Equivalence of K-Functionals and Modulus of Smoothness Generated by the Weinstein Operator

Transcription

1 International Journal of Mathematical Analysis Vol. 11, 2017, no. 7, HIKARI Ltd, Equivalence of K-Functionals and Modulus of Smoothness Generated by the einstein Operator Sami Rebhi Department of Mathematics Issat Kasserine - University of Kairouan Kairouan, Tunisia Copyright c 2017 Sami Rebhi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract On this paper, we study the equivalence between K-functionals and modulus of smoothness tied to a einstein operator. Mathematics Subject Classification: 42B10, 42B30, 42B35 Keywords: einstein operator; Fourier-einstein transfor; translation operator; K-functionals; modulus of smoothness 1 Introduction Given a positive real number r and a positive integer m, the classical modulus of smoothness is defined for a function R 2 by where w m (f, r) = sup 0<h r m h 2 m h = (τ h I) m f, I being the unit operator and τ h stands for the usual translation operator given by τ h f(x) = f(x h).hile the classical K-functional, introduced in [4], is defined by K m (f, r) = inf{ f g 2 r D m g 2 ; g m 2 },

2 338 Sami Rebhi where m 2 be the Sobolev space constructed by the operator D = d dx, m 2 = {f L 2 (R) : D j f L 2 (R), j = 1,..., m}. An oustanding result of the theory of approximation of functions on R, which establishes the equivalence between modulus of smoothness and K functionals, can be formulated as follows: Theorem 1.1. (see[2]) There are two positive constants c 1 and c 2 such that for all f L 2 (R) and r > 0: c 1 w m (f, r) K m (f, r m ) c 2 w m (f, r) IN the classical theory of approximation of functions on R, the modulus of smoothness are basically built by means of the translation operators f f(x y). The translation operator is used for the the construction of modulus of continuity and smoothness which are the fundamental elements of direct and inverse theorems in the approximation theory. Many generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see[5] [6]). In addition to modulus of smoothness, the K-functionals introduced by J.Peetre [4]have turned out to be a simple efficient tool for the description of smoothness properties of functions. The study of the connection between these two quantities is one of the main problems in the theory of approximation of functions. In the classical setting, the equivalence of modulus of smoothness these problems are studied, for example, in [2]. The present paper is organized as follows: In Section 2, we present some preliminary results and notations that will be useful in the sequel and we establish some results associated with the einstein operator. In section 3, the main result is the proof of the theorem on the equivalence of a K functional and the modulus of smoothness constructed by the einstein oprtator. 2 Preliminaries In this section, we provide some facts about harmonic analysis related to the einstein Operator Λ α,d. e cite here, as briefly as possible, only those properties actually required for the discussion. For more details we refer to [1]. In the following we denote by = R d (0, ). x = (x 1, x 2,..., x d, x d1 ).

3 Equivalence of K-functionals and modulus of smoothness 339 S ( ), the Shwartz space of rapidly decreasing functions on, even with respect to the last variable. This space is equipped with the topology defined by the seminorms (N h,k 1 ) h,k>0 given by: N h,k 1 (f) = sup [ (1 x 2 ) k µ f(x) ], h, k > 0. x R d,µ N h d1 µ µ! L p α( ), 1 p, the lebesgue space constituted of mesurable functions on R d such that where f α,p = [ f(x) p dµ α,d (x)] 1 p <, if 1 p <, f α, = ess sup f(x) <, x dµ α,d (x) = x 2α1 d1 (2π) d/2 2 α Γ(α 1) dx S (R d1 ), the strong dual with of the space S ( ). e consider the einstein operator Λ d,α defined on Rd1 by Λ d,α = d1 2 x 2 j=1 j 2α 1 x d1 = d L α, α > 1 x d1 2. here d is the Laplacian operator for the d first variables and L α is the Bessel operator for the last variable defined on (0, ) by L α u = 2 u x 2 d1 2α 1 x d1 u x d1 Definition 2.1. The eigenfunction of the einstein operator is unique solution on with the system 2 u = λ 2 ju(x) if 1 j d, x 2 d1 L α u(x) = λ 2 d1u(x), u(0) = 1, u x d1 (0) = 0, and u x j (0) = iλ j if 1 j d, The eigenfunction denoted by Ψ α,d (λ,.), and given by z C d1, Ψ α,d (λ, z) = e i<z,λ > j α (λ d1 z d1 ) (1)

