On some shift invariant integral operators, univariate case

Transcription

1 ANNALES POLONICI MATHEMATICI LXI ) On some shift invariant integral operators, univariate case by George A. Anastassiou Memphis, Tenn.) and Heinz H. Gonska Duisburg) Abstract. In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is R. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic functions. Finally, four examples of very general specialized operators are presented fulfilling all the above properties; in particular, the inequalities for global smoothness preservation are proven to be sharp. 1. Introduction. In approximating a function f CR) by means of approximation operators L k, it is interesting to examine which properties of f are preserved by the approximants L k f. For example some of these could be: preservation of global smoothness given by the modulus of continuity, shape preservation, and preservation of properties of a probabilistic distribution function. Another important property of the operators L k is the one of shift invariance. Also of interest is their convergence to the unit operator with rates. In this article we introduce a very general family of operators L k see 2) and we study the above characteristics by giving sufficient conditions so they hold, plus we give several examples of such operators. Our research has been mainly motivated by the works of Anastassiou, Cottin and Gonska 1991 Mathematics Subject Classification: 26A15, 26A18, 41A17, 41A25, 41A35, 41A36, 41A55, 41A99, 60E05. Key words and phrases: global smoothness preservation, convergence to the unit with rates, Jackson type inequalities, sharp inequalities, modulus of continuity, integral operators, shift invariant operators, convolution type operators, shape preserving operators, probabilistic distribution function. Research of both authors ported in part by NATO Grant CRG [225]

2 226 G. A. Anastassiou and H. H. Gonska [1 2] and Anastassiou and Yu [3 4]. We would like to mention some results from there. Theorem A [1]). Let X, d) be a compact metric space, and L : CX) CX), L 0, be a bounded linear operator mapping LipX) to LipX) : M>0 Lip M 1; X) such that for all g LipX), Lg Lip c g Lip g Lip : dx,y)>0 ) gx) gy) dx, y) with constant c possibly depending on L, but independent of g. Then for all F CX) and t 0, ) ω 1 Lf; t) L ω 1 f; ct/ L ), where ω 1 is the least concave majorant of ω 1. In the univariate case we get Theorem B [1]). Let I be a compact interval, and L : CI) CI), L 0, be a bounded linear operator mapping C 1 I) to C 1 I). Then the estimate ) of Theorem A is true for all f CI) if the condition is satisfied for all g C 1 I). Lg) I c g I In the multivariate case we obtain Theorem C [2]). Let X be a compact convex subset of R k equipped with the metric d p, 1 p, and let L : CX) CX) be an operator satisfying the assumptions of Theorem A. Then for all f CX) and t 0 we have where ω 1,dp ω 1,dp Lf; t) L + c)ω 1,dp f; t), is the modulus of continuity with respect to d p and k d p x 1,..., x k ), y 1,..., y k )) x i y i p) 1/p, 1 p <, i1 d x 1,..., x k ), y 1,..., y k )) max 1ik x i y i. Definition A. Let ϕ be a bounded right-continuous function on R of compact port [, a], a > 0. Define For f CR), we define ϕ kj x) : 2 k/2 ϕ2 k x j) for k, j Z. Ãkf)x) : f, ϕ kj ϕ kj x) for k Z, j

3 where Shift invariant integral operators 227 f, ϕ kj : Note that Ãkf)x) Ã0f2 k ))2 k x). Theorem D [3]). Assume ft)ϕ kj t) dt. i) j ϕx j) 1, x R; ii) there is a number b R such that ϕx) is non-decreasing if x b and is non-increasing if x b. Let F x) be a continuous probabilistic distribution function on R. Then à k F ) is a probabilistic distribution function and satisfies ÃkF )x) F x) ω 1 F ; a/2 k 1 ), x R, k Z. Moreover, the above inequalities are sharp in the sense that the constant 1 in front of ω 1 is the best possible. Next, if ϕ is continuous and as in Definition A, we have Theorem E [4]). Assume i) j ϕx j) 1 on R; ii) j jϕx j) is a linear function on R; iii) there exist real numbers b 1 and b 2, b 1 b 2, such that ϕx) is convex on, b 1 ] and [b 2, + ) respectively, and ϕx) is concave on [b 1, b 2 ]. Then, for f CR), if f is a convex function on R, Ãkf) is also convex on R and satisfy Ãkf)x) fx) ω 1 f; a/2 k 1 ), x R, k Z. 2. Main results. Let X : C U R) be the space of uniformly continuous real valued functions on R and CR) the space of continuous functions from R into itself. For any f X we have ω 1 f; δ) < +, δ > 0, where ω 1 is the first modulus of continuity with respect to the remum norm. Let {l k } k Z be a sequence of positive linear operators that map X into CR) with the property: 1) l k f; x) l 0 f2 k ); x), x R, f X. For fixed a > 0 we assume that 2) l 0 f; u) fy) ω 1 f; ma + n ) u;y R 2 r for any f X, where m N, n Z +, r Z.

