REVUE D’ANALYSE NUMERIQUE´ ET DE THEORIE´ DE L’APPROXIMATION Tome 32, No 2, 2003, pp. 135–144 ON COMPOUND OPERATORS CONSTRUCTED WITH BINOMIAL AND SHEFFER SEQUENCES∗ MARIA CRACIUN˘ † Abstract. In this note we consider a general compound approximation operator using binomial sequences and we give a representation for its corresponding remainder term. We also introduce a more general compound approximation operator using Sheffer sequences. We provide convergence theorems for both studied operators. MSC 2000. 41A36, 05A40. Keywords. Sequences of binomial type, Sheffer sequences, compound opera- tors. 1. INTRODUCTION A sequence of polynomials (pm (x))m≥0 is called a sequence of binomial type if deg pm = m, ∀m ∈ N, and satisfies the identities m m pm (x + y)= k pk (x) pm−k (1 − x) . k=0 X We will denote by Ea the shift operator defined by (Eap) (x) = p (x + a), for every polynomial p and every real number x. A linear operator T is said to be shift invariant if it commutes with the shift operator Ea, for every real number a. Sequences of binomial type are connected with the notion of theta ope- rators (J. F. Steffensen [27], [28]) which were called delta operators by F. B. Hildebrand [7] and G.-C. Rota and his collaborators [15]. A delta operator Q is a shift invariant operator for which Qx = const. 6= 0. Definition 1. Let Q be a delta operator. A sequence (pm (x))m≥0 is a sequence of basic polynomials for Q (basic sequence, for short) if: i) p0 = 1, ii) pm (0) = 0, if m ≥ 1, iii) Qpm = mpm−1, if m ≥ 1. ∗This work has been supported by the Romanian Academy under grant GAR 15/2003. †“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Ro- mania, e-mail: [email protected]. 136 Maria Cr˘aciun 2 For every delta operator there exists a unique basic sequence. A polynomial sequence is a sequence of binomial type if and only if it is the sequence of basic polynomials for a delta operator. Sequences of binomial type were called poweroids by Steffensen [27], because the action of any delta operator on the binomial sequence, which is its basic sequence, is the same as the action of the derivative D on xm. Definition 2. A sequence of polynomials (sm (x))m≥0 is called a Sheffer sequence for Q if: i) s0 = const. 6= 0, ii) Qsm = msm−1, if m ≥ 1. It is known [15] that if (sm (x))m≥0 is a Sheffer sequence for a delta operator Q with the basic sequence (pm (x))m≥0 then there exists a shift invariant and −1 invertible operator S such that sm = S pm, ∀m ∈ N, so every pair (Q, S) gives us a unique Sheffer sequence. A Sheffer sequence satisfies the relations m m N (1) sm (x + y)= k pk (x) sm−k (1 − x) , ∀m ∈ . k=0 X A Sheffer sequence for the usual derivative D is an Appell sequence. The Umbral Calculus allows a unified and simple study of binomial, Appell and Sheffer sequences. More details about Umbral Calculus can be found in [15], [5] and [6]. T. Popoviciu proposed in [14] the use of binomial sequences in order to construct a class of approximation operators of the form m Q 1 m k (2) T f (x)= p (x) p − (1 − x) f , m pm(1) k k m k m k=0 X for every function f ∈ C [0, 1] . This kind of operators and their generalizations were intensively studied. They interpolate the function f at 0 and 1 and preserve the polynomials of Q degree one. The expressions for Tm en, n ≥ 2, were computed by C. Manole (see [10] and [11]) using the umbral calculus and later by P. Sablonni`ere using the generating function for the binomial sequences (see [16]). We mention that Sablonni`ere called them Bernstein–Sheffer operators while D. D. Stancu called them binomial operators of Tiberiu Popoviciu type. Q Different results regarding the operator Tm were obtained by several au- thors: D. D. Stancu and M. R. Occorsio found representations for the re- Q Q mainder