ANNALES POLONICI MATHEMATICI LXIII.3 (996) L p -convergence of Bernstein Kantorovich-type operators by Michele Campiti (Bari) and Giorgio Metafune (Lecce) Abstract. We study a Kantorovich-type modification of the operators introduced in [] and we characterize their convergence in the L p -norm. We also furnish a quantitative estimate of the convergence. In [] and [2] we introduced a modification of classical Bernstein operators in C([, ]) which we used to approximate the solutions of suitable parabolic problems. These operators are defined by () A n (f)(x) := α n,k x k ( x) n k f(k/n), f C([, ]), x [, ], where the coefficients α n,k satisfy the recursive formulas (2) (3) α n+,k = α n,k + α n,k, k =,..., n, α n, = λ n, α n,n = ϱ n, and (λ n ) n N, (ϱ n ) n N are fixed sequences of real numbers. In [], we investigated convergence and regularity properties of these operators; in particular, we found that (A n ) n N converges strongly in C([, ]) if and only if (λ n ) n N and (ϱ n ) n N converge. In this case A n (f) w f, for every f C([, ]), where (4) w(x) = (λ m x( x) m + ϱ m x m ( x)) is continuous in [,] and analytic in ],[. Connections with semigroup theory and evolution equations, via a Voronovskaya-type formula, have been explored in [2]. 99 Mathematics Subject Classification: 4A35, 4A. Key words and phrases: Kantorovich operators, quantitative estimates. Work performed under the auspices of M.U.R.S.T. (6% and 4%) and G.N.A.F.A. [273]
274 M. Campiti and G. Metafune In this paper we deal with a Kantorovich-type version of the operators () (see [3, p. 3]) and characterize the convergence in the L p -norm giving also a quantitative estimate. Let p < and define an operator K n : L p ([, ]) L p ([, ]) by (5) K n (f)(x) := α n,k x k ( x) n k (n + ) f(t) dt, for every f L p ([, ]) and x [, ], where the coefficients α n,k satisfy (2) and (3). If λ m = ϱ m = for every m =,..., n, then α n,k = ( n k), k =,..., n, whence the operator K n becomes the well-known nth Bernstein Kantorovich operator on L p ([, ]) (see, e.g., [3, p. 3]), in the sequel denoted by U n. We define (6) s(n) := max m n { λ m, ϱ m }, M := sup s(n). n Note that n ( n k) x k ( x) n k = and x k ( x) n k for every x [, ]. Hence by the convexity of the function t t p (p ) and Jensen s inequality applied to the measure (n + )dt, we get, for every f L p ([, ]), n ( ) n K n (f)(x) p s(n) p x k ( x) n k (n + ) f(t) p dt. k Consequently, the equality x k ( x) n k dx = ( ) n, k =,..., n, n + k yields K n (f) p s(n) f p and hence (7) K n (f) p s(n). On the other hand, if we take the function f = sign(λ n ) χ [,/(n+)], then f p = /(n + ) /p and K n p K ( ) /p n(f) p n + = λ n p /p λ n, f p np + from which Analogously, λ n p /p K n p. ϱ n p /p K n p. These last inequalities together with (7) lead us to the following result.
Bernstein Kantorovich-type operators 275 Proposition. The sequence ( K n p ) n N is bounded if and only if the sequences λ = (λ n ) n N and ϱ = (ϱ n ) n N are bounded. In this case (8) sup K n p M. n In the following, we assume that the sequences λ = (λ n ) n N and ϱ = (ϱ n ) n N are bounded. Consequently, the function w defined by (4) satisfies (9) w M. Observe that w is not necessarily continuous on [,]. More precisely, if λ n and ϱ n for every n, then the existence of the limit lim w(x) x + ( lim w(x), respectively) x is equivalent to the existence of the limit ( ) λ +... + λ n ϱ +... + ϱ n lim lim, respectively, n n n n and these two limits coincide (see, e.g. [5, Ch. 7, 5, pp. 226 229]). However, by (9), the operator K : L p ([, ]) L p ([, ]) defined by K(f) = w f for every f L p ([, ]) is continuous in the L p -norm and satisfies K p = w. Before stating our convergence results, we need some elementary formulas for Kantorovich operators. Using the following identities for Bernstein operators: B n () =, B n (id) = id, B n (id 2 ) = n n we obtain by direct computation U n () =, U n (id) = n n + id + 2(n + ), U n (id 2 ) = and consequently, for fixed x [, ], id2 + n id, n(n ) (n + ) 2 id2 + 2n (n + ) 2 id + 3(n + ) 2 () U n ((id x ) 2 )(x) = n (n + ) 2 x( x) + 3(n + ) 2 3n + 2(n + ) 2 4(n + ). Moreover, we give an explicit expression of the function K n () in terms of the assigned sequences (λ n ) n N and (ϱ n ) n N.
