Optimal Simultaneous Approximation via A-Summability

and Applied Analysis 3 t ∈ (a, b), and directly from Lemma 2, if we takeZ ∈ C[a, b] such thatDZ(t) = z(t) for all t ∈ (a, b), we derive that A k,n Dm∘L (x) ≤ A k,n Dm∘L (x) + o (n) (λ −1 k ) , (11) or equivalently A k,n Dm∘L (x) − D m Z (x) ≤ A k,n Dm∘L (x) − D m f (x) + o (n) (λ −1 k ) . (12) Finally we apply Lemma 3 to the fuction Z and obtain the required inequality as follows: A k,n Dm∘L (x) ≥ D m f (x) + o (n) (λ −1 k ) . (13) To prove the conversewe assume the contrary; that is, that D m f is not convexwith respect to 0 1 on (a, b); then there exist three points 1 x2, and s such that a < 1 < s < 2 < b, S (D m f, 1 2 (s) < D m f (s) , (14) where S(Df, 1 2 is the unique function of the space 0 1 which interpolatesD m f at 1 and x2. Now we apply Lemma 1 with h = 2 and g = D m f − S(D m f, 1 2 and derive the existence of ε < 0, a solution ?̂? ofDz ≡ 0 and 1 ∈ 1 2 satisfying 2 (t) + ?̂? (t)≥D m f (t) − S (D m f, 1 2 (t) , t ∈ 1 2 (15) 2 1 + ?̂? 1 = D m f 1 − S (D m f, 1 2 1 . (16) Let us take 2 ̂ Z, ̂ S ∈ C m [a, b] such that DmU2 = u2, D m̂ Z = ?̂? andD̂ S = S(Df, 1 2 on (a, b) and apply then Lemma 2 taking into account (15). This yields that εA k,n Dm∘L2 1 +A k,n Dm∘L ̂ Z 1 ≥ A k,n Dm∘L 1 −A k,n Dm∘L ̂ S 1 + o (n) (λ −1 k ) . (17) After introducing equality (16) we get ε (A k,n Dm∘L2 1 − D m 2 1 +A k,n Dm∘L ̂ Z 1 − D m ̂ Z 1 ≥ A k,n Dm∘L 1 − D m f 1 − (A k,n Dm∘L ̂ S 1 − D m ̂ S 1 + o (n) (λ −1 k ) . (18) Finally, multiplying by k and applying (P2) we obtain the following inequality which contradicts our assumption: ε ≥ A k,n Dm∘L 1 − D m f 1 + o (n) (λ −1 k ) . (19) (b) If A Dm∘L f(x) ≥ A Dm∘L f(x) + o (n) (λ −1 k ) for x ∈ (a, b), then directly from (P0) we have thatA Dm∘L f(x) ≥ D m f(x)+ o (n) (λ −1 k ) and it suffices to use (a) to complete the proof. Proposition 5. Let M ≥ 0 and let f,w ∈ C[a, b]. Then the following items are equivalent (i) MDw ± Df are convex with respect to 0 1 on (a, b), (ii) for each x ∈ (a, b) 󵄨 󵄨 󵄨 󵄨 󵄨 A k,n Dm∘L (x) − D m f (x) 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ M(A k,n Dm∘L (x) − D m w (x)) + o (n) (λ −1 k ) . (20) Proof. It suffices to apply Proposition 4 replacing f byMw± f. With appropriate choices of the function w and applying the results of [11], we give two saturation results; the first one is stated in terms of classic Lipschitz spaces, while the second one puts across the relationship with the asymptotic formula. Theorem 6. Let f ∈ C[a, b]. Then k 󵄨 󵄨 󵄨 󵄨 󵄨 A k,n Dm∘L (x) − D m f (x) 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ M


