Convergence of Fourier series expansion related to free groups

Kuhn, G.

doi:10.1090/S0002-9939-1984-0749884-9

Convergence of Fourier series expansion related to free groups
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Proc. Amer. Math. Soc. 92 (1984), 31-36 Request permission

Abstract:

In $[0,\pi ]$ we consider the complete orthogonal system ${P_n}$ associated to the weight function $\psi = r(2r - 1){\pi ^{ - 1}}{\rm {si}}{{\rm {n}}^2}\theta {({r^2} - (2r - 1){\rm {co}}{{\rm {s}}^2}\theta )^{ - 1}}$ and we study mean and pointwise convergence of series expansions with respect to the system ${P_n}$ in ${L^p}([0,\pi ],d\psi )$. This weight function, and the corresponding system ${P_n}$ arise from the study of Gelfand transforms of radial functions on a finitely generated free group ${F_r}$ and our results can be interpreted in terms of multipliers theory on ${F_r}$.

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