4 340 Sami Rebhi where z = (z, z d1 ), z = (z 1,..., z d ) and j α is the normalized Bessel function of index α, defined by z, C, j α (z) = Γ(α 1) n=0 ( 1) n n!γ(n α 1) (z 2 )2n. Lemma 2.1. x R the following inequalities are fulfilled and the equality is attained only with x = 0. j α (x) 1, (2) 1 j α (x) 2 x. (3) 3. 1 j α (x) c, (4) with x 1, where c > 0 is a certain constant which depends only on α. Proof. Analog of Lemma 2.9 in [3]. Definition 2.2. The Fourier-einstein transform is given for f L 1 α( ) by λ, F α,d R (f)(λ) = f(x)ψ α,d (x, λ)dµ α,d (x), d1 Theorem The Fourier-einstein transform F α,d isomorphism from S ( ) onto itself. 2. Let f L 1 α( ) such that F α,d (f) L1 α( ). Then Lemma 2.2. (see [1]) x, f(x) = 1. For m N and f S ( ), we have F α,d (f)(y)ψ α,d( x, y)dµ α,d (y). is a topological y, F α,d [(Λα,d )m f](y) = ( 1) m y 2m F α,d (f)(y). (5) 2. For f S ( ) and m N, we have λ, (Λ α,d )m [F α,d where Q m (λ) = ( 1) m λ 2m. (f)](λ) = Fα,d (Q mf)(λ), (6)

5 Equivalence of K-functionals and modulus of smoothness 341 Theorem For all f, g S ( ), we have the following Parseval formula f(x)g(x)dµ α,d (x) = F α,d (f)(λ)fα,d (g)(λ)dµ α,d(λ) (7) 2. (Plancherel Formula). For all f S ( ), we have f(x) 2 dµ α,d (x) = F α,d (f)(λ) 2 dµ α,d (λ) (8) extends uniquely to an iso- 3. (Plancherel Theorem ): The transform F α,d metric isomorphism on L 2 α( ). Definition 2.3. The Fourier-einstein transform of a distribution S S ( ) is defined by : φ S ( ) < F α,d (S), φ >=< S, Fα,d (φ) > Proposition 2.1. The transform F α,d onto itself. is a topological isomorphism from S Definition 2.4. The translation operator T x, x, associated with the einstein operator Λ α,d is defined on C ( ), for all y, by: T x f(y) = a α 2 π 0 f(x y, x 2 d1 y2 d1 2x d1y d1 cosθ)(sinθ) 2α dθ, where x y = (x 1 y 1,..., x d y d ) and a α = 2Γ(α1) πγ(α 1 2 ). Lemma Let f L p α( ), 1 p and x. Then T x f belongs to L p α( ) and we have T x f α,p f α,p (9) 2. Let f L p α( ), p = 1or p = 2 and x, we have y, F α,d (T x f)(y) = e i x y j α ( x y )F α,d (f)(y). (10)

6 342 Sami Rebhi 3 Equivalence of K-Functionnals and Modulus of Smoothness Definition 3.1. Let f L 2 α( ) and r > 0. Then The generalized modulus of smoothness is defined by where w m (f, r) 2,α = sup 0<h r m h f 2,α m h f = (T h I) m f, I being the unit operator. The generalized K-functional is defined by K m (f, r) 2,α = inf{ f g 2,α r (Λ α,d )m g 2,α ; g m 2,α}. The next theorem, which is the main result of this paper, establishes the equivalence between the modulus of Smoothness and the K-functional. Theorem 3.1. There are two positive constants c 1 = c 1 (m, α) and c 2 = c 2 (m, α) such that for all f L 2 α( ) and r > 0. c 1 w m (f, r) 2,α K m (f, r m ) 2,α c 2 (M(r)) m w m (f, r) 2,α, where M(r) = sup 0<h<r ( 1 h ). In order to prove Theorem 3.1, we shall need some preliminary results. Lemma 3.1. Let f L 2 α( ) and h > 0. Then F α,d m h f 2,α 2 m f 2,α ( m h f)(λ) = (Ψ α,d (λ, h) 1) m F α,d (f)(λ). (11) Proof. The result follows easily by using (9), (10) and an induction on m. Lemma 3.2. Let f m 2,α, r > 0. The following inequality is true: w m (f, r) 2,α c 2 r m (Λ α,d )m f 2,α Proof. Assume that h ]0, r] and λ 1. By (5), (8) and (11) we have m h f 2,α = F α,d ( m h f)(λ) 2,α = (1 Ψ α,d (λ, h)) m F α,d (f)(λ) 2,α = h m (1 Ψ α,d(λ,h)) m ( 1) m λ 2m F α,d ( h) m λ 2m (f)(λ) 2,α = h m (1 Ψ α,d(λ,h)) m ( h) m λ 2m F α,d ((Λα,d )m f)(λ) 2,α