4 228 G. A. Anastassiou and H. H. Gonska Let ϕ be a real valued function of compact port [, a], ϕ 0, ϕ is Lebesgue measurable and such that 3) One can easily find that 4) Examples. ϕx u) du 1, x R. i) ϕx) : χ [ 1/2,1/2) x) the characteristic function; ii) ϕx) : the hat function. ϕu) du 1. { 1, x [ 1/2, 1/2), 0, elsewhere, { 1 x, 0 x 1, 1 + x, 1 x 0, 0, elsewhere, Let {L k } k Z be the sequence of positive linear operators acting on X and defined by 5) L k f; x) : In particular, 6) L 0 f; x) By 1) we observe that l k f; u)ϕ2 k x u) du. l 0 f; u)ϕx u) du. 7) L k f; x) L 0 f2 k ); 2 k x), x R. If the definition of l k is extended to CR) and l k maps π n into itself, then L k does the same; here π n denotes the space of polynomials of degree at most n Z +. For the sake of convenience we make the following informal definition. Definition 1. Let f α ) : f + α), α R, and φ be an operator. If φf α ) φf) α, then φ is called a shift invariant operator. Proposition 1. Assume that 8) l 0 f2 k + α); 2 k u) l 0 f2 k ); 2 k u + α)), for all k Z, α R fixed, all u R and any f X. Then L k is a shift invariant operator for all k Z.

5 P r o o f. Note that L 0 f)x) From 7) we have Shift invariant integral operators 229 l 0 f)u)ϕx u) du L k f + α); x) L k f α ; x) L 0 f α 2 k ); 2 k x) i.e., L k f α ) L k f)) α. l 0 f)x u)ϕu) du. l 0 f2 k + α))2 k x u)ϕu) du l 0 f2 k + α))2 k x 2 k u))ϕu) du l 0 f2 k )))2 k x 2 k u + α))ϕu) du L 0 f2 k ); 2 k x + α)) L k f; x + α), Next we study the property of global smoothness preservation of the operators L k. Theorem 1. For any f X assume that, for all u R, 9) l 0 f; x u) l 0 f; y u) ω 1 f; x y ), for any x, y R. Then 10) ω 1 L k f; δ) ω 1 f; δ) for any δ > 0. P r o o f. Notice that L 0 f; x) L 0 f; y) by 4) and 9). l 0 f; u)ϕx u) du l 0 f; x u)ϕu) du ϕu) l 0 f; x u) l 0 f; y u) du ) ϕu) du ω 1 f; x y ), u R l 0 f; u)ϕy u) du l 0 f; y u)ϕu) du l 0 f; x u) l 0 f; y u)

6 230 G. A. Anastassiou and H. H. Gonska From 7) we now have L k f; x) L k f; y) L 0 f2 k ); 2 k x) L 0 f2 k ); 2 k y) ω 1 f2 k ); 2 k x y ) ω 1 f; x y ), i.e., global smoothness of L k has been established. The convergence of L k to I as k + I is the unit operator) with rates is studied in the following: Theorem 2. For f X, under the assumption 2), we have 11) L k f; x) fx) ω 1 f; ma + n ) 2 k+r, where m N, n Z +, k, r Z. P r o o f. From 7), 3) and p ϕ [, a], we observe that L k f; x) fx) L 0 f2 k ); 2 k x) f2 k 2 k x)) [l 0 f2 k ); u) f2 k 2 k x))]ϕ2 k x u) du i.e., 2 k x+a [l 0 f2 k ); u) f2 k 2 k x))]ϕ2 k x u) du 2 k x l 0 f2 k ); u) f2 k 2 k x)) 2 k xu2 k x+a 2 k x+a 2 k x ) ϕ2 k x u) du, 12) L k f; x) fx) l 0 f2 k ); u) f2 k 2 k x)). 2 k xu2 k x+a Consider g : f2 k ) X. Hence the right-hand side of 12) equals l 0 g; u) g2 k x) ω 1 g; ma + n ) u 2 k x a 2 r ω 1 f; ma + n ) 2 k+r, the last inequality being valid by 2). Thus we have established 11). R e m a r k 1. i) Assume that l 0 f)x u)/ x exists and is continuous in x, u and ϕu) is continuous in u, for x R, u [, a]. Then, from 6) by the Leibniz rule, we get 13) dl 0 f) dx x) a l 0 f) x x u)ϕu) du.

