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E iÂ1 ; : : : ; e iÂn / D e i m1 Â1 C Ci mn Ân . Then n 1 n cn >0 m 2 T . Let K be a compact neighborhood of 0 in R . m/ m : s f D f m2Zn \ K Example. Œ 1;1/ sN f is the Fourier sum of f . The following theorem (see [111], p. Rn /. Theorem 5. Let K be a compact convex neighborhood of 0 in Rn and 1 < p < 1. The following statements are equivalent. T n /, b D Œ1K . T / and for every 1 < p < 1. Consequently for every interval I of R and every b D Œ1I . Rn / with b T D Œ1C . But for D the unit ball in Rn (n > 1) and for b D Œ1D .

Dixmier, Chap. I, Sect. 3, no. 4, Corollaire 1, p. 42. The next result is Kaplansky’s density theorem. Theorem 2. H/ with B C. H/. S˛ / of B such that: 1: lim˛ S˛ D T strongly, 2: kS˛ k Ä kT k for every ˛. Proof. See Dixmier, [36], Chap. I, Sect. 3, no. 5, Th´eor`eme 3, p. 43–44. Let G be a locally compact group. In this paragraph, we denote by A the set of all /, where is a complex measure with finite support. G// with unit: 2G . / D 2G . G/ . The C following statement is straightforward. 2 G.

G/: Remark. We will extend this result to p 6D 2 for certain classes of locally compact groups. f˛ /. Chapter 3 The Figa–Talamanca Herz Algebra Let G be a locally compact group. G/, is a Banach algebra for the b pointwise product on G. G/. G / 0 Let G be a locally compact group and 1 < p < 1. G/. ıx /Œ p l; Œ p k : Definition 1. Let G be a locally compact group and 1 < p < 1. ln / < 1. G/. G/. nD1 A. 1007/978-3-642-20656-6 3, © Springer-Verlag Berlin Heidelberg 2011 33 34 3 The Figa–Talamanca Herz Algebra Definition 2.