in the approximation formula f(x) = (Tm f)(x) + (Rmf)(x) [24]; V. Q Mihe¸san proved that Tm preserve the Lipschitz constant for a Lipschitz func- tion [12]; D. D. Stancu and A. Vernescu studied bivariate operators of this 3 Compound operators constructed with binomial and Sheffer sequences 137 Q type [26]; O. Agratini considered a generalization of Tm in the Kantorovich sense [1]; L. Lupa¸sand A. Lupa¸sintroduced and studied a modified operator of binomial type replacing x by mx and 1 by m [9], [8]. More details about the role of the binomial polynomials in the Approximation Theory can be found in [2], [8] and [24]. 2. COMPOUND POWEROID OPERATORS Let Q be a delta operator with the basic sequence (pm (x))m≥0 . If pm (1) 6= 0, ∀m ∈ N, for every function f ∈ C [0, 1] we consider the general approxima- tion operator defined by m−sr s Q Q Q k+jr (3) Sm,r,sf (x)= pm−sr,k (x) ps,j (x) f m , k=0 j=0 X X − where pQ (x)= n pk(x)pn−k(1 x) , while s and r are two nonnegative integers n,k k pn(1) satisfying the condition 2sr ≤ m. 0 N Q If pm (0) ≥ 0, ∀m ∈ , then pm (x) ≥ 0, ∀x ∈ [0, 1] , so the operator Sm,r,sf is a positive approximation operator. Different instances of this compound poweroid operator were previously studied by D. D. Stancu and his collaborators as follows: k 1. For Q = D, pk (x)= x , s = 1 the corresponding compound operator was introduced and studied by D. D. Stancu (see [18]); if s is arbitrary, D α,β the operator Sm,r,s is a special case of the operator Lm,r1,...,rs , consid- D α,β ered by D. D. Stancu in [19] (in fact Sm,r,s is obtained from Lm,r1,...,rs when α = β = 0 and r1 = r2 = ... = rs = r); 1 I−E−α α [k,−α] 2. The case obtained for Q = α ∇α = α ,pk (x)= x was studied by D. D. Stancu and J. W. Drane in [23]; 3. D. D. Stancu and A. C. Simoncelli studied in [25] the compound powe- 1 −β 1 −β −α−β α,β roid operator for Q = α E ∇α = α E − E , pk (x) = x (x + α + kβ)[k−1,−α]. They proved that if α = α (m) → 0, mβ (m) → α,β 0, as m → ∞, then (Sm,r,sf) converges uniformly to f on the interval [0, 1] . Using the Peano theorem, the authors also gave a representation α,β of the remainder Rm,r,sf for the approximation formula α,β α,β f (x) = (Sm,r,sf)(x) + (Rm,r,sf)(x) D. D. Stancu considered also a class of linear positive compound operators α,β,γ,δ Sm,r,s f (see [22]) with modified knots defined by the following relation m−sr s α,β,γ,δ α,β α,β k+jr+γ Sm,r,s f (x)= pm−sr,k(x) ps,j (x)f m+δ , k=0 j=0 X X 138 Maria Cr˘aciun 4 where 0 ≤ γ ≤ δ. Q Q If s =0 or r = 0 then Sm,r,0 and Sm,0,s reduce to the binomial operator of Q T. Popoviciu Tm , defined by (2). From the definition of a basic sequence it results that Q 1, if k = 0 Q 1, if k = n pn,k (0) = and pn,k(1) = (0, if k 6= 0 (0, if k 6= n. Q Using these relations we obtain that the polynomial Sm,r,sf interpolates f Q Q at both sides of the interval [0, 1], that is, Sm,r,sf (0) = f(0), Sm,r,sf (1) = f(1). Q Lemma 3. The values of the operator Sm,r,s for the test functions are Q Sm,r,se0 (x)= e0 (x) , Q (4) Sm,r,se1 (x)= e1 (x) , Q 2 Q Sm,r,se2 (x)= x + x (1 − x) Am,s,r, 2 Q 2 2 Q −2 Q (m−sr) d +r s ds Q (Q0) p (1) where A = m−sr and d = 1 − m−1 m−2 . m,s,r m2 m m pm(1) Proof. From the definition of a sequence of binomial type we have that m Q k=0 pm,k (x) = 1, so it follows that P m−sr s Q Q Q Sm,r,se0 (x)= pm−sr,k (x) ps,j (x)=1= e0 (x) , k=0 j=0 X X m−sr s s Q 1 Q Q Q Sm,r,se1 (x)= m pm−sr,k(x) k ps,j (x)+ r jps,j (x) j=0 j=0 Xk=0 h i m−sr P P 1 Q Q Q = m pm−sr,k(x) k Ts e0 (x)+ rs Ts e1 (x) Xk=0 h i 1 Q Q = m (m − sr)(Tm−sre1)(x)+ xrs(Tm−sre0) (x) 1 = m (m − sr)x + rsx = x. Q Hence the operator Sm,r,s preserves the polynomials of degree one. Analogously, we obtain that Q Sm,r,se2 (x)= 1 2 Q 2 2 2 Q = m2 (m − sr) Tm−sre2 (x)+2(m − sr) srx + r s Ts e2 (x) .