On Compound Operators Constructed with Binomial and Sheffer Sequences∗

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 operators.

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 operators (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. Sablonnière using the generating function for the binomial sequences (see [16]). We mention that Sablonnière 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 authors: 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 function [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 approximation 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) . h i 5 Compound operators constructed with binomial and Sheﬀer sequences 139

Q Using the expression found by C. Manole in [11] for Tm e2, Q 2 Q Tm e2 (x)= x + x (1 − x) dm, 0 −2 with dQ = 1 − m−1 (Q ) pm−2(1) , we obtain m m pm(1) Q 2 Q Sm,r,se2 (x)= x + x (1 − x) Am,s,r,

m−sr 2dQ r2s2dQ Q ( ) m−sr+ s where Am,s,r = m2 . D 1 D m+sr(r−1) If Q = D then dm = m and Am,s,r = m2 . ∇α ∇α α 1+αm For Q = α we have dm = (1+α)m so it follows that

∇α 2 α sr (1+αs)+(m−sr)(1+α(m−sr)) Am,s,r = (1+α)m2 . Q From the relations (4) it results that (Sm,r,se2)(x) converges to e2(x) if Q Q Q Am,s,r → 0, as m → ∞. But, if dm → 0, then Am,s,r → 0, so using the Bohman–Korokvin criterion of convergence we have the following result.

Theorem 4. Let Q be a delta operator with the basic sequence (pm (x))m≥0, 0 Q pm (1) 6= 0 and pm (0) ≥ 0, for every positive integer m. If dm → 0 then the Q sequence of linear and positive operators Sm,r,sf converges to the function f, uniformly on the interval [0, 1]. Now we establish an estimate for the order of approximation of a func- Q tion f ∈ C [0, 1] by means of the operator Sm,r,s using the ﬁrst modulus of continuity. Taking into account an inequality proved by O. Shisha and B. Mond (see [17]), we can write

Q 1 Q 2 f(x) − Sm,r,sf (x) ≤ 1+ δ2 Sm,r,s((t − x) ; x) ω1 (f; δ) . h Q i Using the expressions obtained for Sm,r,sei, for i = 0, 1, 2, we obtain that Q 2 Q 1 Sm,r,s((t − x) ; x) = x(1 − x)Am,s,r. Taking into account that x(1 − x) ≤ 4 , Q ∀x ∈ [0, 1] and replacing δ by Am,s,r, we obtain that q Q 5 Q f(x) − Sm,r,sf (x) ≤ 4 ω1 f; Am,s,r . q Example 1. If we consider the delta operator T = ln(I + D), its basic sequence is the sequence of exponential polynomials: m k ϕm (x)= S (m, k) x , Xk=1 140 Maria Cr˘aciun 6 where S (m, k) = [0, 1, . . . , k; em] are the Stirling numbers of second kind. In this case, Manole obtained dT = 1 + m−1 ϕm−1(1) , m m m ϕm(1) and he proved that there exist two positive constants c1 and c2 such that c ln m ≤ ϕm−1(1) ≤ c ln m . 1 m ϕm(1) 2 m

Hence, ϕm−1(1) → 0, as m → ∞, which implies dT → 0. Consequently, ST f ϕm(1) m m,r,s deﬁned by m−sr s − − ST = ϕk(x)ϕm−sr−k(1 x) ϕj (x)ϕs−j (1 x) f k+jr m,r,s ϕm−sr(1) ϕs(1) m k=0 j=0 X X converges to the function f, uniformly on the interval [0, 1] .

3. EVALUATION OF THE REMAINDER

Using Peano’s theorem, the remainder of the approximation formula Q Q (5) f (x)= Sm,r,sf (x)+ Rm,r,sf (x) for f ∈ C2 [0, 1] can be represented as 1 Q Q 00 (6) Rm,r,sf (x)= Gm,r,s(t; x)f (t) dt, Z0 Q where Gm,r,s (t; x) is the Peano kernel deﬁned by Q Q 1 Gm,r,s (t; x)= Rm,r,sϕx (t) , ϕx (t) = (x − t)+ = 2 [x − t + |x − t|] . Since the expression m−sr s Q Q Q k+jr Gm,r,s (t; x) = (x − t)+ − pm−sr,k (x) ps,j (x) m − t + k=0 j=0 X X is negative, one can apply the mean value theorem to the integral from (6) and we obtain that there exists ξ ∈ [0, 1] such that 1 Q 00 Q Rm,r,sf (x)= f (ξ) Gm,r,s (t; x) dt. Z0 Taking f (x)= x2 in the previous relation, we obtain that

1 2 Q 2 2 Q Q 1 Q 1 (m−sr) dm−sr+r s ds Gm,r,s (t; x) dt = 2 Rm,r,se2 (x)= − 2 x (1 − x) m2 , Z0 7 Compound operators constructed with binomial and Sheﬀer sequences 141 so it follows that, for every function f ∈ C2[0, 1], the remainder in formula (5) is of the following form:

Q x(x−1) 2 Q 2 2 Q 00 Rm,r,sf (x)= 2m2 (m − sr) dm−sr + r s ds f (ξ) . h i 4. COMPOUND SHEFFER OPERATORS