7 Shift invariant integral operators 231 Thus L k f)x) L 0 f2 k ))2 k x), k Z, is also differentiable. ii) Let i 1 be an integer, assume that i 1 l 0 f)x u)/ x i 1 and i l 0 f)x u)/ x i exist and are continuous in x, u and ϕu) is continuous in u, for x R, u [, a]. Then 14) d i L 0 f) dx i x) i l 0 f) x i x u)ϕu) du, i.e., L k is i times differentiable. iii) Assuming that l 0 maps X i) the space of i times continuously differentiable functions from X) into C i) R), the space of i times continuously differentiable functions from R into itself, we see that if f X i) then i 1 l 0 f)x u)/ x i 1 and i l 0 f)x u)/ x i are continuous in x, u, where i 1 is an integer. We need the following Lemma 1. If F x, u) is continuous on R [, a], then continuous in x. P r o o f. Trivial. F x, u) du is R e m a r k 2. i) By Lemma 1, we see that if l 0 maps X i) into C i) R), ϕ being continuous on [, a], then L k maps also X i) into C i) R), where i 0 is an integer. ii) Since ϕ 0 and ϕ is continuous on [, a] we have: if i l 0 f) x i x u) 0, i 1 integer, then d i L 0 f) 15) dx i x) 0. iii) Again assume that ϕ is continuous on [, a]. One can easily prove that if f i) 0 and di dx L i 0 f; x) 0 for some i 1, then d i 16) dx i L kf; x) 0, k Z. Note that if l 0 f is an increasing continuous function and ϕ 0 is continuous on [, a], one can easily see tha L 0 f is increasing and continuous. The above motivates the following Problem 1. If f and l 0 f are continuous probabilistic) distribution functions and ϕ 0 is continuous on [, a], can we conclude then that L 0 f is a continuous distribution function? Note that any continuous distribution function is in X : C U R). The answer to this problem is in the affirmative, see Theorem 4 which follows.

8 232 G. A. Anastassiou and H. H. Gonska To prove Theorem 4 we need Definition 2. Let G, ψ be functions from R 2, R respectively) to R. We say that lim Gx, y) ψx) y y 0 uniformly with respect to x iff for every ε > 0, there is δ > 0 such that for all x, if y y 0 < δ then Gx, y) ψx) < ε y > δ if y 0 ± ). The next is a well-known basic result of Real Analysis. Theorem 3. Assume that β Gx, y) dx exists for all y R. If α lim y y0 Gx, y) ψx) y 0 can be ± ) uniformly with respect to x, then 17) lim y y 0 β α Gx, y) dx β α ψx) dx β α lim Gx, y) dx. y y 0 Again, here we assume that ϕ 0 is continuous on [, a], p ϕ [, a], ϕx u) du 1 for all x R, and f and l 0f are distribution functions that are continuous from R into R. Obviously by 1), l k f is a continuous distribution function too. We aim to prove that L k f)x) is a continuous distribution function, where Gx, u) du, k Z, 18) Gx, u) : l k f)2 k x u)ϕu). Note that G is continuous in x, u) R [, a], hence by Lemma 1, L k f is also continuous, and lim Gx, u) ϕu), 0 respectively) u [, a]. x ± Still we need to prove that respectively). Theorem 3. lim x ± Gx, u) du lim Gx, u) du 1, 0 x ± The last is true from the next Lemmas 2 and 3, 4) and Lemma 2. lim x + Gx, u) ϕu), uniformly with respect to u. P r o o f. Let ε > 0 be given, and observe that C a s e i): Gx, u) ϕu) ϕu) l k f)2 k x u) 1. u > a. Then Gx, u) ϕu) 0, by p ϕ [, a].

