Return to Article Details On approximation by binomial operators of Tiberiu Popoviciu type

REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATION
Tome XXVII, № 1, 1998, pp. 167-181

ON APPROXIMATION BY BINOMIAL OPERATORS OF TIBERIU POPOVICIU TYPE

D. D. STANCU, M. R. OCCORSIO

1. INTRODUCTION

  1. It is known that an important class of polynomial sequences ( p m ) m ≥ 0 p m m ≥ 0 (p_(m))_(m >= 0)\left(p_{m}\right)_{m \geq 0}(pm)m≥0 occurring in combinatorics and analysis is represented by the sequences of binomial type (B.T.), for which we have p 0 = 1 , deg p m = m p 0 = 1 , deg p m = m p_(0)=1,degp_(m)=mp_{0}=1, \operatorname{deg} p_{m}=mp0=1,degpm=m and the following equalities
(1.1) p m ( u + v ) = ∑ k = 0 m ( m k ) p k ( u ) p m − k ( v ) (1.1) p m ( u + v ) = ∑ k = 0 m   ( m k ) p k ( u ) p m − k ( v ) {:(1.1)p_(m)(u+v)=sum_(k=0)^(m)((m)/(k))p_(k)(u)p_(m-k)(v):}\begin{equation*} p_{m}(u+v)=\sum_{k=0}^{m}\binom{m}{k} p_{k}(u) p_{m-k}(v) \tag{1.1} \end{equation*}(1.1)pm(u+v)=∑k=0m(mk)pk(u)pm−k(v)
are identically satisfied in u u uuu and v v vvv for any nonnegative integer m m mmm.
One can obtain sequences of B.T. by using generating functions methods and by using operatorial or umbral methods.
It is known (cf., e.g., A. M. Garsia [2]) that ( p m p m p_(m)p_{m}pm ) is a sequence of B.T. if and only if it is defined by a generating relation
(1.2) [ ϕ ( t ) ] x = e x φ ( t ) = ∑ m = 0 ∞ p m ( x ) t m m ! , (1.2) [ ϕ ( t ) ] x = e x φ ( t ) = ∑ m = 0 ∞   p m ( x ) t m m ! , {:(1.2)[phi(t)]^(x)=e^(x varphi(t))=sum_(m=0)^(oo)p_(m)(x)(t^(m))/(m!)",":}\begin{equation*} [\phi(t)]^{x}=e^{x \varphi(t)}=\sum_{m=0}^{\infty} p_{m}(x) \frac{t^{m}}{m!}, \tag{1.2} \end{equation*}(1.2)[ϕ(t)]x=exφ(t)=∑m=0∞pm(x)tmm!,
where
(1.3) ϕ ( t ) = 1 + a 1 t + a 2 t 2 + … , φ ( t ) = c 1 t + c 2 t 2 + … ( c 1 ≠ 0 ) . (1.3) ϕ ( t ) = 1 + a 1 t + a 2 t 2 + … , φ ( t ) = c 1 t + c 2 t 2 + … c 1 ≠ 0 . {:(1.3)phi(t)=1+a_(1)t+a_(2)t^(2)+dots","varphi(t)=c_(1)t+c_(2)t^(2)+dotsquad(c_(1)!=0).:}\begin{equation*} \phi(t)=1+a_{1} t+a_{2} t^{2}+\ldots, \varphi(t)=c_{1} t+c_{2} t^{2}+\ldots \quad\left(c_{1} \neq 0\right) . \tag{1.3} \end{equation*}(1.3)ϕ(t)=1+a1t+a2t2+…,φ(t)=c1t+c2t2+…(c1≠0).
Sequences of B.T. have remarkable algebraic-combinatorial properties and many practical applications.
In 1931 the great Romanian mathematician Tiberiu Popoviciu [12] already had the wonderful idea to indicate a method for constructing linear polynomial operators, useful in constructive theory of functions, by means of sequences of B.T.
If in the identity (1.1) we set u = x u = x u=xu=xu=x and v = 1 − x v = 1 − x v=1-xv=1-xv=1−x, we obtain the identity
(1.4) ∑ k = 0 m ( m k ) p k ( x ) p m − k ( 1 − x ) = p m ( 1 ) (1.4) ∑ k = 0 m   ( m k ) p k ( x ) p m − k ( 1 − x ) = p m ( 1 ) {:(1.4)sum_(k=0)^(m)((m)/(k))p_(k)(x)p_(m-k)(1-x)=p_(m)(1):}\begin{equation*} \sum_{k=0}^{m}\binom{m}{k} p_{k}(x) p_{m-k}(1-x)=p_{m}(1) \tag{1.4} \end{equation*}(1.4)∑k=0m(mk)pk(x)pm−k(1−x)=pm(1)
It suggests the introduction of an operator T m T m T_(m)T_{m}Tm, of Tiberiu Popoviciu [12], associated to a function f : [ 0 , 1 ] → R f : [ 0 , 1 ] → R f:[0,1]rarr Rf:[0,1] \rightarrow Rf:[0,1]→R, by means of the formula
(1.5) ( T m f ) ( x ) = 1 p m ( 1 ) ∑ k = 0 m ( m k ) p k ( x ) p m − k ( 1 − x ) f ( k m ) , (1.5) T m f ( x ) = 1 p m ( 1 ) ∑ k = 0 m   ( m k ) p k ( x ) p m − k ( 1 − x ) f k m , {:(1.5)(T_(m)f)(x)=(1)/(p_(m)(1))sum_(k=0)^(m)((m)/(k))p_(k)(x)p_(m-k)(1-x)f((k)/(m))",":}\begin{equation*} \left(T_{m} f\right)(x)=\frac{1}{p_{m}(1)} \sum_{k=0}^{m}\binom{m}{k} p_{k}(x) p_{m-k}(1-x) f\left(\frac{k}{m}\right), \tag{1.5} \end{equation*}(1.5)(Tmf)(x)=1pm(1)∑k=0m(mk)pk(x)pm−k(1−x)f(km),
where x ∈ [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1] and m = N m = N m=Nm=\mathbb{N}m=N, with the assumption: p m ( 1 ) ≠ 0 p m ( 1 ) ≠ 0 p_(m)(1)!=0p_{m}(1) \neq 0pm(1)≠0.
It should be mentioned that, in fact, in front of the above sum he chose the factor 1 / a m 1 / a m 1//a_(m)1 / a_{m}1/am, but it is easy to see that we have a m = p m ( 1 ) a m = p m ( 1 ) a_(m)=p_(m)(1)a_{m}=p_{m}(1)am=pm(1) (we only have to replace in (1.2) x = 1 x = 1 x=1x=1x=1 and then to identify the coefficients of t m t m t^(m)t^{m}tm ).
According to a result of Tiberiu Popoviciu [12], found later also by P. Sablonnière [16], we have p n ( x ) ≥ 0 p n ( x ) ≥ 0 p_(n)(x) >= 0p_{n}(x) \geq 0pn(x)≥0 on [ 0 , 1 ] ( n = 0 , 1 , … , m ) [ 0 , 1 ] ( n = 0 , 1 , … , m ) [0,1](n=0,1,dots,m)[0,1](n=0,1, \ldots, m)[0,1](n=0,1,…,m) if and only if the coefficients c k c k c_(k)c_{k}ck from (1.3) are nonnegative. In this case the linear operator T m T m T_(m)T_{m}Tm is of positive type.
It is obvious that in the case of binomial polynomials p r ( x ) = e r ( x ) == x r ′ ( r ∈ N 0 ) T m p r ( x ) = e r ( x ) == x r ′ r ∈ N 0 T m p_(r)(x)=e_(r)(x)==x^(r^('))(r inN_(0))T_(m)p_{r}(x)=e_{r}(x)= =x^{r^{\prime}}\left(r \in \mathbb{N}_{0}\right) T_{m}pr(x)=er(x)==xr′(r∈N0)Tm reduces to the operator B m B m B_(m)B_{m}Bm of Bernstein.
In this paper we shall consider some more general binomial-type operators and we shall study their approximation properties, including the estimation of the orders of approximation by means of the first and second moduli of smoothness, as well as representations of the remainder term in approximation formulas by the Tiberiu Popoviciu-type operators.