Introduction
The notion of almost convergence of a sequence introduced by Lorentz [1] in 1948 entered the Korovkin-type approximation theory (see [2]) through the papers of King and Swetits [3] and Mohapatra [4].A step forward was given by Swetits [5] in 1979 who applied in the theory the more general notion of A-summability that Bell [6] had introduced a few years earlier.
After Swetits, within a shape preserving approximation setting and using as well A-summability, one finds in the literature two recent papers of the authors, [7,8], where they studied, on one hand, qualitative and quantitative Korovkintype results, and on the other, results on asymptotic formulae.In this paper we continue this line of work which naturally takes us to the topic of saturation.Indeed, after having established an asymptotic formula, a natural way to keep on is to study optimal results to control the goodness of the approximation errors.Here saturation enters the picture.Now, before detailing our specific aim, we present the general framework of the paper which includes the definition of Asummability.
Let A := { () } = { ()   } be a sequence of infinite matrices with nonnegative real entries; then a sequence of real numbers {  } is said to be A-summable to ℓ if (whenever the series below converges for all  and ) ()     = ℓ uniformly for  ∈ N = {1, 2, . ..} .( uniformly in . The asymptotic formula (3) informs us that the order of convergence of A ,   ∘L () towards   () is not better than  −1  if the right-hand side of (3) is different from 0. Thus,  −1  is called the optimal order of convergence, and those functions that possess it form the saturation class.As for our specific aim with this paper, the results of Section 2 give us information about this saturation class, while Section 3 is devoted to state a sort of converse result of asymptotic formulae.We follow the line of two respective papers of two of the authors, namely [9,10], which at the same time have their foundations on two outstanding papers of Lorentz and Schumaker [11] and Berens [12].The last section of the paper contains some applications.Now we close this one with some remarks and notation that we will use throughout the paper.
In this respect, we refer the reader to [11] to recall the class Lip   1,  ≥ 0, formed by those functions , differentiable on (, ), fulfilling where Finally, if  ()  is a double sequence of real numbers such that lim  → +∞  ()  = 0 uniformly in  ∈ N and   is another sequence of real numbers with lim  → +∞   = 0, then we use the notation

Saturation Results
In this section we obtain local saturation results in the approximation process of A ,   ∘L () towards   ().Firstly we state without proof three lemmas; Lemma 1 coincides with [10, Lemma 1], Lemma 2 follows the same pattern as [10, Lemma 2], and finally Lemma 3 is a very direct consequence of (P1).
Then there exist a real number  < 0, a solution of the differential equation (4) on , say , and a point  ∈ ( 1 ,  2 ) such that ℎ() + () = () and, for all  ∈ ( 1 ,  2 ), ℎ() + () ≥ ().Lemma 2. Let ,  ∈   [, ] and let  ∈ (, ).Assume that there exists a neighborhood   of  where    ≤   .Then Lemma 3.  ∈   [, ] is a solution of the differential equation (4) in some neighborhood of  ∈ (, ) if and only if The following two propositions, of interest by themselves, prepare the way to prove the announced results.An important role is played by the notion of convexity with respect to the extended complete Tchebychev system { 0 ,  1 } that here we relate to the monotonic convergence of the process and allows us to compare the degree of approximation for two different functions.
only if for each  ∈ (, ) Proof.(a) Let  ∈ (, ).Assume that    is convex with respect to { 0 ,  1 } on (, ) and let  ∈ ⟨ 0 ,  1 ⟩ such that (here  1 + denotes the right first derivative operator).Then, from [11, Lemma 2.2], we have that () ≤   () for all  ∈ (, ), and directly from Lemma 2, if we take  ∈   [, ] such that   () = () for all  ∈ (, ), we derive that or equivalently Finally we apply Lemma 3 to the fuction  and obtain the required inequality as follows: To prove the converse we assume the contrary; that is, that    is not convex with respect to { 0 ,  1 } on (, ); then there exist three points  1 ,  2 , and  such that where (  ,  1 ,  2 ) is the unique function of the space ⟨ 0 ,  1 ⟩ which interpolates    at  1 and  2 .Now we apply Lemma 1 with ℎ =  2 and  =    − (  ,  1 ,  2 ) and derive the existence of  < 0, a solution ẑ of D ≡ 0 and After introducing equality (16) we get Finally, multiplying by   and applying (P2) we obtain the following inequality which contradicts our assumption: With appropriate choices of the function  and applying the results of [11], we give two saturation results; the first one is stated in terms of classic Lipschitz spaces, while the second one puts across the relationship with the asymptotic formula.

Converse Result of the Asymptotic Formula
This section is devoted to give a converse result of the asymptotic formula stated in (3).It turns to be an extension of the results of [12].A rough statement of the problem would read as follows: under the general framework of the paper, assume the existence of a function  such that for  ∈   [, ], Is +2-times differentiable at ? is it true that D(  ) = ?
The answer, affirmative in certain sense, represents the content of this section.We will make use of two lemmas.We state them without proof as they resemble closely [10, Lemmas 3,4].
Lemma 9. Let  ∈ (, ) and let  ∈   [, ] such that for all  ∈ (, )   () =  0 () ∫ whose existence is guaranteed from the theory of Lebesgue integration (see e.g., [14]).In particular it follows that lim sup from where the proof follows recalling the definition of Ψ at the top of the proof, the one of D in (4), and finally using (P2).

Applications
In this section we illustrate the use of some of the results of the paper.We will make use of the asymptotic formulae obtained in [8, Section 3] to state some saturation results for the classical Bernstein operators and for a modification of them.Here we consider almost convergence, as a particular case of A-summability.We refer the reader to [8, Subsections 3.1, 3.2] for further details.