9 C a s e ii): Shift invariant integral operators 233 u a. Here ϕu) A. Then Gx, u) ϕu) ϕu) l k f)2 k x u) 1 A l k f)2 k x u) 1 A1 l k f)2 k x u)) A1 l k f)2 k x a)), by l k f being a distribution function. Also lim l kf)z) 1, z + i.e., for ε > 0 there is a δ real) such that z > δ implies 1 l k f)z) < ε. Now choose δ for the uniform convergence) as equal to δ + a)/2 k. Then for x > δ one has 2 k x a > δ. Hence 1 l k f)2 k x a) < ε, x > δ for x + ). Now take ε/a instead of ε, and do the same things again. Lemma 3. lim x Gx, u) 0, uniformly with respect to u. P r o o f. Let ε > 0 be given, and observe that Gx, u) 0 ϕu) l k f)2 k x u). C a s e i): u > a. Then Gx, u) 0. C a s e ii): u a. Here ϕu) A. Then Gx, u) ϕu) l k f)2 k x u) A l k f)2 k x u) Al k f)2 k x u) Al k f)2 k x + a), by l k f being a distribution function. We know that lim l kf)z) 0, z i.e., for ε > 0 there is a δ > 0 such that z < δ implies l k f)z) < ε. Now choose δ δ + a)/2 k > 0. Then for x < δ one has 2 k x + a < δ. Thus l k f)2 k x + a) < ε, x < δ. Now take ε/a instead of ε, and repeat the same things. Now our final main result has been proven and is stated next. Theorem 4. Let l k be positive linear operators from X into CR) as in 1) and f be a probabilistic) distribution function from R into itself that is continuous. Assume that l k f is also a continuous distribution function. Assume furthermore that ϕ 0 is continuous on [, a], a > 0, pϕ)

10 234 G. A. Anastassiou and H. H. Gonska [, a], and ϕx u) du 1 for all x R. Then the operator 19) L k f; x) l k f)2 k x u)ϕu) du, k Z, which is the same as in 5), when applied to f as above produces a continuous probabilistic) distribution function from R into itself. 3. Applications. Next we present four examples of shift invariant integral operators where l k is specified. It will be shown that they satisfy exactly the theory presented in 2. The basic function ϕ will always be as in 2. In particular, for the operators A k ) k Z, to be defined next, ϕ will be assumed additionally to be an even continuous function. The properties of our specific operators will be presented according to the order of the properties of the general operators L k, k Z, in 2. For each k Z, we define i) 20) A k f)x) : where 21) r f k u) : 2k is continuous in u, i.e., here r f k u)ϕ2k x u) du, ft)ϕ2 k t u) dt 22) l k f; u) r f k u), u R. ii) 23) B k f)x) : i.e., here fu/2 k )ϕ2 k x u) du, 24) l k f; u) fu/2 k ) is continuous in u R. iii) 25) L k f)x) : c f k u)ϕ2k x u) du,

11 where Shift invariant integral operators k u+1) 26) c f k u) : 2k ft) dt is continuous in u, i.e., here 2 k u 27) l k f; u) c f k u), u R. iv) 28) Γ k f)x) : where 29) γ f k u) : n i.e., here j0 γ f k u)ϕ2k x u) du, u w j f 2 k + j ) 2 k, n N, w j 0, n 30) l k f; u) γ f k u) w j 1, is continuous in u R. First observe that 1) is satisfied by all l k see 22), 24), 27), 30)) corresponding to the operators A k, B k, L k, Γ k, k Z. Thus 7) is true for all A k, B k, L k, Γ k : 31) Note that A k f; x) A 0 f2 k ); 2 k x), B k f; x) B 0 f2 k ); 2 k x), L k f; x) L 0 f2 k ); 2 k x), Γ k f; x) Γ 0 f2 k ); 2 k x), for all k Z. 32) A k 1) B k 1) L k 1) Γ k 1) 1. In case the above specific operators are defined on the whole CR) and if π n is the space of polynomials of degree n, then 33) A k π n ) π n, B k π n ) π n, L k π n ) π n, Γ k π n ) π n. 34) j0 Proposition 2. A k, B k, L k, Γ k are shift invariant operators. P r o o f. Here we apply Proposition 1. i) For A k operators ϕ even): Note that l 0 f; x) r f 0 x) ft)ϕt x) dt