2. USE OF UMBRAL METHODS FOR OBTAINING BINOMIAL-TYPE POLYNOMIALS

  1. The umbral methods are associated with the names of S . Pincherle, E. T. Bell, J. L. W. Jensen, J. I. Sylvester, J. F. Steffensen, G.-C. Rota, R. Mullin, S. Roman, A. M. Garsia, and others.
The operators Ω Ω Omega\OmegaΩ, considered in 1902 by Jensen [6], include the shift operator E a E a E^(a)E^{a}Ea, defined by ( E a f ) ( x ) = f ( x + a ) E a f ( x ) = f ( x + a ) (E^(a)f)(x)=f(x+a)\left(E^{a} f\right)(x)=f(x+a)(Eaf)(x)=f(x+a), the central mean operator μ h μ h mu_(h)\mu_{h}μh, defined by ( μ h f ) ( x ) = 1 2 [ f ( x + h 2 ) + f ( x − h 2 ) ] μ h f ( x ) = 1 2 f x + h 2 + f x − h 2 (mu_(h)f)(x)=(1)/(2)[f(x+(h)/(2))+f(x-(h)/(2))]\left(\mu_{h} f\right)(x)=\frac{1}{2}\left[f\left(x+\frac{h}{2}\right)+f\left(x-\frac{h}{2}\right)\right](μhf)(x)=12[f(x+h2)+f(x−h2)] and the integration operator.
An operator T T TTT which commutes with all shift operators is called a shiftinvariant operator, i.e., T E a = E a T T E a = E a T TE^(a)=E^(a)TT E^{a}=E^{a} TTEa=EaT.
A special class of omega operators is represented by the theta operators θ θ theta\thetaθ (a term introduced in 1927 by Steffensen [23]) or delta operators, denoted by Q Q QQQ (a term suggested in 1956 by E. B. Hildebrand [5]) and used intensively by G.-C. Rota and his collaborators ([2], [11], [14] and [15]).
A theta operator θ θ theta\thetaθ is a shift-invariant operator for which θ e 1 θ e 1 thetae_(1)\theta e_{1}θe1 is a nonzero constant.
Such operators possess many of the properties of the derivative operator D D DDD. They are sometimes called reductive operators.
Here are some typical examples: the forward, backward and central difference operators Δ h , ∇ h , δ h Δ h , ∇ h , δ h Delta_(h),grad_(h),delta_(h)\Delta_{h}, \nabla_{h}, \delta_{h}Δh,∇h,δh, the prederivative operator D h = Δ h / h D h = Δ h / h D_(h)=Delta_(h)//hD_{h}=\Delta_{h} / hDh=Δh/h (we consider that D 0 D 0 D_(0)D_{0}D0 is the derivative operator D D DDD ). Another example is represented by the Abel operator A a = D E a = E a D A a = D E a = E a D A_(a)=DE^(a)=E^(a)DA_{a}=D E^{a}=E^{a} DAa=DEa=EaD, which in the case p n ( x ; a ) = x ( x − n a ) n − 1 p n ( x ; a ) = x ( x − n a ) n − 1 p_(n)(x;a)=x(x-na)^(n-1)p_{n}(x ; a)=x(x-n a)^{n-1}pn(x;a)=x(x−na)n−1 leads us to the formula: A a p n ( x ; a ) = n x ( x − ( n − 1 ) a ) n − 2 A a p n ( x ; a ) = n x ( x − ( n − 1 ) a ) n − 2 A_(a)p_(n)(x;a)=nx(x-(n-1)a)^(n-2)A_{a} p_{n}(x ; a)=n x(x-(n-1) a)^{n-2}Aapn(x;a)=nx(x−(n−1)a)n−2.
It should be noted that each theta operator θ θ theta\thetaθ can be expressed as a power series in the derivative operator (see [24] and [2]):
θ = ∑ k = 1 ∞ c k D k ( c 1 ≠ 0 ) θ = ∑ k = 1 ∞   c k D k c 1 ≠ 0 theta=sum_(k=1)^(oo)c_(k)D^(k)quad(c_(1)!=0)\theta=\sum_{k=1}^{\infty} c_{k} D^{k} \quad\left(c_{1} \neq 0\right)θ=∑k=1∞ckDk(c1≠0)
It is easy to see that: (i) for every theta operator we have θ c = 0 θ c = 0 theta c=0\theta c=0θc=0, where c c ccc is a constant; (ii) if p n p n p_(n)p_{n}pn is a polynomial of degree n n nnn, then θ p n θ p n thetap_(n)\theta p_{n}θpn is of degree n − 1 n − 1 n-1n-1n−1.
3. A great part of applied mathematics is concerned with the study of various special polynomials.
A sequence of polynomials ( p m p m p_(m)p_{m}pm ) is called by I. M. Sheffer [17] and by GianCarlo Rota [15] and his collaborators, the sequence of basic polynomials for a theta operator θ θ theta\thetaθ if we have: (i) p 0 ( x ) = 1 p 0 ( x ) = 1 p_(0)(x)=1p_{0}(x)=1p0(x)=1; (ii) p n ( 0 ) = 0 ( n ≥ 1 ) p n ( 0 ) = 0 ( n ≥ 1 ) p_(n)(0)=0(n >= 1)p_{n}(0)=0(n \geq 1)pn(0)=0(n≥1); (iii) θ p n = n p n − 1 θ p n = n p n − 1 thetap_(n)=np_(n-1)\theta p_{n}=n p_{n-1}θpn=npn−1.
These polynomials were called by Steffensen [24] poweroids, considering that they represent an extension of the mathematical notion of power.
The following two results can be easily proved (see [11]): (i) if ( p n ) p n (p_(n))\left(p_{n}\right)(pn) is a basic sequence of polynomials for some theta operator, then it is a B.T. sequence; (ii) if ( p n p n p_(n)p_{n}pn ) is a sequence of B.T., then it is a basic sequence for a theta operator.
It is easy to show, by induction, that every theta operator has a unique sequence of basic polynomials associated with it.
Steffensen [24] was the first who observed that the property of the polynomial sequence e n ( x ) = x n e n ( x ) = x n e_(n)(x)=x^(n)e_{n}(x)=x^{n}en(x)=xn to be of binomial type can be extended to an arbitrary sequence of basic polynomials associated to a theta operator.
Here are some illustrative examples: (i) if θ θ theta\thetaθ is the derivative operator D D DDD, then p m ( x ) = x m p m ( x ) = x m p_(m)(x)=x^(m)p_{m}(x)=x^{m}pm(x)=xm; (ii) if θ θ theta\thetaθ is the prederivative operator D h = h − 1 Δ h D h = h − 1 Δ h D_(h)=h^(-1)Delta_(h)D_{h}=h^{-1} \Delta_{h}Dh=h−1Δh, then we arrive
at the factorial power: p m ( x ) = x ( x − h ) … ( x − ( m − 1 ) h ) p m ( x ) = x ( x − h ) … ( x − ( m − 1 ) h ) p_(m)(x)=x(x-h)dots(x-(m-1)h)p_{m}(x)=x(x-h) \ldots(x-(m-1) h)pm(x)=x(x−h)…(x−(m−1)h); (iii) if θ = E β 1 − E − α α θ = E β 1 − E − α α theta=E^(beta)(1-E^(-alpha))/(alpha)\theta=E^{\beta} \frac{1-E^{-\alpha}}{\alpha}θ=Eβ1−E−αα, then we have p m ( x ; β , α ) = x ( x + α + β m ) [ m − 1 , − α ] p m ( x ; β , α ) = x ( x + α + β m ) [ m − 1 , − α ] p_(m)(x;beta,alpha)=x(x+alpha+beta m)^([m-1,-alpha])p_{m}(x ; \beta, \alpha)=x(x+\alpha+\beta m)^{[m-1,-\alpha]}pm(x;β,α)=x(x+α+βm)[m−1,−α].
