${L^2}$ norm with Jacobi weight, on the set ${L_n}$ or on its subset, are investigated.

References

P. Erdös, On extremal properties of the derivatives of polynomials, Ann. of Math. (2) 41 (1940), 310–313. MR 1945, DOI 10.2307/1969005
P. Erdős and A. K. Varma, An extremum problem concerning algebraic polynomials, Acta Math. Hungar. 47 (1986), no. 1-2, 137–143. MR 836404, DOI 10.1007/BF01949134
G. G. Lorentz, The degree of approximation by polynomials with positive coefficients, Math. Ann. 151 (1963), 239–251. MR 155135, DOI 10.1007/BF01398235
G. G. Lorentz, Derivatives of polynomials with positive coefficients, J. Approximation Theory 1 (1968), 1–4. MR 231957, DOI 10.1016/0021-9045(68)90052-x

On a problem of D. I. Mendeleev

62

Gradimir V. Milovanović, An extremal problem for polynomials with nonnegative coefficients, Proc. Amer. Math. Soc. 94 (1985), no. 3, 423–426. MR 787886, DOI 10.1090/S0002-9939-1985-0787886-8
John T. Scheick, Inequalities for derivatives of polynomials of special type, J. Approximation Theory 6 (1972), 354–358. MR 342909, DOI 10.1016/0021-9045(72)90041-x
Gábor Szegö, On some problems of approximations, Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 3–9 (English, with Russian summary). MR 173895
A. K. Varma, An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside $[-1,+1]$, Proc. Amer. Math. Soc. 55 (1976), no. 2, 305–309. MR 396878, DOI 10.1090/S0002-9939-1976-0396878-7
A. K. Varma, An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside $[-1,+{}1]$. II, Proc. Amer. Math. Soc. 69 (1978), no. 1, 25–33. MR 473124, DOI 10.1090/S0002-9939-1978-0473124-9
A. K. Varma, Some inequalities of algebraic polynomials having real zeros, Proc. Amer. Math. Soc. 75 (1979), no. 2, 243–250. MR 532144, DOI 10.1090/S0002-9939-1979-0532144-7
A. K. Varma, Derivatives of polynomials with positive coefficients, Proc. Amer. Math. Soc. 83 (1981), no. 1, 107–112. MR 619993, DOI 10.1090/S0002-9939-1981-0619993-0
A. K. Varma, Some inequalities of algebraic polynomials having all zeros inside $[-1,\,1]$, Proc. Amer. Math. Soc. 88 (1983), no. 2, 227–233. MR 695248, DOI 10.1090/S0002-9939-1983-0695248-5