12 236 G. A. Anastassiou and H. H. Gonska Thus for a R we have l 0 f2 k + a); 2 k u) ft)ϕx t) dt fx t)ϕt) dt. f2 k 2 k u t) + a)ϕt) dt fu + a 2 k t)ϕt) dt f2 k 2 k u + a) 2 k t)ϕt) dt f2 k [2 k u + a) t])ϕt) dt l 0 f2 k ); 2 k u + a)). Here r f 0 satisfies 8). Therefore A k is a shift invariant operator. ii) For B k operators: Here l 0 f f. Thus Then l 0 f2 k + a); 2 k u) f2 k 2 k u + a) iii) For L k operators: Here l 0 f2 k + a); 2 k u) iv) For Γ k operators: fu + a) l 0 f2 k ); 2 k u + a)). l 0 f, x) 2 k u+1 2 k u 2 k u+a)+1 2 k u+a) l 0 f; u) x+1 x ft) dt. f2 k t + a) dt j0 f2 k t) dt l 0 f2 k ); 2 k u + a)). w j f u + j ). n

13 Thus l 0 f2 k + a); 2 k u) Shift invariant integral operators 237 w j f2 k 2 k u + j/n) + a) j0 w j f2 k 2 k u + a) + j/n)) j0 l 0 f2 k ); 2 k u + a)). Next we show that the operators A k, B k, L k, Γ k have the property of global smoothness preservation. 35) Theorem 5. For all f C U R) and all δ > 0 we have ω 1 A k f; δ) ω 1 f; δ), ω 1 L k f; δ) ω 1 f; δ), ω 1 B k f; δ) ω 1 f; δ), ω 1 Γ k f; δ) ω 1 f; δ). P r o o f. Here we apply Theorem 1. It is enough to prove 9). i) For A k operators ϕ even): From 34) we have l 0 f; x u) l 0 f; y u) fx u t) fy u t))ϕt) dt where the last equality holds by 4). ii) For B k operators: Here l 0 f f, therefore fx u t) fy u t) ϕt) dt ω 1 f; x y )ϕt) dt ω 1 f; x y ), l 0 f; x u) l 0 f; y u) fx u) fy u) ω 1 f; x y ). iii) For L k operators: We have l 0 f; x u) l 0 f; y u) c f 0 x u) cf 0 y u) x u+1 y u+1 ft) dt ft) dt x u y u fw u + x) dw 1 0 fw u + y) dw fw u + x) fw u + y) dw ω 1 f; x y ).

14 238 G. A. Anastassiou and H. H. Gonska iv) For Γ k operators: We have l 0 f; x u) l 0 f; y u) γ f 0 x u) γf 0 y u) w j fx u + j/n) fy u + j/n)) j0 w j fx u + j/n) fy u + j/n) j0 ω 1 f; x y ) w j ω 1 f; x y ). Theorem 6. Inequalities 35) are sharp, in the sense that they hold as equalities when fx) x C U R). Here j0 P r o o f. i) For A k operators ϕ even): Note that Therefore A k f)x) r f k u) 2k If fx) x then r f k u)ϕ2k x u) du r f k 2k x u)ϕu) du ft)ϕ2 k t u) dt 2 k τ f 2 k u τ f 2 k r f k 2k x u) r f k 2k x u) r f k 2k y u) ) ϕu τ) dτ ) ϕτ) dτ. f r f k 2k x u)ϕu) du. ft)ϕu 2 k t) dt u τ f 2 k f x u + τ ) 2 k ϕτ) dτ. ) ϕτ) dτ x u + τ ) 2 k f y u + τ )) 2 k ϕτ) dτ x y)ϕτ) dτ x y,

15 Shift invariant integral operators 239 by 4). Thus A k f)x) A k f)y) r f k 2k x u) r f k 2k y u))ϕu) du x y)ϕu) du x y, proving that ω 1 A k id); δ) ω 1 id; δ), δ > 0, where id stands for the identity map. ii) For B k operators: Note that ) u B k f)x) f ϕ2 k x u) du 2 k 2 k ) x u f ϕu) du 2 k f x u2 ) k ϕu) du. When fx) x we get B k f)x) B k f)y) f x u2 ) k f y u2 )) k ϕu) du x y)ϕu) du x y. iii) For L k operators: Note that L k f)x) c f k u)ϕ2k x u) du c f k 2k x u)ϕu) du c f k 2k x u)ϕu) du. Here that is, 2 k u+1) 2 c f k u) 2k ft) dt 2 k 2 k u 0 k f t + u2 ) k dt, k 2 c f k 2k x u) 2 k f t + x u2 ) k dt. 0