4. The classical Taylor formula can be extended to arbitrary theta operators.
If T T TTT is a shift-invariant operator and ( p n p n p_(n)p_{n}pn ) is the corresponding basic polynomial sequence, then we have
T = ∑ k = 0 ∞ a k k ! θ k , where a k = [ T p k ( x ) ] x = 0 T = ∑ k = 0 ∞   a k k ! θ k ,  where  a k = T p k ( x ) x = 0 T=sum_(k=0)^(oo)(a_(k))/(k!)theta^(k),quad" where "quada_(k)=[Tp_(k)(x)]_(x=0)T=\sum_{k=0}^{\infty} \frac{a_{k}}{k!} \theta^{k}, \quad \text { where } \quad a_{k}=\left[T p_{k}(x)\right]_{x=0}T=∑k=0∞akk!θk, where ak=[Tpk(x)]x=0
The Taylor expansion formula for a polynomial P m P m P_(m)P_{m}Pm, of degree m m mmm, is given by
(2.1) P m ( x + y ) = ∑ k = 0 m ( 0 k P m ) ( y ) k ! p k ( x ) (2.1) P m ( x + y ) = ∑ k = 0 m   0 k P m ( y ) k ! p k ( x ) {:(2.1)P_(m)(x+y)=sum_(k=0)^(m)((0^(k)P_(m))(y))/(k!)p_(k)(x):}\begin{equation*} P_{m}(x+y)=\sum_{k=0}^{m} \frac{\left(0^{k} P_{m}\right)(y)}{k!} p_{k}(x) \tag{2.1} \end{equation*}(2.1)Pm(x+y)=∑k=0m(0kPm)(y)k!pk(x)
where θ θ theta\thetaθ is a theta operator with the basic sequence ( p k p k p_(k)p_{k}pk ).
Examples. If θ = A − β = E − β D θ = A − β = E − β D theta=A_(-beta)=E^(-beta)D\theta=A_{-\beta}=E^{-\beta} Dθ=A−β=E−βD, then we obtain the Abel basic polynomials p k ( x ) = x ( x + k β ) k − 1 p k ( x ) = x ( x + k β ) k − 1 p_(k)(x)=x(x+k beta)^(k-1)p_{k}(x)=x(x+k \beta)^{k-1}pk(x)=x(x+kβ)k−1 and the Taylor expansion becomes
(2.2) P m ( x + y ) = ∑ k = 0 m P m ( k ) ( y − β k ) k ! ⋅ x ( x + k β ) k − 1 . (2.2) P m ( x + y ) = ∑ k = 0 m   P m ( k ) ( y − β k ) k ! ⋅ x ( x + k β ) k − 1 . {:(2.2)P_(m)(x+y)=sum_(k=0)^(m)(P_(m)^((k))(y-beta k))/(k!)*x(x+k beta)^(k-1).:}\begin{equation*} P_{m}(x+y)=\sum_{k=0}^{m} \frac{P_{m}^{(k)}(y-\beta k)}{k!} \cdot x(x+k \beta)^{k-1} . \tag{2.2} \end{equation*}(2.2)Pm(x+y)=∑k=0mPm(k)(y−βk)k!⋅x(x+kβ)k−1.
If we choose P m ( x ) = x m P m ( x ) = x m P_(m)(x)=x^(m)P_{m}(x)=x^{m}Pm(x)=xm, we obtain
(2.3) ( x + y ) m = ∑ k = 0 m ( m k ) x ( x + k β ) k − 1 ( y − k β ) m − k , (2.3) ( x + y ) m = ∑ k = 0 m   ( m k ) x ( x + k β ) k − 1 ( y − k β ) m − k , {:(2.3)(x+y)^(m)=sum_(k=0)^(m)((m)/(k))x(x+k beta)^(k-1)(y-k beta)^(m-k)",":}\begin{equation*} (x+y)^{m}=\sum_{k=0}^{m}\binom{m}{k} x(x+k \beta)^{k-1}(y-k \beta)^{m-k}, \tag{2.3} \end{equation*}(2.3)(x+y)m=∑k=0m(mk)x(x+kβ)k−1(y−kβ)m−k,
which is the Abel classical combinatorial identity.
When we select P m ( x ) = x ( x + m β ) m − 1 P m ( x ) = x ( x + m β ) m − 1 P_(m)(x)=x(x+m beta)^(m-1)P_{m}(x)=x(x+m \beta)^{m-1}Pm(x)=x(x+mβ)m−1 we have ( θ k P m ) ( y ) == m [ k ] ( y + ( m − k ) β ) m − 1 θ k P m ( y ) == m [ k ] ( y + ( m − k ) β ) m − 1 (theta^(k)P_(m))(y)==m^([k])(y+(m-k)beta)^(m-1)\left(\theta^{k} P_{m}\right)(y)= =m^{[k]}(y+(m-k) \beta)^{m-1}(θkPm)(y)==m[k](y+(m−k)β)m−1 and formula (2.2) leads us to the Abel-Jensen combinatorial formula
(2.4) ( x + y ) ( x + y + m β ) m − 1 = ∑ k = 0 m ( m k ) x ( x + k β ) k − 1 y ( y + ( m − k ) β ) m − 1 − k (2.4) ( x + y ) ( x + y + m β ) m − 1 = ∑ k = 0 m   ( m k ) x ( x + k β ) k − 1 y ( y + ( m − k ) β ) m − 1 − k {:(2.4)(x+y)(x+y+m beta)^(m-1)=sum_(k=0)^(m)((m)/(k))x(x+k beta)^(k-1)y(y+(m-k)beta)^(m-1-k):}\begin{equation*} (x+y)(x+y+m \beta)^{m-1}=\sum_{k=0}^{m}\binom{m}{k} x(x+k \beta)^{k-1} y(y+(m-k) \beta)^{m-1-k} \tag{2.4} \end{equation*}(2.4)(x+y)(x+y+mβ)m−1=∑k=0m(mk)x(x+kβ)k−1y(y+(m−k)β)m−1−k
which can be obtained from (1.1) if we use the following sequence of binomial type: p n ( t ) = t ( t + n β ) n − 1 p n ( t ) = t ( t + n β ) n − 1 p_(n)(t)=t(t+n beta)^(n-1)p_{n}(t)=t(t+n \beta)^{n-1}pn(t)=t(t+nβ)n−1.
If we consider a generalized Vandermonde formula of the following form
( x + y + m β ) [ m , h ] = ∑ k = 0 m ( m k ) x ( x + k β − h ) [ k − 1 , h ] ( y + ( m − k ) β ) [ m − k , h ] ( x + y + m β ) [ m , h ] = ∑ k = 0 m   ( m k ) x ( x + k β − h ) [ k − 1 , h ] ( y + ( m − k ) β ) [ m − k , h ] (x+y+m beta)^([m,h])=sum_(k=0)^(m)((m)/(k))x(x+k beta-h)^([k-1,h])(y+(m-k)beta)^([m-k,h])(x+y+m \beta)^{[m, h]}=\sum_{k=0}^{m}\binom{m}{k} x(x+k \beta-h)^{[k-1, h]}(y+(m-k) \beta)^{[m-k, h]}(x+y+mβ)[m,h]=∑k=0m(mk)x(x+kβ−h)[k−1,h](y+(m−k)β)[m−k,h]
and we take h = 0 h = 0 h=0h=0h=0, we obtain the Abel-Jensen combinatorial formula
(2.5) ( x + y + m β ) m = ∑ k = 0 m ( m k ) x ( x + k β ) k − 1 ( y + ( m − k ) β ) m − k (2.5) ( x + y + m β ) m = ∑ k = 0 m   ( m k ) x ( x + k β ) k − 1 ( y + ( m − k ) β ) m − k {:(2.5)(x+y+m beta)^(m)=sum_(k=0)^(m)((m)/(k))x(x+k beta)^(k-1)(y+(m-k)beta)^(m-k):}\begin{equation*} (x+y+m \beta)^{m}=\sum_{k=0}^{m}\binom{m}{k} x(x+k \beta)^{k-1}(y+(m-k) \beta)^{m-k} \tag{2.5} \end{equation*}(2.5)(x+y+mβ)m=∑k=0m(mk)x(x+kβ)k−1(y+(m−k)β)m−k
We mention that Cheney and Sharma have used in [1] the combinatorial formulas (2.4) and (2.5) for constructing two Bernstein-type operators P m P m P_(m)P_{m}Pm and Q m Q m Q_(m)Q_{m}Qm, depending on the nonnegative parameter β = o ( n − 1 ) β = o n − 1 beta=o(n^(-1))\beta=o\left(n^{-1}\right)β=o(n−1).

