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Nonexistence of Chebyshev-type quadratures on infinite intervals

Author: Walter Gautschi
Journal: Math. Comp. 29 (1975), 93-99
MSC: Primary 65D30
MathSciNet review: 0368392
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Abstract: Quadrature rules on semi-infinite and infinite intervals are considered involving weight functions of the Laguerre and Hermite type. It is shown that such quadrature rules cannot have equal coefficients and real nodes unless the algebraic degree of accuracy is severely limited.

References [Enhancements On Off] (What's this?)

  • [1] L. A. ANDERSON & W. GAUTSCHI, "Optimal weighted Chebyshev-type quadrature formulas" (To be published.)
  • [2] S. BERNSTEIN, "Sur un système d'équations indéterminées," J. Math. Pures Appl., v. 17, 1938, pp. 179-186.
  • [3] Luigi Gatteschi, Sulla non esistenza di certe formule di quadratura, Univ. e Politec. Torino Rend. Sem. Mat. 24 (1964/1965), 157–172 (Italian). MR 0187394
  • [4] V. I. Krylov, Mechanical quadratures with equal coefficients for the integrals ∫^{∞}₀ 𝑒^{-𝑥}𝑓(𝑥)𝑑𝑥 and ∫_{-∞}𝑒^{-𝑥²}𝑓(𝑥)𝑑𝑥, Dokl. Akad. Nauk BSSR 2 (1955), 187–192 (Russian). MR 0109978
  • [5] Herbert E. Salzer, Equally weighted quadrature formulas over semi-infinite and infinite intervals, J. Math. and Phys. 34 (1955), 54–63. MR 0069586,
  • [6] Herbert S. Wilf, The possibility of Tschebycheff quadrature on infinite intervals, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 209–213. MR 0125380

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Article copyright: © Copyright 1975 American Mathematical Society

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