7 Equivalence of K-functionals and modulus of smoothness 343 According to Lemma 2.1, we have the inequality (1 Ψ α,d(λ,h)) m 2 m. λ 2m h m Then we deduce m h f 2,α h m 2 m (Λ α,d )m f 2,α Calculating the supremum with respect to all h ]0, r], we obtain w m (f, r) 2,α c 2 r m (Λ α,d )m f 2,α, where c 2 = 2 m. Definition 3.2. For any function f L 2 α( ) and any number ν > 0 let us define the function P ν (f)(x) := ν ν F α,d (f)(λ)ψ α,d( x, λ)dµ α,d (λ) = (F α,d ) 1 (F α,d (λ)χ ν(λ)), where χ ν (λ) is the characteristic function of the segment [ ν, ν], (F α,d ) 1 is the einstein transform inverse. One can easily prove that the function P ν (f) is infinitely differentiable and belongs to all classes m 2,α, m N. Lemma 3.3. There is a positive constant c 3 such that for any f L 2 α( ) and ν > 0. f P ν (f) 2,α c 3 m 1/ν f 2,α Proof. Using the Plancherel formula, we obtain f P ν (f) 2 2,α = F α,d (f P νf) 2 2,α = (1 χ ν (λ))f α,d (f)(λ) 2 2,α = 1 χ ν (λ) 2 F α,d (f)(λ) 2 dµ α,d (λ) By lemma 2.1 there is a constant c 1 > 0 such that 1 Ψ α,d (x, λ/ν) c 1. For all λ R with λ ν. From this, (11) we get f P ν (f) 2 2,α c 2m 1 λ ν 1 Ψ α,d(x, λ/ν) 2m F α,d (f)(λ) 2 dµ α,d (λ) = c 2m 1 F α,d ( m 1/ν f)(λ) 2m dµ α,d (λ) = c 2m 1 m 1/ν f 2 2,α. e get the inequality f P ν (f) 2,α c 3 m 1/ν f 2,α, where c 3 = 1 (c 1 ) m.

8 344 Sami Rebhi Corollary 3.1. There is a positive constant c 3 such that for any f L 2 α( ) and ν > 0. f P ν (f) 2,α c 3 w m (f, 1/ν) 2,α Lemma 3.4. There is a positive constant c 4 such that for any f L 2 α( ) and ν > 0. Proof. (Λ α,d )m (P ν (f)) 2,α c 4 ν 2m m 1/ν f 2,α (Λ α,d )m (P ν (f)) 2,α = F α,d ((Λα,d )m (P ν (f))) 2,α = ( 1) m λ 2m χ ν (λ)f α,d (f)(λ) 2,α λ 2m χ ν(λ) = (1 Ψ α,d (1 Ψ (x,λ/ν)) m α,d (x, λ/ν)) m F α,d (f)(λ) 2,α λ Note that sup 2m χ ν(λ) λ R 1 Ψ α,d ν 2m (λ/ν) sup m (x,λ/ν) m λ ν 1 Ψ α,d = ν 2m t sup m (x,λ/ν) m t 1 Let c 4 = sup t 1 t m 1 Ψ α,d (x,t) m Corollary 3.2. There is a positive constant c 4 such that for any f L 2 q(r q ) and ν > 0. Proof. of Theorem 3.1 (Λ α,d )m (P ν (f)) 2,α c 4 ν 2m w m (f, 1/ν) 2,α 1. Let h ]0, r], g m 2,α. Using lemma 3.2 and lemma 3.1, we have m h f 2,α = m h (f g g) 2,α 1 Ψ α,d (x,t) m m h (f g) 2,α m h (g) 2,α 2 m f g 2,α c 2 h m (Λ α,d )m (g) 2,α c 5 ( f g 2,α r m (Λ α,d )m (g) 2,α ), where c 5 = max{2 m, c 2 }. Calculating the supremum with respect to h ]0, r] and the infimum with respect to all possible functions g m 2,α, we obtain w m (f, r) 2,α c 5 K m (f, r m ) 2,α.

9 Equivalence of K-functionals and modulus of smoothness Since P ν (f) m 2,α, by the definition of a K functional we have K m (f, r m ) 2,α f P ν (f) 2,α r m (Λ α,d )m (P ν (f)) 2,α Using Corollaries 3.1 and 3.2, we get K m (f, r m ) 2,α c 3 w m (f, 1/ν) 2,α c 4 (rν) m ν m w m (f, 1/ν) 2,α. Since ν is an arbitrary positive value, choosing ν = 1, we obtain r K m (f, r m ) 2,α c 6 (M(r)) m w m (f, 1/ν) 2,α, where c 6 = c 3 c 4. This concludes the proof. References [1] H. Ben Mohamed, Neji Bettaibi and S. Hamidou Jah, Sobolev Type Spaces Associated with the einstein Operator, Int. Journal of Math. Analysis, 5 (2011), no. 28, [2] H. Berens, P.L. Butzer, Semi-Groups of Operators and Approximation, Die Grundlehren der Math. issenschaften in Einzeldarstellungen, Vol. 145, Springer, Berlin, [3] E. S. Belkina and S.S. Platonov, Equivalence of K-functionals and Modulus of Smoothness Constructed by Generalized Dunkl Translations, Russian Mathematics, 52 (2008), [4] J. Peetre, A Theory of Interpolation of Normed Spaces, Notes de Universidade de Brasilia, [5] Mk. Potapov, Application of the Operator of Generalized Translation in Approximation Theory, Vestnik Moskovskogo Universiteta, Seriya Matematika, Mekhanika, 3 (1998), [6] S.S. Platonov, Generalized Bessel Translations and Certain Problems of the Theory of Approximation of Functions in the Metrics of L 2,α. I, Trudy Petrozavodskogo Gosudarstvennogo Universiteta, Seriya Matematika, 7 (2000), Received: March 9, 2017; Published: March 28, 2017