Direct and inverse theorems of approximation theory in L 2 (R d, w l (x)dx)

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1 MATEMATIKA, 2017, Volume 33, Number 2, c Penerbit UTM Press. All rights reserve Direct an inverse theorems of approximation theory in L 2 (, w l (x)x) 1 aouan Daher, 2 Salah El Ouaih an 3 Mohame El Hamma 1,2,3 Department of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco 1 rjaher024@gmail.com, 2 salahwaih@gmail.com, 3 m-elhamma@yahoo.fr Abstract In this paper, we prove analogues of irect an some inverse theorems for the Dunkl harmonic analysis, using the function with boune spectrum an generalize spherical mean operator. Keywors Generalize continuity moulus; Bernstein theorem; Jackson theorem; best approximation. AMS Mathematics Subject Classification 42B37, 42B10 1 Introuction Yet by the year 1912, S. Bernstein obtaine the estimate inverse to Jakson s inequality in the space of continuous functions for some special cases [1], later Stechkin [2], Timan [3], prove such inverse estimates, incluing the case of the space L p, 1 < p <. Approximation problems for functions in the space L 2 (, w l (x)x), where w l is a weight function invariant uner the action of an associate reflection groups, using the function with boune spectrum, are stuie in this paper. Applying the Dunkl transform, Dunkl Laplacian operator an generalize spherical mean operator, we obtain analogs of the Bernstein inequality for function with boune spectrum, irect an inverse theorem of Jackson type [4],[2],[5], where the moulus of smoothness is constructe on the basis of generalize spherical mean operator. 2 The Dunkl transform an its basic properties Dunkl [3] efine a family of first-orer ifferential-ifference operators relate to some reflection groups. These operators generalize in a certain manner the usual ifferentiation an have gaine consierable interest in various fiels of mathematics an also in physical applications. The theory of Dunkl operators provies generalizations of various multivariable analytic structures. Among others, we cite the exponential function, the Fourier transform an the translation operator. For more etails about these operators see [6], [7], [1] an [8]. Let be a root system in, W the corresponing reflection group, + a positive subsystem of an l a non-negative an W-invariant function efine on. The Dunkl operator is efine for f C 1 ( ) by D j f(x) = f (x) + f(x) f(σ α (x)) l(α)α j, x. x j α, x α + Here, is the usual Eucliean scalar prouct on with the associate norm. an σ α the reflection with respect to the hyperplane H α orthogonal to α. We consier the weight

2 178 aouan Daher, Salah El Ouaih an Mohame El Hamma function w l (x) = α + α, x 2l(α), where w l is W-invariant an homogeneous of egree 2γ where γ = α + l(α). We let η be the normalize surface measure on the unit sphere S 1 in an set η l (y) = w l (y)η(y). Then η l is a W-invariant measure on S 1, an we let l = η l (S 1 ). The Dunkl kernel E l on has been introuce by Dunkl in [9]. For y the function x E(x, y) can be viewe as the solution on of the following initial problem: { Dj u(x, y) = y j u(x, y), if 1 j, u(0, y) = 0, for all y, This kernel has a unique holomorphic extension to C C. ösler has prove in [8] the following integral representation for the Dunkl kernel, E l (x, z) = e y,z µ x (y), x, z C, where µ x is a probability measure on with support in the close ball B(0, x ) of center 0 an raius x. Proposition 1 [6] Let z, w C an λ C (i) E l (z, 0) = 1,E l (z, w) = E l (w, z), E l (λz, w) = E l (z, λw). (ii) For all ν = (ν 1,..., ν ) N,x,z C, we have where D ν ze k (x; z) x ν exp( x ez, D ν z = ν ν z 1... ν z, ν = ν ν. In particular D ν ze l (ix; z) x ν for all x, z. We enote by L 2 l ( ) = L 2 (, w l (x)x) the space of measurable functions on such that ( )1 f 2,l = f(x) 2 2 w l (x)x, an D l the Dunkl Laplacian efine by D l = Dj. 2 i=1

