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On the Lebesgue Constants

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We present the solution of a classical problem of approximation theory about the sharp asymptotics of Lebesgue constants or the norms of Fourier–Laplace projections on the real sphere \( {\mathbbm{S}}^d \), in complex Pd(ℂ) and quaternionic Pd(ℍ) projective spaces, and in the Cayley elliptic plane P16(Cay). In particular, these results supplement the sharp asymptotics established by Fejer (1910) in the case of \( {\mathbbm{S}}^1 \) and by Gronwall (1914) in the case of \( {\mathbbm{S}}^2 \).

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References

  1. 1.

    B. Bordin, A. Kushpel, J. Levesley, and S. Tozoni, “Estimates of n-widths of Sobolev’s classes on compact globally symmetric spaces of rank 1,” J. Funct. Anal., 202, 307–326 (2003).

  2. 2.

    E. Cartan, “Sur la determination d’un systeme orthogonal complet dans un espace de Riemann symetrique clos,” Rend. Circ. Mat. Palermo, 53, 217–252 (1929).

  3. 3.

    V. K. Dzyadyk, S. Yu. Dzyadyk, and A. S. Prypik, “Asymptotic behavior of Lebesgue constants in trigonometric interpolation,” Ukr. Mat. Zh., 33, No. 6, 736–744 (1981); English translation:Ukr. Math. J., 33, No. 6, 553–559 (1981).

  4. 4.

    A. Erdélyi (editor), Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1953).

  5. 5.

    L. Fejer, “Lebesguesche Konstanten und divergente Fourierreihen,” J. Reine Angew. Math., 138, 22–53 (1910).

  6. 6.

    R. Gangolli, “Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters,” Ann. Inst. H. Poincaré, 3, 121–225 (1967).

  7. 7.

    T. H. Gronwall, “On the degree of convergence of Laplace series,” Trans. Amer. Math. Soc., 15, No. 1, 1–30 (1914).

  8. 8.

    S. Helgason, “The radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds,” Acta Mat., 113, 153–180 (1965).

  9. 9.

    S. Helgason (editor), Differential Geometry and Symmetric Spaces, Academic Press, New York (1962).

  10. 10.

    T. Koornwinder, “The addition formula for Jacobi polynomials and spherical harmonics,” SIAM J. Appl. Math., 25, No. 2, 236–246 (1973).

  11. 11.

    A. K. Kushpel, “Uniform convergence of orthogonal expansions on the real projective spaces,” in: Approximation Theory and Contingent Problems, Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2008), pp. 191–208.

  12. 12.

    G. Szegö (editor), Orthogonal Polynomials, American Mathematical Society, New York (1939).

  13. 13.

    H. C. Wang, “Two-point homogeneous spaces,” Ann. Math., 55, 177–191 (1952).

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Correspondence to A. K. Kushpel.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 8, pp. 1073–1081, August, 2019.

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Kushpel, A.K. On the Lebesgue Constants. Ukr Math J 71, 1224–1233 (2020) doi:10.1007/s11253-019-01709-5

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