3. GENERALIZED BINOMIAL OPERATORS OF TIBERIU POPOVICIU TYPE

  1. Let ( p n ) p n (p_(n))\left(p_{n}\right)(pn) be a basic sequence of polynomials for some theta operator. It follows that it is a B.T. sequence and we can write the identity (1.4).
Assuming that p m ( 1 ) ≠ 0 p m ( 1 ) ≠ 0 p_(m)(1)!=0p_{m}(1) \neq 0pm(1)≠0, for any m ∈ N m ∈ N m inNm \in \mathbb{N}m∈N, we define an operator of binomial type T m γ , δ T m γ , δ T_(m)^(gamma,delta)T_{m}^{\gamma, \delta}Tmγ,δ, associated to a function f : [ 0 , 1 ] → R f : [ 0 , 1 ] → R f:[0,1]rarrRf:[0,1] \rightarrow \mathbb{R}f:[0,1]→R, by the following formula
(3.1) T m γ , δ ( f ( t ) ; x = ( T m γ , δ f ) ( x ) = = 1 p m ( 1 ) ∑ k = 0 m ( m k ) p k ( x ) p m − k ( 1 − x ) f ( k + γ m + δ ) (3.1) T m γ , δ f ( t ) ; x = T m γ , δ f ( x ) = = 1 p m ( 1 ) ∑ k = 0 m   ( m k ) p k ( x ) p m − k ( 1 − x ) f k + γ m + δ {:[(3.1)T_(m)^(gamma,delta)(f(t);x=(T_(m)^(gamma,delta)f)(x)=:}],[=(1)/(p_(m)(1))sum_(k=0)^(m)((m)/(k))p_(k)(x)p_(m-k)(1-x)f((k+gamma)/(m+delta))]:}\begin{gather*} T_{m}^{\gamma, \delta}\left(f(t) ; x=\left(T_{m}^{\gamma, \delta} f\right)(x)=\right. \tag{3.1}\\ =\frac{1}{p_{m}(1)} \sum_{k=0}^{m}\binom{m}{k} p_{k}(x) p_{m-k}(1-x) f\left(\frac{k+\gamma}{m+\delta}\right) \end{gather*}(3.1)Tmγ,δ(f(t);x=(Tmγ,δf)(x)==1pm(1)∑k=0m(mk)pk(x)pm−k(1−x)f(k+γm+δ)
where x ∈ [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1] and γ γ gamma\gammaγ and δ δ delta\deltaδ are parameters satisfying the relations: 0 ≤ γ ≤ δ 0 ≤ γ ≤ δ 0 <= gamma <= delta0 \leq \gamma \leq \delta0≤γ≤δ.
As we have pointed out, we have p j ( x ) ≥ 0 p j ( x ) ≥ 0 p_(j)(x) >= 0p_{j}(x) \geq 0pj(x)≥0 on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], for j = 0 , 1 , … , m j = 0 , 1 , … , m j=0,1,dots,mj=0,1, \ldots, mj=0,1,…,m, if the coefficients c k c k c_(k)c_{k}ck from (1.3) are nonnegative.
It is obvious that our operator reproduces the constants, since according to (1.4) we have T m γ , δ e 0 = e 0 T m γ , δ e 0 = e 0 T_(m)^(gamma,delta)e_(0)=e_(0)T_{m}^{\gamma, \delta} e_{0}=e_{0}Tmγ,δe0=e0.
Now we want to see if we can choose the parameters γ γ gamma\gammaγ and δ δ delta\deltaδ such that the polynomial T m γ , δ f T m γ , δ f T_(m)^(gamma,delta)fT_{m}^{\gamma, \delta} fTmγ,δf is interpolatory at the endpoints of the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].
It is easily verified that we have
( T m γ , δ f ) ( 0 ) = f ( γ m + δ ) , ( T m γ , δ f ) ( 1 ) = f ( m + γ m + δ ) T m γ , δ f ( 0 ) = f γ m + δ , T m γ , δ f ( 1 ) = f m + γ m + δ (T_(m)^(gamma,delta)f)(0)=f((gamma)/(m+delta)),quad(T_(m)^(gamma,delta)f)(1)=f((m+gamma)/(m+delta))\left(T_{m}^{\gamma, \delta} f\right)(0)=f\left(\frac{\gamma}{m+\delta}\right), \quad\left(T_{m}^{\gamma, \delta} f\right)(1)=f\left(\frac{m+\gamma}{m+\delta}\right)(Tmγ,δf)(0)=f(γm+δ),(Tmγ,δf)(1)=f(m+γm+δ)
Consequently, if we take γ = δ = 0 γ = δ = 0 gamma=delta=0\gamma=\delta=0γ=δ=0 we can see that the Tiberiu Popoviciu polynomial T m f = T m 0 , 0 f T m f = T m 0 , 0 f T_(m)f=T_(m)^(0,0)fT_{m} f=T_{m}^{0,0} fTmf=Tm0,0f is interpolatory in x = 0 x = 0 x=0x=0x=0 and x = 1 x = 1 x=1x=1x=1, that is
(3.2) ( T m f ) ( 0 ) = f ( 0 ) , ( T m f ) ( 1 ) = f ( 1 ) (3.2) T m f ( 0 ) = f ( 0 ) , T m f ( 1 ) = f ( 1 ) {:(3.2)(T_(m)f)(0)=f(0)","quad(T_(m)f)(1)=f(1):}\begin{equation*} \left(T_{m} f\right)(0)=f(0), \quad\left(T_{m} f\right)(1)=f(1) \tag{3.2} \end{equation*}(3.2)(Tmf)(0)=f(0),(Tmf)(1)=f(1)
In order to study the convergence of the sequence ( T m γ , δ f T m γ , δ f T_(m)^(gamma,delta)fT_{m}^{\gamma, \delta} fTmγ,δf ), we also need to find its value for the monomials e 1 e 1 e_(1)e_{1}e1 and e 2 e 2 e_(2)e_{2}e2.
In the case f = e 1 f = e 1 f=e_(1)f=e_{1}f=e1 we have
( T m γ , δ e 1 ) ( x ) = 1 p m ( 1 ) ∑ k = 0 m k + γ m + δ ( m k ) p k ( x ) p m − k ( 1 − x ) = = m m + δ ( T m e 1 ) ( x ) + γ m + δ ( T m e 0 ) ( x ) = m x + γ m + δ T m γ , δ e 1 ( x ) = 1 p m ( 1 ) ∑ k = 0 m   k + γ m + δ ( m k ) p k ( x ) p m − k ( 1 − x ) = = m m + δ T m e 1 ( x ) + γ m + δ T m e 0 ( x ) = m x + γ m + δ {:[(T_(m)^(gamma,delta)e_(1))(x)=(1)/(p_(m)(1))sum_(k=0)^(m)(k+gamma)/(m+delta)((m)/(k))p_(k)(x)p_(m-k)(1-x)=],[quad=(m)/(m+delta)(T_(m)e_(1))(x)+(gamma)/(m+delta)(T_(m)e_(0))(x)=(mx+gamma)/(m+delta)]:}\begin{gathered} \left(T_{m}^{\gamma, \delta} e_{1}\right)(x)=\frac{1}{p_{m}(1)} \sum_{k=0}^{m} \frac{k+\gamma}{m+\delta}\binom{m}{k} p_{k}(x) p_{m-k}(1-x)= \\ \quad=\frac{m}{m+\delta}\left(T_{m} e_{1}\right)(x)+\frac{\gamma}{m+\delta}\left(T_{m} e_{0}\right)(x)=\frac{m x+\gamma}{m+\delta} \end{gathered}(Tmγ,δe1)(x)=1pm(1)∑k=0mk+γm+δ(mk)pk(x)pm−k(1−x)==mm+δ(Tme1)(x)+γm+δ(Tme0)(x)=mx+γm+δ
since in a recent paper [22] it has been proved that T m e 1 = e 1 T m e 1 = e 1 T_(m)e_(1)=e_(1)T_{m} e_{1}=e_{1}Tme1=e1. Hence we have
(3.3) T m γ , δ e 0 = e 0 , ( T m γ , δ e 1 ) ( x ) = x + γ − δ x m + δ (3.3) T m γ , δ e 0 = e 0 , T m γ , δ e 1 ( x ) = x + γ − δ x m + δ {:(3.3)T_(m)^(gamma,delta)e_(0)=e_(0)","quad(T_(m)^(gamma,delta)e_(1))(x)=x+(gamma-delta x)/(m+delta):}\begin{equation*} T_{m}^{\gamma, \delta} e_{0}=e_{0}, \quad\left(T_{m}^{\gamma, \delta} e_{1}\right)(x)=x+\frac{\gamma-\delta x}{m+\delta} \tag{3.3} \end{equation*}(3.3)Tmγ,δe0=e0,(Tmγ,δe1)(x)=x+γ−δxm+δ