3 Direct an inverse theorems of approximation theory 179 The scalar prouct in the Hilbert space L 2 l ( ) obeys the formula (f, g) := f(x)g(x)w l (x)x, f, g L 2 l ( ). By the partial integration one can verify the correlation (D l f, g) = (g, D l f). for any functions f, g D (D enotes the set of infinitely ifferentiable functions with a compact support). As usual, we enow the space D with a topology; this turns it into a topological vector space [10]. Let D stan for the set of generalize functions, i.e., linear continuous functionals on the space D. We enote the value of a functional f D on a function ϕ D by f, ϕ. The space L 2 l ( ) is embee into D, provie that for f L 2 l ( ) an ϕ D we put f, ϕ := f(x)ϕ(x)w l (x)x. One can exten (in a natural way) the action of the Dunkl Laplacian operator D l onto the space of generalize functions D, putting D l f, ϕ := f, D l ϕ, f D, ϕ D. In particular, the action of the operator D l f is efine for any function f L 2 l ( ) but, generally speaking, D l f is a generalize function. The Dunkl transform is efine for f L 1 l ( ) = L 1 (, w l (x)x) F(f)(ξ) = f(ξ) = c 1 l f(x)e l ( iξ, x)w l (x)x, where the constant c l is given by c l = e x 2 2 wl (z)z. The inverse Dunkl transform is efine by the formula f(x) = f(ξ)el (ix, ξ)w l (ξ)ξ, x. From [11], we have that if f L 2 l ( ) D l f(ξ) = ξ 2 f(ξ). (1) The Dunkl transform shares several properties with its counterpart in the classical case. We mention here, in particular that Parseval theorem hols in L 2 l ( ). As in the classical case, a generalize translation operator is efine in the Dunkl [5, 12]. Namely, for f L 2 l ( ) an x we efine τ x (f) to be the unique function in L 2 l ( ) satisfying τ x f(y) = E l (ix, y) f(y) a.e y.

4 180 aouan Daher, Salah El Ouaih an Mohame El Hamma Form to Parseval theorem an proposition 1.1, we see that τ x f 2,l f 2,l for all x. For α > 1 2, let j α(x) be a normalize Bessel function of the first kin, i.e., j α (x) = 2α Γ(α + 1)J α (x) x α, where J α (x) is a Bessel function of the first kin [13]. The function J α (x) is infinitely ifferentiable, J α (0) = 1. Proposition 2 ([14] or [15]) For x the following inequalities are fulfille (i) j α (x) 1. (ii) 1 j α (x) c 0 with x 1, where c 0 > 0 is a certain constant which epen only on α. (iii) 1 j α (x) c 1 x 2, where c 1 is a constant. The generalize spherical mean value of f L 2 l ( ) is efine by M h f(x) = 1 l S 1 τ x f(hy)η l (y), x, h > 0. We have M h f 2,l f 2,l. (2) Proposition 3 ([16]) Let f L 2 l ( ) an fix h > 0. Then M h f L 2 l ( ) an M h f(ξ) = j γ+ 2 1(h ξ ) f(ξ), ξ. (3) Let the function f L 2 l ( ). We efine ifferences of the orer k (k = 1, 2,...) with a step h > 0. k hf(x) = (I M h ) k f(x), where I is the unit operator. For any positive integer k, we efine the generalize moule of smoothness of the kth orer by the formula ω k (f, δ) 2,l = sup k h f 2,l, δ > 0. 0<h δ Let W k 2,l be the Sobolev space constructe by the operator D l, i.e., W k 2,l = {f L2 l ( ) : D j l f L2 l ( ); j = 1, 2,..., k}, where Dl 0f = f, Dj l f = D l(d j 1 l f). For any f L 2 l ( ) an any number ν > 0, let us efine the function P ν (f)(x) = F 1 ( f(ξ)χ ν (ξ)),

5 Direct an inverse theorems of approximation theory 181 where χ ν (ξ) = 1 if ξ ν an χ ν (ξ) = 0 if ξ > ν, F 1 is the inverse Dunkl transform. One can easily prove that the function P ν (f)(x) is infinitely ifferentiable an belongs to all classes W2,l k, k = 1, 2,... A function f L 2 l ( ) is calle a function with boune spectrum of orer ν > 0 if f(ξ) = 0 for ξ > ν. The set of all such functions is enote by I ν. The best approximation of a function f L 2 l ( ) by functions in I ν is the quantity E ν (f) 2,l := inf g I ν f g 2,l. 3 Bernstein s inequality an Jackson s irect theorems Bernstein s Theorem 1 If f I ν, then D l f I ν an D l f 2,l ν 2 f 2,l. (4) Proof From equality (1), we have D l f I ν if f I ν Formula (1) an Parseval theorem gives D l f 2 2,l = D l f(x) 2 w l (x)x = D l f(ξ) 2 w l (ξ)ξ = ξ 4 f(ξ) 2 w l (ξ)ξ ξ ν ν 4 f(ξ) 2 w l (ξ)ξ = ν 4 f(x) 2 w l (x)x = ν 4 f 2 2,l. D l f 2,l ν 2 f 2,l. Jackson s Theorem 2 Suppose that f W2,l m (m = 1, 2,...), then for all ν > 0, where c 2 = c (k+m) 0 c m 1 E ν (f) 2,l c 2 ν 2m ω k (D m l f, 1/ν) 2,l, (5) is a constant.