16 240 G. A. Anastassiou and H. H. Gonska If fx) x, then and so again k 2 c f k 2k x u) c f k 2k y u) 2 k x y) dt L k f)x) L k f)y) 0 k 2 x y)2 k dt x y, 0 x y)ϕu) du x y. iv) For Γ k operators: As before we obtain Γ k f; x) w j f x u 2 k + j ) 2 k ϕu) du. n If fx) x, then j0 Γ k f; x) Γ k f; y) j0 w j j0 x y)ϕu) du n ) x y) w j ϕu) du x y. The operators A k, B k, L k, Γ k, k Z, converge to the unit operator I with rates as given below. Theorem 7. For k Z, ) a 36) A k f; x) fx) ω 1 f; 2 k 1, 37) B k f; x) fx) ω 1 f; a ), 2 k L k f; x) fx) ω 1 f; a + 1 ) 38) 2 k, and 39) Γ k f; x) fx) ω 1 f; a + 1 ) 2 k. P r o o f. i) For A k operators ϕ even): By 34) and 3), l 0 f, u) fy)

17 Shift invariant integral operators 241 ω 1 f; 2a) ft)ϕu t) dt ft) fy) ϕu t) dt ω 1 f; t y )ϕu t) dt ϕu t) dt ω 1 f; 2a). proving 2). Hence by Theorem 2, we obtain 36). ii) For B k operators: Here l 0 f, u) fu) and fy)ϕu t) dt l 0 f; u) fy) fu) fy) ω 1 f; a), and we use Theorem 2. iii) For L k operators: Here u+1 l 0 f, u) fy) ft) dt fy) u and we use Theorem 2 again. u+1 u u+1 u ft) fy) dt ω 1 f; t y ) dt ω 1 f; t + u y ) dt ω 1 f; t + u y ) dt ω 1 f; 1 + u y ) ω 1 f; 1 + a),

18 242 G. A. Anastassiou and H. H. Gonska iv) For Γ k operators: Here l 0 f, u) fy) w j fu + j/n) fy) j0 w j fu + j/n) fy) j0 w j ω 1 f; u + j/n y ) j0 w j ω 1 f; j/n + u y ) j0 n w j )ω 1 f; 1 + a) ω 1 f; a + 1). j0 Again, an application of Theorem 2 completes the proof. Some final comments follow. R e m a r k 3. According to Remark 2 we have: i) Here we assume ϕ 0, ϕ continuous on [, a], p ϕ [, a]. Let f X i), i 0 an integer. If f i) 0, then and so r f 0 )i) 0, c f 0 )i) 0, γ f 0 )i) 0 A 0 f) i) 0, B 0 f) i) 0, L 0 f) i) 0, Γ 0 f) i) 0. ii) If f i) 0, i 0 an integer, ϕ 0, ϕ continuous on [, a], p ϕ [, a], then A k f) i) 0, B k f) i) 0, L k f) i) 0, Γ k f) i) 0, for any k Z. An application of Theorem 4 comes next: iii) Let ϕ 0, ϕ continuous on [, a], p ϕ [, a], ϕx u) du 1, x R. Then the operators A k, B k, L k, Γ k k Z) map continuous probabilistic distribution functions into continuous probabilistic distribution functions. Acknowledgements. The authors gratefully acknowledge the hospitality of the Mathematical Research Institute at Oberwolfach where this article was started and finished during two most pleasant visits.

19 Shift invariant integral operators 243 References [1] G. Anastassiou, C. Cottin and H. Gonska, Global smoothness of approximating functions, Analysis ), [2],,, Global smoothness preservation by multivariate approximation operators, in: Israel Mathematical Conference Proc. 4, Weizmann Science Press, 1991, [3] G. Anastassiou and X. M. Yu, Monotone and probabilistic wavelet approximation, Stochastic Anal. Appl ), [4],, Convex and coconvex-probabilistic wavelet approximation, ibid., DEPARTMENT OF MATHEMATICAL SCIENCES THE UNIVERSITY OF MEMPHIS MEMPHIS, TENNESSEE 38152, U.S.A. DEPARTMENT OF MATHEMATICS UNIVERSITY OF DUISBURG D DUISBURG, GERMANY Reçu par la Rédaction le Révisé le