Going on to e 2 ( x ) = x 2 e 2 ( x ) = x 2 e_(2)(x)=x^(2)e_{2}(x)=x^{2}e2(x)=x2 we obtain
( T m γ , δ e 2 ) ( x ) = 1 p m ( 1 ) ∑ k = 0 m ( k + γ m + δ ) 2 p k ( x ) p m − k ( 1 − x ) = = 1 ( m + δ ) 2 [ m 2 ( T m e 2 ) ( x ) + 2 m γ ( T m e 2 ) ( x ) + γ 2 ] T m γ , δ e 2 ( x ) = 1 p m ( 1 ) ∑ k = 0 m   k + γ m + δ 2 p k ( x ) p m − k ( 1 − x ) = = 1 ( m + δ ) 2 m 2 T m e 2 ( x ) + 2 m γ T m e 2 ( x ) + γ 2 {:[(T_(m)^(gamma,delta)e_(2))(x)=(1)/(p_(m)(1))sum_(k=0)^(m)((k+gamma)/(m+delta))^(2)p_(k)(x)p_(m-k)(1-x)=],[quad=(1)/((m+delta)^(2))[m^(2)(T_(m)e_(2))(x)+2m gamma(T_(m)e_(2))(x)+gamma^(2)]]:}\begin{aligned} & \left(T_{m}^{\gamma, \delta} e_{2}\right)(x)=\frac{1}{p_{m}(1)} \sum_{k=0}^{m}\left(\frac{k+\gamma}{m+\delta}\right)^{2} p_{k}(x) p_{m-k}(1-x)= \\ & \quad=\frac{1}{(m+\delta)^{2}}\left[m^{2}\left(T_{m} e_{2}\right)(x)+2 m \gamma\left(T_{m} e_{2}\right)(x)+\gamma^{2}\right] \end{aligned}(Tmγ,δe2)(x)=1pm(1)∑k=0m(k+γm+δ)2pk(x)pm−k(1−x)==1(m+δ)2[m2(Tme2)(x)+2mγ(Tme2)(x)+γ2]
Therefore
(3.4) ( T m γ , δ e 2 ) ( x ) = γ 2 ( m + δ ) 2 + 2 m γ x ( m + δ ) 2 + m 2 ( m + δ ) 2 ( T m e 2 ) ( x ) (3.4) T m γ , δ e 2 ( x ) = γ 2 ( m + δ ) 2 + 2 m γ x ( m + δ ) 2 + m 2 ( m + δ ) 2 T m e 2 ( x ) {:(3.4)(T_(m)^(gamma,delta)e_(2))(x)=(gamma^(2))/((m+delta)^(2))+(2m gamma x)/((m+delta)^(2))+(m^(2))/((m+delta)^(2))(T_(m)e_(2))(x):}\begin{equation*} \left(T_{m}^{\gamma, \delta} e_{2}\right)(x)=\frac{\gamma^{2}}{(m+\delta)^{2}}+\frac{2 m \gamma x}{(m+\delta)^{2}}+\frac{m^{2}}{(m+\delta)^{2}}\left(T_{m} e_{2}\right)(x) \tag{3.4} \end{equation*}(3.4)(Tmγ,δe2)(x)=γ2(m+δ)2+2mγx(m+δ)2+m2(m+δ)2(Tme2)(x)
It remains to evaluate ( T m e 2 ) ( x ) T m e 2 ( x ) (T_(m)e_(2))(x)\left(T_{m} e_{2}\right)(x)(Tme2)(x).
In this sense we mention two results.
The first one was found by the former graduate student of the first author - C. Manole [9]:
(3.5) ( T m e 2 ) ( x ) = x 2 + x ( 1 − x ) m + x ( 1 − x ) a m (3.5) T m e 2 ( x ) = x 2 + x ( 1 − x ) m + x ( 1 − x ) a m {:(3.5)(T_(m)e_(2))(x)=x^(2)+(x(1-x))/(m)+x(1-x)a_(m):}\begin{equation*} \left(T_{m} e_{2}\right)(x)=x^{2}+\frac{x(1-x)}{m}+x(1-x) a_{m} \tag{3.5} \end{equation*}(3.5)(Tme2)(x)=x2+x(1−x)m+x(1−x)am
where
(3.6) a m = m − 1 m [ 1 − p n − 1 ( θ ′ ) p m − 2 ( 1 ) ] (3.6) a m = m − 1 m 1 − p n − 1 θ ′ p m − 2 ( 1 ) {:(3.6)a_(m)=(m-1)/(m)[1-p_(n)^(-1)(theta^('))p_(m-2)(1)]:}\begin{equation*} a_{m}=\frac{m-1}{m}\left[1-p_{n}^{-1}\left(\theta^{\prime}\right) p_{m-2}(1)\right] \tag{3.6} \end{equation*}(3.6)am=m−1m[1−pn−1(θ′)pm−2(1)]
θ ′ θ ′ theta^(')\theta^{\prime}θ′ being the Pincherle derivative (see, e.g. [11]) of the theta operator θ θ theta\thetaθ, for which ( p m ) p m (p_(m))\left(p_{m}\right)(pm) is a basic sequence.
The second result has been recently obtained by P . Sablonnière [16]
(3.7) ( T m e 2 ) ( x ) = x 2 + x ( 1 − x ) m b m , b m = 1 + ( m − 1 ) r m − 2 ( 1 ) p m ( 1 ) , (3.7) T m e 2 ( x ) = x 2 + x ( 1 − x ) m b m , b m = 1 + ( m − 1 ) r m − 2 ( 1 ) p m ( 1 ) , {:(3.7)(T_(m)e_(2))(x)=x^(2)+(x(1-x))/(m)b_(m)","quadb_(m)=1+(m-1)(r_(m-2)(1))/(p_(m)(1))",":}\begin{equation*} \left(T_{m} e_{2}\right)(x)=x^{2}+\frac{x(1-x)}{m} b_{m}, \quad b_{m}=1+(m-1) \frac{r_{m-2}(1)}{p_{m}(1)}, \tag{3.7} \end{equation*}(3.7)(Tme2)(x)=x2+x(1−x)mbm,bm=1+(m−1)rm−2(1)pm(1),
the sequence ( r m r m r_(m)r_{m}rm ) being generated by the expansion
(3.8) h ′ ′ ( t ) e x h ( t ) = ∑ m = 0 ∞ r m ( x ) t m m ! (3.8) h ′ ′ ( t ) e x h ( t ) = ∑ m = 0 ∞   r m ( x ) t m m ! {:(3.8)h^('')(t)e^(xh(t))=sum_(m=0)^(oo)r_(m)(x)(t^(m))/(m!):}\begin{equation*} h^{\prime \prime}(t) \mathrm{e}^{x h(t)}=\sum_{m=0}^{\infty} r_{m}(x) \frac{t^{m}}{m!} \tag{3.8} \end{equation*}(3.8)h′′(t)exh(t)=∑m=0∞rm(x)tmm!
If we take into account (3.3), (3.4) and (3.5)-(3.6) or (3.7)-(3.8) we can state
THEOREM 3.1. If f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] and
(3.9) lim m → ∞ a m = 0 , or lim m → ∞ r m − 2 ( 1 ) p m ( 1 ) = 0 (3.9) lim m → ∞   a m = 0 ,  or  lim m → ∞   r m − 2 ( 1 ) p m ( 1 ) = 0 {:(3.9)lim_(m rarr oo)a_(m)=0","quad" or "quadlim_(m rarr oo)(r_(m-2)(1))/(p_(m)(1))=0:}\begin{equation*} \lim _{m \rightarrow \infty} a_{m}=0, \quad \text { or } \quad \lim _{m \rightarrow \infty} \frac{r_{m-2}(1)}{p_{m}(1)}=0 \tag{3.9} \end{equation*}(3.9)limm→∞am=0, or limm→∞rm−2(1)pm(1)=0
then the sequence of polynomials of binomial type ( T m γ , δ f ) T m γ , δ f (T_(m)^(gamma,delta)f)\left(T_{m}^{\gamma, \delta} f\right)(Tmγ,δf), where 0 ≤ γ ≤ δ 0 ≤ γ ≤ δ 0 <= gamma <= delta0 \leq \gamma \leq \delta0≤γ≤δ, converges to the function f f fff, uniformly on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].
For proving it, we have to take into account the fact that the relations (3.3), (3.4) and (3.5)-(3.6), or (3.7)-(3.8) and the assumption (3.9) perinit us to make use of the convergence criterion of Bohman-Korovkin.