6 182 aouan Daher, Salah El Ouaih an Mohame El Hamma Proof Using the Parseval equality, we have f P ν (f) 2 2,l = = = f P ν (f) 2 2,l = f(x) P ν (f)(x) 2 w l (x)x f(ξ) P ν (f)(ξ) 2 w l (ξ)ξ (1 χ ν (ξ)) f(ξ) 2 w l (ξ)ξ. ξ >ν f(ξ) 2 w l (ξ)ξ. (6) By Proposition 2, we have 1 j γ+ 1( ξ /ν) c 0 2 for ξ ν., from (3) an Parseval equality we euce that f P ν (f) 2 1 2,l (1 j c 2(k+m) γ ( ξ /ν))2(k+m) f(ξ) 2 w l (ξ)ξ 1 = F((I M c 2(k+m) 1/ν ) k+m f)(ξ) 2 w l (ξ)ξ 0 1 = (I M 1/ν ) k+m f(x) 2 w l (x)x = c 2(k+m) 0 1 c 2(k+m) 0 (I M 1/ν ) k+m f 2 2,l. f P ν (f) 2,l c (k+m) 0 (I M 1/ν ) k+m f 2,l. (7) Proposition 2, Parseval equality an formula (1) show that (I M 1/ν )f 2 2,l = (I M 1/ν )f(x) 2 w l (x)x = F((I M 1/ν )f)(ξ) 2 w l (ξ)ξ = (1 j γ+ 2 1( ξ /ν))2 f(ξ) 2 w l (ξ)ξ c 2 1ν 4 ξ 4 f(ξ) 2 w l (ξ)ξ = c 2 1ν 4 D l f(ξ) 2 w l (ξ)ξ = c 2 1ν 4 D l f(x) 2 w l (x)x = c 2 1 ν 4 D l f 2 2,l.

7 Direct an inverse theorems of approximation theory 183 (I M 1/ν )f 2,l c 1 ν 2 D l f 2,l. (8) Successive applications of (8) to the right-han sie of (7) result in f P ν (f) 2,l c (k+m) 0 c m 1 ν 2m (I T 1/ν ) k Dl m f 2,l c 2 ν 2m ω k (Dl m f, 1/ν) 2,l, where c 2 = c (k+m) 0 c m 1 which implies (5) hols, the theorem is prove. Proposition 4 The moulus of smoothness ω k (f, t) 2,l has the following properties. (i) ω k (f + g, t) 2,l ω k (f, t) 2,l + ω k (g, t) 2,l. (ii) ω k (f, t) 2,l 2 k f 2,l. (iii) If f W k 2,l, then ω k (f, t) 2,l t 2k D k f 2,l, where c 3 = c k 1 is a constant. Proof Property (i) follow from the efinition of ω k (f, t) 2,l. Property (ii) follow from the fact that M h f 2,l f 2,l. Assume that h (0, t]. From formulas (1), (3) an Parseval equality, we have k hf 2 2,l = k h f 2 2,l = (1 j γ 2 1(h ξ ))2k f(ξ) 2 w l (ξ)ξ, (9) Dl k f 2 2,l = D l kf 2 2,l = ξ 4k f(ξ) 2 w l (ξ)ξ. (10) Formula (9) implies the equality (1 j k hf 2 2,l = h 4k γ 2 1(h ξ ))2k h 4k ξ 4k ξ 4k f(ξ) 2 w l (ξ)ξ. From Proposition 2 an Parseval equality we obtain k h f 2 2,l c 2k 1 h4k ξ 4k f(ξ) 2 w l (ξ)ξ = c 2k 1 h4k D k l f 2,l = c 2k 1 h4k D k l f 2 2,l. k hf 2,l c k 1h 2k D k l f 2,l. Calculating the supremum with respect to all h (0, t], we obtain ω k (f, t) 2,l t 2k D k l f 2,l, where c 3 = c k 1.