276 M. Campiti and G. Metafune Lemma 2. We have n () K n () = (λ m x( x) m + ϱ m x m ( x)) + λ n ( x) n + ϱ n x n. P r o o f. We proceed by induction on n ; if n =, () is obviously true. Supposing () true for n, we have by (2) and (3), K n+ () = n+ α n+,k x k ( x) n+ k = λ n+ ( x) n+ + ϱ n+ x n+ + (α n,k + α n,k )x k ( x) n+ k k= = λ n+ ( x) n+ + ϱ n+ x n+ n + ( x) α n,k x k ( x) n k + x α n,k x k ( x) n k k= = (λ n+ λ n )( x) n+ + (ϱ n+ ϱ n )x n+ + ( x) α n,k x k ( x) n k + x α n,k x k ( x) n k = (λ n+ λ n )( x) n+ + (ϱ n+ ϱ n )x n+ + K n () = (λ n+ λ n )( x) n+ + (ϱ n+ ϱ n )x n+ n + (λ m x( x) m + ϱ m x m ( x)) + λ n ( x) n + ϱ n x n n = λ n+ ( x) n+ + ϱ n+ x n+ + (λ m x( x) m + ϱ m x m ( x)) + λ n ( x) n x + ϱ n x n ( x) = λ n+ ( x) n+ + ϱ n+ x n+ + (λ m x( x) m + ϱ m x m ( x)) and hence () holds for n +. Theorem 3. The following statements are equivalent: (a) For every f L p ([, ]), the sequence (K n (f)) n N converges in the L p -norm; (b) The sequences (λ n ) n N and (ϱ n ) n N are bounded.
Bernstein Kantorovich-type operators 277 Moreover, if statement (a) or equivalently (b) is satisfied, then (2) lim n K n(f) w f p = for every f L p ([, ]). P r o o f. By the Banach Steinhaus theorem and Proposition, we only have to prove the implication (b) (a). By Proposition again, the sequence (K n ) n N is equibounded in the L p -norm, and therefore it is sufficient to show that lim n K n (f) w f p = for every f C([, ]). If f C([, ]), then (i) K n (f) w f p K n (f) f K n () + f K n () w p. By () and the inequality f(t) f(x) (+δ 2 (t x) 2 )ω(f, δ), where ω(f, δ) is the modulus of continuity of f, we get K n (f)(x) f(x) K n ()(x) ( ) n M x k ( x) n k (n + ) k Mω(f, δ) ( ) n x k ( x) n k (n + ) k ( Mω(f, δ) + ) δ 2. 4(n + ) Taking δ = / n +, we obtain K n (f) f K n () 5 4 Mω (f, f(t) f(x) dt ( + n + ). ) (t x)2 δ 2 dt Finally, we estimate the second term on the right-hand side of (i). By Lemma 2, we have K n ()(x) w(x) = λ n ( x) n + ϱ n x n (λ m x( x) m + ϱ m x m ( x)) m=n = ( x) n (λ n λ n+m )x( x) m + x n (ϱ n ϱ n+m )x m ( x) m= 2M(( x) n + x n ); m=
278 M. Campiti and G. Metafune this yields ( f K n () w f p 2M and the proof is complete. 4M = 4M ( (( x) n + x n ) p dx) /p f x np dx) /p f ( ) /p f np + It is well known that if f L p ([, ]) and x [, ] is a Lebesgue point for f, i.e., then (see [3, p. 3]) lim δ δ δ f(x + t) f(x) dt =, (3) lim n U n(f)(x) = f(x). In particular, lim U n(f) = f a.e. n Next we prove an analogous result for the operators K n. Proposition 4. If λ = (λ n ) n N and ϱ = (ϱ n ) n N are bounded sequences and f L p ([, ]), then (4) lim n K n(f)(x) = w(x) f(x) at every Lebesgue point x ], [. Consequently, lim n K n (f) = w f a.e. P r o o f. Let x ], [ be a Lebesgue point for f. Then K n (f)(x) f(x) K n ()(x) ( ) n M x k ( x) n k (n + ) k = MU n (u x )(x), f(t) f(x) dt where u x (t) := f(t) f(x). Since x is a Lebesgue point for u x and u x (x) =, by (3) it follows that lim n U n (u x )(x) = and hence lim n K n (f)(x) f(x) K n ()(x) =. Moreover, lim n f(x) K n ()(x) w(x) = since x ], [ and therefore (4) follows.
