Real Analytic Version of Lévy s Theorem

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E extracta mathematicae Vol. 30, Núm. 2, 153 159 (2015) Real Analytic Version of Lévy s Theorem A. El Kinani, L. Bouchikhi Université Mohammed V, Ecole Normale Supérieure de Rabat, B.P. 5118, 10105 Rabat (Morocco) abdellah_elkinani@yahoo.fr Presented by Alfonso Montes Received July 24, 2014 Abstract: We obtain real analytic version of the classical theorem of Lévy on absolutely convergent power series. Whence, as a consequence, its harmonic version. Key words: Fourier series, Lévy s theorem, weight function, weihgted algebra, commutative Banach algebra, Hermitian Banach algebra, Gelfand space, functional calculus, real analytic function, harmonic function. AMS Subject Class. (2010): 46J10, 46H30 1. Introduction Let A be a complex Banach algebra with the involution x x and unit e. The spectrum of an element x of A will be denoted by Spx. An element h of A is called hermitian if h = h. The set of all Hermitian elements of A will be denoted by H(A). We say that the Banach algebra A is Hermitian if the spectrum of every element of H(A) is real ([9]). For scalars λ, we often write simply λ for the element λe of A. Let p ]1, + [. We say that ω is a weight on Z if ω : Z [1, + [, is a map satisfying c(ω) = ω(n) 1 1 p < +. (1) We consider the following weighted space: A p (ω) = { f : R C : f(t) = a n e int, a n l p (Z, ω) }. Endowed with the norm. p,ω defined by: ( f p,ω = a n p ω(n) ) 1 p 153, for every f A p (ω),

154 a. el kinani, l. bouchikhi the space A p (ω) becomes a Banach space. Moreover, if there exists a constant γ = γ(ω) > 0 such that ω 1 1 p ω 1 1 p γω 1 1 p (2) then (A p (ω),. p,ω ) is closed under pointwise multiplication and it is a commutative semi-simple Banach algebra with unity element ê given by ê(t) = 1 (t R) ([4]). For the weight function ω on Z satisfying (2) and ω(n + m) ω(n)ω(m), for every n, m Z, (3) it is also shown in ([4]), that the character space of (A p (ω),. p,ω ) can be identified with the closed annulus: Γ ω (ρ 1, ρ 2 ) = {ξ C : ρ 1 (ω) ξ ρ 2 (ω)}, in such a way that each character has the form f a n ξ n for some ξ Γ ω (ρ 1, ρ 2 ), where f = a n u n A p (ω) with u(t) = e it, for every t R. For ρ 1 and ρ 2, they are given by: where σ 1 = sup { 1 np ln(ω(n)), n 1 } ρ 1 = e σ 2 and ρ 2 = e σ 1 and σ 2 = inf { 1 np ln(ω( n)), n 1 }. The real analytic functional calculus is defined and studied in [1]. To make the paper self-contained, we recall the fundamental properties of this calculus. Let U be an open subset of R 2 and F : U C be real analytic function. Then there exists an open subset V, of C 2, and an holomorphic function F : V C such that V R 2 = U and F U = F. For the construction of V, we have V = Ω x, where Ω x is an open of C 2 x U centered at x. We denote by Λ 0 (U) the set of all open subset V described us above and we consider, in Λ 0 (U), the order given in the following way: V W W V.

real analytic version of lévy s theorem 155 For V Λ 0 (U), we denote by O(V ) the set of holomorphic functions on V. Now we consider the family (O(V )) V Λ0 (U) of algebras and for every V, W Λ 0 (U) with V W, let π W,V : O(V ) O(W ) : F F W The family of algebras (O(V )) V Λ0 (U) with the maps π W,V is an inductive system of algebras and it is denoted by (O(V ), π W,V ). Let lim (O(V ), π W,V ) its inductive limit. We shall denote this simply by lim O(V ) and we have: lim O(V ) = V Λ 0 (U) O(V ) In the sequel, we denote bya(u) the algebra of real analytic functions on U. By lemma 2.1.1 of [1], the map Ψ : A(U) lim O(V ) : f Ψ(f) is an isomorphism algebra. Now let A be a commutative and unital Hermitian Banach algebra (with continuous involution) and a A. Then a = h+ik with h, k H(A). Put a = (h, k) and Sp A a the joint spectrum of (h, k). We denote by Θ a the map that defined the holomorphic functional calculus for a. One has Sp A (h, k) Sp A h Sp A k R 2. By the identification R 2 C, via the map (x, y) x + iy, we can consider that Sp A a Sp A (h, k) and this motivates the following definition: Definition 1.1. ([1], définition 2.1.2) Let A be a commutative and unital Hermitian Banach with continuous involution, a A, U an open subset, of R 2, containing Sp A a and f A(U). We denote by f(a) the element of A defined by: f(a) = Θ a (Ψ(f)) = Ψ(f) (h, k), where a = h + ik and a = (h, k) with h, k H(A). The fundamental properties of this functional calculus are contained in the following result:

