The Lebesgue constants on projective spaces


Abstract: We give the solution of a classical problem of Approximation Theory on sharp asymptotic of the Lebesgue constants or norms of the Fourier-Laplace projections on the real projective spaces $\mathrm{P}^{d}(\mathbb{R})$. In particular, these results extend sharp asymptotic found by Fejer [2] in the case of $\mathbb{S}^{1}$ in 1910 and by Gronwall [4] in 1914 in the case of $\mathbb{S}^{2}$. The case of spheres, $\mathbb{S}^{d}$, complex and quaternionic projective spaces, $\mathrm{P}^{d}(\mathbb{C})$, $% \mathrm{P}^{d}(\mathbb{H})$ and the Cayley elliptic plane $\mathrm{P}^{16}(% \mathrm{Cay})$ was considered by Kushpel [8].

Keywords: Lebesgue constant, Fourier-Laplace projection, Jacoby polynomia

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