4. EVALUATION OF THE ORDERS OF APPROXIMATION

  1. Now we shall evaluate the order of approximation of a function f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] by means of the operator T m T m T_(m)T_{m}Tm of Tiberiu Popoviciu.
For this purpose we can use two inequalities, which can be seen in H. H. Gonska and J. Meier [3], for the evaluation of the orders of approximation by using the first and second order moduli of smoothness ω 1 ω 1 omega_(1)\omega_{1}ω1 and ω 2 ω 2 omega_(2)\omega_{2}ω2.
If we approximate f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] by means of T m f T m f T_(m)fT_{m} fTmf, we obtain
| f ( x ) − ( T m f ) ( x ) | ≤ [ 1 + 1 δ 2 T m ( ( t − x ) 2 ; x ) ] ω 1 ( f ; δ ) , δ ∈ R + f ( x ) − T m f ( x ) ≤ 1 + 1 δ 2 T m ( t − x ) 2 ; x ω 1 ( f ; δ ) , δ ∈ R + |f(x)-(T_(m)f)(x)| <= [1+(1)/(delta^(2))T_(m)((t-x)^(2);x)]omega_(1)(f;delta),quad delta inR_(+)\left|f(x)-\left(T_{m} f\right)(x)\right| \leq\left[1+\frac{1}{\delta^{2}} T_{m}\left((t-x)^{2} ; x\right)\right] \omega_{1}(f ; \delta), \quad \delta \in \mathbb{R}_{+}|f(x)−(Tmf)(x)|≤[1+1δ2Tm((t−x)2;x)]ω1(f;δ),δ∈R+
By using the relations T m e 0 = e 0 , T m e 1 = e 1 T m e 0 = e 0 , T m e 1 = e 1 T_(m)e_(0)=e_(0),T_(m)e_(1)=e_(1)T_{m} e_{0}=e_{0}, T_{m} e_{1}=e_{1}Tme0=e0,Tme1=e1 and (3.5)-(3.6), we get
| f ( x ) − ( T m f ) ( x ) | ≤ [ 1 + 1 δ 2 ⋅ x ( 1 − x ) m ( 1 + m a m ) ] ω 1 ( f ; δ ) f ( x ) − T m f ( x ) ≤ 1 + 1 δ 2 ⋅ x ( 1 − x ) m 1 + m a m ω 1 ( f ; δ ) |f(x)-(T_(m)f)(x)| <= [1+(1)/(delta^(2))*(x(1-x))/(m)(1+ma_(m))]omega_(1)(f;delta)\left|f(x)-\left(T_{m} f\right)(x)\right| \leq\left[1+\frac{1}{\delta^{2}} \cdot \frac{x(1-x)}{m}\left(1+m a_{m}\right)\right] \omega_{1}(f ; \delta)|f(x)−(Tmf)(x)|≤[1+1δ2⋅x(1−x)m(1+mam)]ω1(f;δ)
If we choose δ = A x ( 1 − x ) m ( A > 0 ) δ = A x ( 1 − x ) m ( A > 0 ) delta=Asqrt((x(1-x))/(m))(A > 0)\delta=A \sqrt{\frac{x(1-x)}{m}}(A>0)δ=Ax(1−x)m(A>0), we obtain
| f ( x ) − ( T m f ) ( x ) | ≤ [ 1 + m a m A 2 ] ω 1 ( f ; A x ( 1 − x ) m ) f ( x ) − T m f ( x ) ≤ 1 + m a m A 2 ω 1 f ; A x ( 1 − x ) m |f(x)-(T_(m)f)(x)| <= [1+(ma_(m))/(A^(2))]omega_(1)(f;Asqrt((x(1-x))/(m)))\left|f(x)-\left(T_{m} f\right)(x)\right| \leq\left[1+\frac{m a_{m}}{A^{2}}\right] \omega_{1}\left(f ; A \sqrt{\frac{x(1-x)}{m}}\right)|f(x)−(Tmf)(x)|≤[1+mamA2]ω1(f;Ax(1−x)m)
This inequality permits us to see that T m f T m f T_(m)fT_{m} fTmf is interpolatory at x = 0 x = 0 x=0x=0x=0 and x = 1 x = 1 x=1x=1x=1.