Bernstein Kantorovich-type operators 279 Finally, we state a quantitative estimate of the convergence in terms of the averaged modulus of smoothness τ(f, δ) p defined by (5) τ(f, δ) p := ( ) /p ω(f, x, δ) p dx for every f L p ([, ]), p <, and δ >, where (6) ω(f, x, δ) := sup{ f(t + h) f(t) t, t + h [x δ/2, x + δ/2] [, ]}. Denote by M([, ]) the space of all bounded measurable real functions on [,]. If L : M([, ]) M([, ]) is a positive operator satisfying L() = and (7) d = id 2 +L(id 2 ) 2 id L(id), it is well known that (8) L(f) f p 748τ(f, d) p for every f M([, ]) and p < (see, e.g., [4, Theorem 4.3]). In the case of Bernstein Kantorovich operators, the preceding inequality yields (9) U n (f) f p 748τ ( f, ) n + If L() is strictly positive, we may apply (8) to the operator L/L() and we have (2) L(f) f L() p L() L(f) L() f 748 L() τ(f, δ) p, p where (2) δ = id 2 L() + L(id 2 ) 2 id L(id) L(). Theorem 5. Assume that the sequences λ = (λ n ) n N and ϱ = (ϱ n ) n N are bounded. Then, for every n and f M([, ]), ( ) ( ) /p (22) K n (f) w f p Cτ f, + 4M f, n + np + where the constant C depends only on λ and ϱ (e.g., C = 683M). (i) P r o o f. For every f M([, ]), we write K n (f) w f p K n (f) f K n () p + f K n () w p and we estimate separately the two right-hand terms. p p.
28 M. Campiti and G. Metafune If c > M, then we consider K n,c = K n + c I which satisfies K n,c () > and (ii) K n,c (f) f K n,c () = K n (f) f K n (). By (9) and (2) we have (iii) K n,c (f) f K n,c () p 748 K n,c () τ(f, δ) p, where, by (), δ = id 2 K n,c () + K n,c (id 2 ) 2 id K n,c (id) K n,c () = id 2 K n () + K n (id 2 ) 2 id K n (id) K n,c () = sup K n ((id x ) 2 )(x) K n,c ()(x) x M c M sup U n ((id x ) 2 )(x) x M 4(c M)(n + ). Choosing c = 5 4 M, we obtain δ /(n + ) and K n,c() 9 4M. Consequently, by (ii) and (iii), it follows that K n (f) f K n () p 683Mτ ( f, ) n + The second term on the right-hand side of the inequality (i) has been already estimated in the proof of Theorem 3. p. References [] M. Campiti and G. Metafune, Approximation properties of recursively defined Bernstein-type operators, preprint, 994. [2],, Evolution equations associated with recursively defined Bernstein-type operators, preprint, 994. [3] G. G. Lorentz, Bernstein Polynomials, 2nd ed., Chelsea, New York, 986. [4] B. Sendov and V. A. Popov, The Averaged Moduli of Smoothness, Pure Appl. Math., Wiley, 988. [5] E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford, 939. Department of Mathematics Department of Mathematics University of Bari University of Lecce Via E. Orabona, 4 Via Arnesano 725 Bari, Italy 73 Lecce, Italy Reçu par la Rédaction le 6..994 Révisé le 2.4.995