By Antoine Derighetti (auth.)

This quantity is dedicated to a scientific examine of the Banach algebra of the convolution operators of a in the neighborhood compact team. encouraged by means of classical Fourier research we give some thought to operators on Lp areas, arriving at an outline of those operators and Lp models of the theorems of Wiener and Kaplansky-Helson.

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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.