Let Q be a delta operator with the basic sequence (pm(x)), S a shift invari- −1 ant and invertible operator and sm = S pm a Sheffer sequence. We can also generalize the operator defined in (3), by considering another compound approximation operator containing a Sheffer sequence (additionally to the basic sequence pm):

m−sr s Q,S Q,S Q,S k+jr (7) Sm,r,sf (x)= wm−sr,k (x) ws,j (x) f m , k=0 j=0 X X − where wQ,S (x)= n pk(x)sn−k(1 x) . n,k k sn(1) When S = I this operator reduces to the operator deﬁned by (3). Q,S For s = 0, Sm,r,0 is in fact the operator that we studied in our paper [3], Q,S m Q,S k Lm f (x)= k=0 wm,k (x) f m . We remind that for this operator we have obtained the following expressions for the test functions P Q,S Lm e0 = e0 Q,S (8) Lm e1 (x)= ame1 (x) Q,S 2 Lm e2 (x)= bmx + x (am − bm − cm) , where

0 −2 − 0 0 −1 0 −2 Q S 1 p (Q ) s (1) [(Q ) sm−2](1) ( ) ( ) m−2 a = m−1 , b = m−1 , c = m−1 . m sm(1) m m sm(1) m m h sm(1) i

0 N Q,S If pn (0) ≥ 0 and sn (0) ≥ 0, ∀n ∈ , then wn,k (x) ≥ 0, ∀x ∈ [0, 1] (see [3]) Q,S and so the operator Sm,r,sf is a positive approximation operator. Q,S For the operator (7) we have Sm,r,sf (0) = f(0). SQ,S In the following we compute the values of the operator m,r,s for the test functions. From the convolution-type relation (1) satisﬁed by a Sheﬀer sequence, it is obvious that Q,S Sm,r,se0 = e0. 142 Maria Cr˘aciun 8

For e1 we have m−sr Q,S Q,S k Q,S rs Q,S Sm,r,se1 (x)= wm−sr,k(x) m Ls e0 (x)+ m Ls e1 (x) k=0 X h i m−sr Q,S asrsx Q,S = m Lm−sre1 (x)+ m Lm−sre0 (x), and using the relations (8) we obtain that

Q,S (m−sr)am−sr+rsas Sm,r,se1 (x)= x m .

Finally, for e2 we have Q,S 1 2 Q,S Sm,r,se2 (x)= m2 (m − sr) Lm−sre2 (x) h Q,S Q,S + 2rs(m − sr) Lm− sre1 (x) Ls e1 (x) 2 2 Q,S + r s Ls e2 (x) . Using again the relations (8) we can rewrite thei last expression as Q,S Sm,r,se2 (x)= 1 2 2 2 2 = m2 x (m − sr) bm−sr + 2sr(m − sr)asam−sr + s r bs n 2 2 2 + x (m − sr) (am−sr − bm−sr − cm−sr)+ s r (as − bs − cs) Q,S o In [3] we proved that if Lm is a positive operator then

1−bm 2 0 ≤ cm ≤ min 2 ,am − am . So, if limm→∞ am = limm→∞ bm = 1, then limm→∞ cm = 0. Taking into account the previous relations and applying the Bohman–Korokvin criterion convergence we can state the following result.

Theorem 5. If f ∈ C [0, 1] and limm→∞ am = limm→∞ bm = 1, then the sequence of compound operators constructed with Sheﬀer sequences deﬁned by (7) converges uniformly to f on the interval [0, 1] . Examples. 1. If we consider the special case when Q = D, then in the D,S expression of Sm,r,sf, instead of sm, there appears an Appell sequence Am. Because the Pincherle derivative of D is the identity operator I, we have

Am−1(1) m−1 m−1 a = , b = a a − , c = a (1 − a − ) , m Am(1) m m m m 1 m m m m 1

D,S Am−1(1) so the condition for the convergence of the operator Sm,r,s is lim = 1. m→∞ Am(1) 2. If we take the Gould delta operator 1 −β 1 −β −α−β G = α E 5α = α E − E 9 Compound operators constructed with binomial and Sheﬀer sequences 143 and the invertible operator α+β 0 1 α S = E G = α (α + β) I − βE then the corresponding basic sequence and Sheﬀer sequence are [m−1,−α] [m,−α] pm (x)= x (x + α + mβ) , resp. sm (x) = (x + mβ) , so in this case m−k,−α wG,S(x)= 1 m x (x + α + kβ)[k−1,−α] 1 − x + (m − k) β [ ]. m,k (1+mβ)[m,−α] k We mention that the operator m G,S G,S k Lm f (x)= wm,k (x)f m k=0 X was studied by G. Moldovan (see for example [13]). mβ mβ If α → 0, mβ → 0, α → 0, as m → ∞, or mα → 0, mβ → 0, α → c, G,S as m → ∞, then the sequence of operators Lm converges uniformly to f on [0, 1] . G,S It can be easily proved that, in the same conditions, the operator Sm,r,s converges also uniformly to f on [0, 1] .

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Received by the editors: April 13, 2003.