156 a. el kinani, l. bouchikhi Proposition 1.2. ([1]) 1. The mapping f f(a) is a homomorphism of A(U) into A that extends the involutive homomorphism from h(u) into A, where h(u) is the set of all harmonic functions on U. 2. Spectral mapping theorem": Let f(t) = Sp A f(a) = f(sp A a), for every f A(U). a n e int be a periodic function such that a n < +. If F is an holomorphic function defined on an open set containing the image of f, then F (f) can be developed in trigonometric series F (f) (t) = c n e int such that c n < +. This result due to P. Lévy ([7]) generalizes the famous theorem of N. Wiener ([10]) which states that the reciprocal of a nowhere vanishing absolutely convergent trigonometric series is also an absolutely convergent trigonometric series. In this paper, we consider the general case of a weight ω on Z which satisfies (2), (3) and lim (ω( n )) 1 n = 1. (4) We then consider f A p (ω) and F an analytic function in two real variables on a neighborhood U of Spf. In this case, we obtain a weighted analogues of Lévy s theorem which states that F (f) can be developed in trigonometric series F (f) (t) = c n e int such that c n p ω(n) < +. To proceed, we consider the Banach algebra (A p (ω),. p,ω ) endowed with the involution f f defined by: f (t) = a n e int, for every f A p (ω). We prove that (A p (ω),. p,ω ) is Hermitian. In the particular case where F is a harmonic function in a neighborhood of f(r), we prove that the expression of F (f) is also given by the Poisson integral formula ([1]). 2. Real analytic version of Levy s theorem Now we are ready to generalize Levy s theorem for real analytic functions.

real analytic version of lévy s theorem 157 Theorem 2.1. (Real analytic version of Lévy s theorem) Let p ]1, + [ and ω be a weight on Z satisfying (2), (3) and (4). Let f(t) = a n e int be a periodic function such a n p ω(n) < +. Let F be an analytic function in two real variables on an open U containing the image of f, then the function F (f) also can be developed in a trigonometric series F (f) (t) = c n e int such that c n p ω(n) < +. Proof. We consider the Banach algebra (A p (ω),. p,ω ) endowed with the involution f f defined by: f (t) = a n e int, for every f A p (ω). One can prove that the map f f is an algebra involution on (A p (ω),. p,ω ). Moreover, it is continuous for the algebra is semi-simple. By the real analytic functional calculus given by Definition 1.1, the proof will be completed by proving that the last involution is hermitian in (A p (ω),. p,ω ). By hypothesis, lim (ω( n )) 1 n = 1. Then the character space M(A p (ω)) of (A p (ω),. p,ω ) can be identified with [0, 2π] in such a way that each character is an evaluation at some t 0 [0, 2π]. This implies that Spf = {f(t) : t [0, 2π]}, for every f A p (ω). Now, it is clear, that f(t) = a n e int, t R, is a hermitian element of A p (ω) if and only, if a n = a n, for every n Z and so Sp(f) R. Whence (A p (ω),. p,ω ) is Hermitian with continuous involution. This completes the proof. Remark 2.2. Actually, the reader can prove that the algebra ( A p (ω), ). p,ω is Hermitian if and only if lim (ω( n )) 1 n = 1. Indeed if the algebra (A p (ω),. p,ω ) is Hermitian. Let f : t a n e int be a hermitian

158 a. el kinani, l. bouchikhi element of (A p (ω),. p,ω ). Then Sp(f) R. Hence where Φ ζ (f) = Φ ζ (f), for every ζ Γ ω (ρ 1, ρ 2 ), Φ ζ (f) = a n ζ n and Φ ζ (f) = n ζ a n, for every ζ Γ ω (ρ 1, ρ 2 ). It follows that ζ = 1, for every ζ Γ ω (ρ 1, ρ 2 ). This yields ρ 1 = ρ 2 = 1, and one obtains that lim (ω( n )) 1 n = 1. Harmonic functions are particular real analytic functions. In this case, we have the following: Corollary 2.3. (Harmonic version of Lévy s theorem) Let p ]1, + [ and ω be a weight on Z satisfying (2), (3) and (4). Let f(t) = a n e int be a periodic function such a n p ω(n) < +. Let U be an open subset of C, z 0 U such that D(z 0, r) U (r > 0) and f(r) D(z 0, r). If F h(u), then F (f) = 1 F (z)re[(z + f 2z 0 )(z f) 1 ] dz 2π z z 0 =r r can be developed in a trigonometric series F (f) (t) = c n p ω(n) < +. c n e int such that

real analytic version of lévy s theorem 159 References [1] M. Akkar, A. El Kinani, M. Oudadess, Calculus fonctionnels harmonique et analytique réel, Ann. Sci. Math. Québec 12 (2) (1988), 151 169. [2] S.J. Bhatt, H.V. Dedania, Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent Fourier series, Proc. Indian Acad. Sci. Math. Sci. 113 (2) (2003), 179 182. [3] A. El Kinani, A version of Wiener s and Lévy s theorems, Rend. Circ. Mat. Palermo (2) 57 (3) (2008), 343 352. [4] A. El Kinani, L. Bouchikhi, A weighted algebra analogues of Wiener s and Lévy s theorems, Rend. Circ. Mat. Palermo (2) 61 (3) (2012), 331 341. [5] A. El Kinani, L. Bouchikhi, Wiener s and Levy s theorems for some weighted power series, Rend. Circ. Mat. Palermo (to appear). [6] I. Gelfand, Normierte Ringe, Rec. Math. [Mat. Sbornik] 9(51) (1941), 3 24. [7] P. Lévy, Sur la convergence absolue des séries de Fourier, Compositio Math. 1 (1935), 1 14. [8] S. Mazur, Sur les anneaux linéaires, C. R. Acad. Sci. Paris 207 (1938), 1025 1027. [9] V. Ptàk, Banach algebras with involution, Manuscripta Math. 6 (1972), 245 290. [10] N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1) (1932), 1 100.

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