8 184 aouan Daher, Salah El Ouaih an Mohame El Hamma 4 Proofs of the inverse theorems Proposition 5 For j 1 we have Proof Note that 2 2k(j 1) E (f) 2,l η= 1 +1 η= 1 +1 η 2k 1 E η (f) 2,l. η 2k 1 ( 1 ) 2k 1 1 = 2 2k(j 1). Since E η (f) 2,l is monotonically ecreasing, we conclue that 2 2k(j 1) E (f) 2,l η= 1 +1 η 2k 1 E η (f) 2,l. Proposition 6 For n N we have Proof Note that n (j + 1) 2k 1 2 k E n (f) 2,l c 4 n (j + 1) 2k 1 E j (f) 2,l. n ( n ) 2k 1 n (j + 1) 2k = 2 2k. j n 2 1 Since E j (f) 2,l is monotonically ecreasing, we conclue that 2 k E n (f) 2,l c 4 n (j + 1) 2k 1 E j (f) 2,l. Proposition 7. If Φ ν I ν such that f Φ ν 2,l = E ν (f) 2,l For every ν N, then D k l Φ +1 D k l Φ 2,l 2 2k(j+1)+1 E (f) 2,l. In particular D k l Φ 1 2,l = D k l Φ 1 D k l Φ 0 2,l 2 4k+1 E 0 (f) 2,l. Proof By Theorem 1 an the fact that E ν (f) 2,l is monotone ecreasing with respect to ν, we obtain D k l Φ +1 D k l Φ 2,l 2 2k(j+1) Φ +1 Φ 2,l = 2 2k(j+1) (f Φ ) (f Φ +1) 2,l 2 2k(j+1) (E (f) 2,l + E +1(f) p,α ) 2,l 2 2k(j+1)+1 E (f) 2,l.

9 Direct an inverse theorems of approximation theory 185 an D k l Φ 1 D k l Φ 0 2,l Φ 1 Φ 0 2,l = (f Φ 1 ) (f Φ 0 ) 2,l E 1 (f) 2,l + E 0 (f) 2,l 2E 0 (f) 2,l 2 4k+1 E 0 (f) 2,l. The following theorems are analogues of the classical inverse theorems of approximation theory [2,5]. Theorem 3 For every function f L 2 l ( ) an every positive integer n we have ω k (f, 1 n ) 2,l c n (j + 1) 2k 1 E j (f) 2,l, where c = c(k, α) is a positive constant. Proof Let 2 m n < 2 m+1 for any integer m 0. For every ν 0, let Φ ν be an element of best approximation to f in the space I ν, that is, Φ ν I ν an f Φ ν 2,l = E ν (f) 2,l. By formulas (i) an (ii) of Proposition 4, we obtain ω k (f, 1 n ) 2,l ω k (f Φ 2 m+1, 1 n ) 2,l + ω k (Φ 2 m+1, 1 n ) 2,l 2 k f Φ 2 m+1 2,l + ω k (Φ 2 m+1, 1 n ) 2,l. ω k (f, 1 n ) 2,l 2 k E 2 m+1(f) 2,l + ω k (Φ 2 m+1, 1 n ) 2,l 2 k E n (f) 2,l + ω k (Φ 2 m+1, 1 n ) 2,l. (11) Now with the ai of Proposition 5 an 7 an formula (iii) of Proposition 3, we conclue that ω k (Φ 2 m+1, 1 n ) 2,l Bk Φ 2 m+1 2,l n n n B k Φ 1 B k Φ 0 2,l + 2 4k+1 E 0 (f) 2,l + 2k 24k+1 2k 24k+1 2k 24k+1 E 0 (f) 2,l + m B k Φ +1 B k Φ 2,l m 2 2k(j+1)+1 E (f) 2,l m 2 2k(j 1) E (f) 2,l E 0 (f) 2,l + E 1 (f) 2,l + E 0 (f) 2,l + E 1 (f) 2,l + m j=1 η= m j=2 η 2k 1 E η (f) 2,l (j + 1) 2k 1 E j (f) 2,l.