Now we can state

THEOREM 4.1. If f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] and we approximate f f fff by T m f T m f T_(m)fT_{m} fTmf, then we can give the following estimation of the order of approximation, by means of the first order modulus of continuity
‖ f − T m f ‖ ≤ ( 5 4 + m 4 a m ) ω 1 ( f ; 1 m ) . f − T m f ≤ 5 4 + m 4 a m ω 1 f ; 1 m . ||f-T_(m)f|| <= ((5)/(4)+(m)/(4)a_(m))omega_(1)(f;(1)/(sqrtm)).\left\|f-T_{m} f\right\| \leq\left(\frac{5}{4}+\frac{m}{4} a_{m}\right) \omega_{1}\left(f ; \frac{1}{\sqrt{m}}\right) .‖f−Tmf‖≤(54+m4am)ω1(f;1m).
In the case of Bernstein polynomials we have a m = 0 a m = 0 a_(m)=0a_{m}=0am=0 and this inequality reduces to an inequality of Tiberiu Popoviciu [13], but with the coefficient 5 4 5 4 (5)/(4)\frac{5}{4}54 obtained by G. G. Lorentz in his book [7].
7. If we use the second order modulus of smoothness, we oblain
∣ f ( x ) − T m f ) ( x ) | ≤ 3 2 [ 1 + 1 2 δ 2 T m ( ( t − x ) 2 ; x ) ] ω 2 ( f ; δ ) ∣ f ( x ) − T m f ( x ) ≤ 3 2 1 + 1 2 δ 2 T m ( t − x ) 2 ; x ω 2 ( f ; δ ) {:∣f(x)-T_(m)f)(x)| <= (3)/(2)[1+(1)/(2delta^(2))T_(m)((t-x)^(2);x)]omega_(2)(f;delta):}\left.\mid f(x)-T_{m} f\right)(x) \left\lvert\, \leq \frac{3}{2}\left[1+\frac{1}{2 \delta^{2}} T_{m}\left((t-x)^{2} ; x\right)\right] \omega_{2}(f ; \delta)\right.∣f(x)−Tmf)(x)|≤32[1+12δ2Tm((t−x)2;x)]ω2(f;δ)
or, more explicitly,
∣ f ( x ) − T m f ) ( x ) | ≤ 3 2 [ 1 + 1 2 δ 2 ⋅ x ( 1 − x ) m ( 1 + a m ) ] ω 2 ( f ; δ ) ∣ f ( x ) − T m f ( x ) ≤ 3 2 1 + 1 2 δ 2 ⋅ x ( 1 − x ) m 1 + a m ω 2 ( f ; δ ) {:∣f(x)-T_(m)f)(x)| <= (3)/(2)[1+(1)/(2delta^(2))*(x(1-x))/(m)(1+a_(m))]omega_(2)(f;delta):}\left.\mid f(x)-T_{m} f\right)(x) \left\lvert\, \leq \frac{3}{2}\left[1+\frac{1}{2 \delta^{2}} \cdot \frac{x(1-x)}{m}\left(1+a_{m}\right)\right] \omega_{2}(f ; \delta)\right.∣f(x)−Tmf)(x)|≤32[1+12δ2⋅x(1−x)m(1+am)]ω2(f;δ)
By choosing δ = A x ( 1 − x ) m ( A > 0 ) δ = A x ( 1 − x ) m ( A > 0 ) delta=Asqrt((x(1-x))/(m))(A > 0)\delta=A \sqrt{\frac{x(1-x)}{m}}(A>0)δ=Ax(1−x)m(A>0), we obtain
∣ f ( x ) − T m f ) ( x ) | ≤ 3 2 [ 1 + 1 + m a m 2 A 2 ] ω 2 ( f ; x ( 1 − x ) m ) ∣ f ( x ) − T m f ( x ) ≤ 3 2 1 + 1 + m a m 2 A 2 ω 2 f ; x ( 1 − x ) m {:∣f(x)-T_(m)f)(x)| <= (3)/(2)[1+(1+ma_(m))/(2A^(2))]omega_(2)(f;sqrt((x(1-x))/(m))):}\left.\mid f(x)-T_{m} f\right)(x) \left\lvert\, \leq \frac{3}{2}\left[1+\frac{1+m a_{m}}{2 A^{2}}\right] \omega_{2}\left(f ; \sqrt{\frac{x(1-x)}{m}}\right)\right.∣f(x)−Tmf)(x)|≤32[1+1+mam2A2]ω2(f;x(1−x)m)
If we take into account that on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] we have x ( 1 − x ) ≤ 1 4 x ( 1 − x ) ≤ 1 4 x(1-x) <= (1)/(4)x(1-x) \leq \frac{1}{4}x(1−x)≤14 and we select A = 2 A = 2 A=2A=2A=2, we can state
THEOREM 4.2. By using the second order modulus of smoothness we can write the inequality
‖ f − T m f ‖ ≤ 3 16 ( 9 + m a m ) ω 2 ( f ; 1 m ) f − T m f ≤ 3 16 9 + m a m ω 2 f ; 1 m ||f-T_(m)f|| <= (3)/(16)(9+ma_(m))omega_(2)(f;(1)/(sqrtm))\left\|f-T_{m} f\right\| \leq \frac{3}{16}\left(9+m a_{m}\right) \omega_{2}\left(f ; \frac{1}{\sqrt{m}}\right)‖f−Tmf‖≤316(9+mam)ω2(f;1m)
Since in the case of Bernstein polynomial we have a m = 0 a m = 0 a_(m)=0a_{m}=0am=0, we arrive at an inequality of the form
‖ f − B m f ‖ ≤ C ω 2 ( f ; 1 m ) f − B m f ≤ C ω 2 f ; 1 m ||f-B_(m)f|| <= Comega_(2)(f;(1)/(sqrtm))\left\|f-B_{m} f\right\| \leq C \omega_{2}\left(f ; \frac{1}{\sqrt{m}}\right)‖f−Bmf‖≤Cω2(f;1m)
where C = 27 16 = 1 , 6875 C = 27 16 = 1 , 6875 C=(27)/(16)=1,6875C=\frac{27}{16}=1,6875C=2716=1,6875. This value of the constant C C CCC was obtained in another way in [4].

