By Berge C.

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There are many possible avenues for the exploration of the Gibbs phenomenon. For example Weyl [6], [7] studied the behavior of the partial sums of spherical harmonic expansions of functions defined on the sphere and having a jump discontinuity along a smooth curve on the sphere. See Colzani and Vignati [2] for an investigation along the same lines in the setting of multiple Fourier integrals. One can also consider other summability methods such as Bochner–Riesz summability. This is the theme of Golubov’s work [3] and the 4 George Benke work of Cheng [1].

Since we assume G to be σ-compact and H to be discrete, H is countable. So → − if f is only defined pointwise almost everywhere, the same is still true for f . , following. 20. For p1 , p2 ∈ [1, ∞], we define Lp1 ,p2 (H, G/H) to be the Banach space of measurable functions on G with norm f Lp1 ,p2 (H,G/H) = → − f (g ) ℓp1 (H) Lp2 (G/H) f (g+h) = G/H p1 1/p2 p2 /p1 dµG/H (g ) , h∈H with the usual convention for pi = ∞. → − 2 In other words, f ∈ Lp1 ,p2 (H, G/H) if and only if f ∈ Lploc (G, ℓp1 (H)).

Weak*, weak)-continuous operators only. (ii) Assume that G is nondiscrete. Then we have for 1 ≤ p1 < p2 ≤ ∞ that M·H ⊥ Lp1 (G/H), Lp2 (G) = {0}. Proof. (i) Let T ∈ M·H ⊥ Lp1 (G/H), Lp2 (G) and set ϕ = T (1). By assumption, we have T (e2πi ηg ) = T (Mη 1) = Mη T (1) = e2πiηg ϕ for all η ∈ H ⊥ . Since trigonometric polynomials are dense in Lp1 (G/H), we have T = Qϕ .