10 186 aouan Daher, Salah El Ouaih an Mohame El Hamma ω k (Φ 2 m+1, 1 n ) 2,l c 5 2 m (j + 1) 2k 1 E j (f) 2,l. (12) Thus from (11) an (12) we erive the estimate ω k (f, 1 n ) 2,l 2 k E n (f) 2,l + c n 5 (j + 1) 2k 1 E j (f) 2,l. (13) By Proposition 6 an formula (13), we have ω k (f, 1 n ) 2,l c n (j + 1) 2k 1 E j (f) 2,l. Theorem 4 Suppose that f L 2 l ( ) an j 2m 1 E j (f) 2,l <. j=1 Then f W2,l m an, for every positive integer n, we have ω k (B m f, 1 n ) 2,l C 1 n (j + 1) 2(k+m) 1 E j (f) 2,l + j=n+1 j 2m 1 E j (f) 2,l, where C = c(k, m, α) is a positive constant. Proof. Let 2 m n < 2 m+1 for any integer m 0. For every positive integer r m, we consier the series Dl r Φ 1 + (Dl r Φ +1 Dl r Φ ). (14) It follows from Propositions 7 an 5 that the series (14) converges in the norm of L 2 l ( ) because Dl r Φ +1 Dr l Φ 2,l 2 2r(j+1)+1 E (f) 2,l j=1 = 2 2r+1 E 1 (f) 2,l + c 2 2 4r+1 2 2r(j 1) E (f) 2,l 2 4r+1 E 1 (f) 2,l + 2 2r(j 1) E (f) 2,l 2 4r+1 E 1 (f) 2,l + j=1 η= r+1 j 2r 1 E j (f) 2,l <. j=1 η 2r 1 E η (f) 2,l j=1

11 Direct an inverse theorems of approximation theory 187 Note that f = Φ 1 + (Φ +1 Φ ). (15) where the series (15) converges in L 2 l ( ) an, a fortiori, in the space D of istributions. Since the operator D l is a linear continuous operator on D, the equality Dl r f = Dl r Φ 1 + (Dl r Φ +1 Dl r Φ ), (16) hols in the space D. Since the right-han sie of (16) belongs to L 2 l ( ) for r m, we see that f belongs to the Sobolev space W m 2,l. In particular, Dm l f L 2 l ( ). By formula (i) of Proposition 4, we obtain ω k (D m l f, 1 n ) 2,l ω k (D m l f D m l Φ 2 s+1, 1 n ) 2,l + ω k (D m l Φ 2 s+1, 1 n ) 2,l. Using Propositions 4, 5 an 7 we get ω k (D m l f D m l Φ 2 s+1, 1 n ) 2,l 2 k D m l f D m l Φ 2 s+1 2,l 2 k j=s+1 2 k j=s+1 2 k+4m+1 D m l Φ +1 D m l Φ 2,l 2 2m(j+1)+1 E (f) 2,l j=s+1 2 k+4m+1 2 k+4m+1 2 2m(j 1) E (f) 2,l j=s+1 η= 1 +1 j=2 s +1 j 2m 1 E j (f) 2,l. η 2m 1 E η (f) 2,l ω k (Dl m f Dl m Φ 2 s+1, 1 n ) 2,l c 6 j 2m 1 E j (f) 2,l. (17) j=2 s +1 Now with the ai of Propositions 5 an 7 an by formula (iii) of Proposition 4, we conclue

12 188 aouan Daher, Salah El Ouaih an Mohame El Hamma that ω k (D m l Φ 2 s+1, 1 n ) 2,l n Dm+k 2k l Φ 2 s+1 2,l D m+k l Φ 1 D m+k l Φ 0 2,l + 2 4(k+m)+1 E 0 (f) 2,l + 24(k+m)+1 24(k+m)+1 24(k+m)+1 E 0 (f) 2,l + ω k (D m l Φ 2 s+1, 1 n ) 2,l c 7 j=n+1 s D m+k l Φ +1 D m+k l Φ 2,l s 2 2(k+m)(j+1)+1 E (f) 2,l s 2 2(k+m)(j 1) E (f) 2,l E 0 (f) 2,l + E 1 (f) 2,l + s j=1 η= 1 +1 η 2(k+m) 1 E η (f) 2,l 2 s E 0 (f) 2,l + E 1 (f) 2,l + (j + 1) 2(k+m) 1 E j (f) 2,l. 2 s j=2 (j + 1) 2(k+m) 1 E j (f) 2,l. (18) Thus from (17) an (18) we erive the estimate ω k (Dl m f, 1 n ) 2,l C j 2m 1 E j (f) 2,l + 1 n (j + 1) 2(k+m) 1 E j (f) 2,l. Acknowlegemnt The authors woul like to thank the referee for his valuable comments an suggestions. eferences [1] Bernstein, S. Sur l orre e la meilleure approximation es fonctions continues par es polynomes e egrè onnè. Mem.Aca.oy.Belgigue, 2 me sèrie : [2] Stechkin, S. B. On the orer of best approximation of continuous functions (ussian). Izv.Aka.Nauk.SS, Ser.Math : [3] Timan, M. F. Best approximation an moulus of smoothness of functions prescribe on the entier real axis, (ussian). Izv.Vyssh. Uchebn. Zave Matematika :

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