5. A BINOMIAL TYPE OPERATOR DEPENDING ON FOUR PARAMETERS

  1. If we consider the basic polynomials of binomial type p n α , β ( x ) == x ( x + α + n β ) [ n − 1 , − α ] p n α , β ( x ) == x ( x + α + n β ) [ n − 1 , − α ] p_(n)^(alpha,beta)(x)==x(x+alpha+n beta)^([n-1,-alpha])p_{n}^{\alpha, \beta}(x)= =x(x+\alpha+n \beta)^{[n-1,-\alpha]}pnα,β(x)==x(x+α+nβ)[n−1,−α], depending on two nonnegative parameters, we can write the identity
∑ k = 0 m ( m k ) x ( x + α + k β ) [ k − 1 , − α ] y ( y + α + ( m − k ) β ) [ m − 1 , − α ] = = ( x + y ) ( x + y + α + m β ) [ m − 1 , − α ] ∑ k = 0 m   ( m k ) x ( x + α + k β ) [ k − 1 , − α ] y ( y + α + ( m − k ) β ) [ m − 1 , − α ] = = ( x + y ) ( x + y + α + m β ) [ m − 1 , − α ] {:[sum_(k=0)^(m)((m)/(k))x(x+alpha+k beta)^([k-1,-alpha])y(y+alpha+(m-k)beta)^([m-1,-alpha])=],[=(x+y)(x+y+alpha+m beta)^([m-1,-alpha])]:}\begin{gathered} \sum_{k=0}^{m}\binom{m}{k} x(x+\alpha+k \beta)^{[k-1,-\alpha]} y(y+\alpha+(m-k) \beta)^{[m-1,-\alpha]}= \\ =(x+y)(x+y+\alpha+m \beta)^{[m-1,-\alpha]} \end{gathered}∑k=0m(mk)x(x+α+kβ)[k−1,−α]y(y+α+(m−k)β)[m−1,−α]==(x+y)(x+y+α+mβ)[m−1,−α]
If we set y = 1 − x y = 1 − x y=1-xy=1-xy=1−x we obtain
(5.1) ∑ k = 0 m ( m k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k , − α ] = = ( 1 + α + m β ) [ m − 1 , − α ] (5.1) ∑ k = 0 m   ( m k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k , − α ] = = ( 1 + α + m β ) [ m − 1 , − α ] {:[(5.1)sum_(k=0)^(m)((m)/(k))x(x+alpha+k beta)^([k-1,-alpha])(1-x)(1-x+alpha+(m-k)beta)^([m-1-k,-alpha])=],[=(1+alpha+m beta)^([m-1,-alpha])]:}\begin{gather*} \sum_{k=0}^{m}\binom{m}{k} x(x+\alpha+k \beta)^{[k-1,-\alpha]}(1-x)(1-x+\alpha+(m-k) \beta)^{[m-1-k,-\alpha]}= \tag{5.1}\\ =(1+\alpha+m \beta)^{[m-1,-\alpha]} \end{gather*}(5.1)∑k=0m(mk)x(x+α+kβ)[k−1,−α](1−x)(1−x+α+(m−k)β)[m−1−k,−α]==(1+α+mβ)[m−1,−α]
By starting from this equality we can construct a linear positive operator Q m α , β , γ , δ Q m α , β , γ , δ Q_(m)^(alpha,beta,gamma,delta)Q_{m}^{\alpha, \beta, \gamma, \delta}Qmα,β,γ,δ, depending on four parameters, defined, for any function f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1], by the formula
(5.2) ( Q m α , β , γ , δ , f ) ( x ) = ∑ k = 0 m q m , k α , β ( x ) f ( f + γ m + δ ) , (5.2) Q m α , β , γ , δ , f ( x ) = ∑ k = 0 m   q m , k α , β ( x ) f f + γ m + δ , {:(5.2)(Q_(m)^(alpha,beta,gamma,delta),f)(x)=sum_(k=0)^(m)q_(m,k)^(alpha,beta)(x)f((f+gamma)/(m+delta))",":}\begin{equation*} \left(Q_{m}^{\alpha, \beta, \gamma, \delta}, f\right)(x)=\sum_{k=0}^{m} q_{m, k}^{\alpha, \beta}(x) f\left(\frac{f+\gamma}{m+\delta}\right), \tag{5.2} \end{equation*}(5.2)(Qmα,β,γ,δ,f)(x)=∑k=0mqm,kα,β(x)f(f+γm+δ),
where 0 ≤ γ ≤ δ 0 ≤ γ ≤ δ 0 <= gamma <= delta0 \leq \gamma \leq \delta0≤γ≤δ and
(5.3) ( 1 + α + m β ) [ m − 1 , − α ] ⋅ q m , k α , β ( x ) = = ( m k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k , − α ] (5.3) ( 1 + α + m β ) [ m − 1 , − α ] ⋅ q m , k α , β ( x ) = = ( m k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) m − 1 − k , − α {:[(5.3)(1+alpha+m beta)^([m-1,-alpha])*q_(m,k)^(alpha,beta)(x)=],[=((m)/(k))x(x+alpha+k beta)^([k-1,-alpha])(1-x)(1-x+alpha+(m-k)beta)^([m-1-k_(,)-alpha])]:}\begin{gather*} (1+\alpha+m \beta)^{[m-1,-\alpha]} \cdot q_{m, k}^{\alpha, \beta}(x)= \tag{5.3}\\ =\binom{m}{k} x(x+\alpha+k \beta)^{[k-1,-\alpha]}(1-x)(1-x+\alpha+(m-k) \beta)^{\left[m-1-k_{,}-\alpha\right]} \end{gather*}(5.3)(1+α+mβ)[m−1,−α]⋅qm,kα,β(x)==(mk)x(x+α+kβ)[k−1,−α](1−x)(1−x+α+(m−k)β)[m−1−k,−α]
In the case α = γ = δ = 0 α = γ = δ = 0 alpha=gamma=delta=0\alpha=\gamma=\delta=0α=γ=δ=0 it reduces to an operator Q m Q m Q_(m)Q_{m}Qm of Cheney-Sharma [1], while for β = γ = δ = 0 β = γ = δ = 0 beta=gamma=delta=0\beta=\gamma=\delta=0β=γ=δ=0 one obtains an operator S m α S m α S_(m)^(alpha)S_{m}^{\alpha}Smα of D. D. Stancu [18].
Since in the points x = 0 x = 0 x=0x=0x=0 and x = 1 x = 1 x=1x=1x=1 we have f ( γ m + δ ) f γ m + δ f((gamma)/(m+delta))f\left(\frac{\gamma}{m+\delta}\right)f(γm+δ), respectively f ( m + γ m + δ ) f m + γ m + δ f((m+gamma)/(m+delta))f\left(\frac{m+\gamma}{m+\delta}\right)f(m+γm+δ), we can see that the polynomial
(5.4) ( Q m α , β f ) ( x ) = ∑ k = 0 m q m , k α , β ( x ) f ( k m ) (5.4) Q m α , β f ( x ) = ∑ k = 0 m   q m , k α , β ( x ) f k m {:(5.4)(Q_(m)^(alpha,beta)f)(x)=sum_(k=0)^(m)q_(m,k)^(alpha,beta)(x)f((k)/(m)):}\begin{equation*} \left(Q_{m}^{\alpha, \beta} f\right)(x)=\sum_{k=0}^{m} q_{m, k}^{\alpha, \beta}(x) f\left(\frac{k}{m}\right) \tag{5.4} \end{equation*}(5.4)(Qmα,βf)(x)=∑k=0mqm,kα,β(x)f(km)
has the property that
( Q m α , β f ) ( 0 ) = f ( 0 ) , ( Q m α , β f ) ( 1 ) = f ( 1 ) Q m α , β f ( 0 ) = f ( 0 ) , Q m α , β f ( 1 ) = f ( 1 ) (Q_(m)^(alpha,beta)f)(0)=f(0),quad(Q_(m)^(alpha,beta)f)(1)=f(1)\left(Q_{m}^{\alpha, \beta} f\right)(0)=f(0), \quad\left(Q_{m}^{\alpha, \beta} f\right)(1)=f(1)(Qmα,βf)(0)=f(0),(Qmα,βf)(1)=f(1)
Hence, it is expected that the approximation formula
(5.5) f ( x ) = ( Q m α , β f ) ( x ) + ( R m α , β f ) ( x ) (5.5) f ( x ) = Q m α , β f ( x ) + R m α , β f ( x ) {:(5.5)f(x)=(Q_(m)^(alpha,beta)f)(x)+(R_(m)^(alpha,beta)f)(x):}\begin{equation*} f(x)=\left(Q_{m}^{\alpha, \beta} f\right)(x)+\left(R_{m}^{\alpha, \beta} f\right)(x) \tag{5.5} \end{equation*}(5.5)f(x)=(Qmα,βf)(x)+(Rmα,βf)(x)
has the degree of exactness equal with one.
According to the identity (5.1), we can see that
(5.6) Q m α , β , γ , δ e 0 = e 0 (5.6) Q m α , β , γ , δ e 0 = e 0 {:(5.6)Q_(m)^(alpha,beta,gamma,delta)e_(0)=e_(0):}\begin{equation*} Q_{m}^{\alpha, \beta, \gamma, \delta} e_{0}=e_{0} \tag{5.6} \end{equation*}(5.6)Qmα,β,γ,δe0=e0
In the case e 1 ( x ) = x e 1 ( x ) = x e_(1)(x)=xe_{1}(x)=xe1(x)=x we have
(5.7) ( Q m α , β , γ , δ e 1 ) ( x ) = γ m + δ + m m + δ ( Q m α , β e 1 ) ( x ) (5.7) Q m α , β , γ , δ e 1 ( x ) = γ m + δ + m m + δ Q m α , β e 1 ( x ) {:(5.7)(Q_(m)^(alpha,beta,gamma,delta)e_(1))(x)=(gamma)/(m+delta)+(m)/(m+delta)(Q_(m)^(alpha,beta)e_(1))(x):}\begin{equation*} \left(Q_{m}^{\alpha, \beta, \gamma, \delta} e_{1}\right)(x)=\frac{\gamma}{m+\delta}+\frac{m}{m+\delta}\left(Q_{m}^{\alpha, \beta} e_{1}\right)(x) \tag{5.7} \end{equation*}(5.7)(Qmα,β,γ,δe1)(x)=γm+δ+mm+δ(Qmα,βe1)(x)
In accordance with (5.3) and (5.4), we can write
( 1 + α + m β ) [ m − 1 , − α ] ( Q m α , β e 1 ) ( x ) = = ∑ k = 1 m k m ( m k ) x ( x + α + k β ) [ k − 1 ; − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k , − α ] = = ∑ k = 1 m ( m − 1 k − 1 ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k , − α ] . ( 1 + α + m β ) [ m − 1 , − α ] Q m α , β e 1 ( x ) = = ∑ k = 1 m   k m ( m k ) x ( x + α + k β ) [ k − 1 ; − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k , − α ] = = ∑ k = 1 m   ( m − 1 k − 1 ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k , − α ] . {:[(1+alpha+m beta)^([m-1,-alpha])(Q_(m)^(alpha,beta)e_(1))(x)=],[=sum_(k=1)^(m)(k)/(m)((m)/(k))x(x+alpha+k beta)^([k-1;-alpha])(1-x)(1-x+alpha+(m-k)beta)^([m-1-k,-alpha])=],[=sum_(k=1)^(m)((m-1)/(k-1))x(x+alpha+k beta)^([k-1,-alpha])(1-x)(1-x+alpha+(m-k)beta)^([m-1-k,-alpha]).]:}\begin{gathered} (1+\alpha+m \beta)^{[m-1,-\alpha]}\left(Q_{m}^{\alpha, \beta} e_{1}\right)(x)= \\ =\sum_{k=1}^{m} \frac{k}{m}\binom{m}{k} x(x+\alpha+k \beta)^{[k-1 ;-\alpha]}(1-x)(1-x+\alpha+(m-k) \beta)^{[m-1-k,-\alpha]}= \\ =\sum_{k=1}^{m}\binom{m-1}{k-1} x(x+\alpha+k \beta)^{[k-1,-\alpha]}(1-x)(1-x+\alpha+(m-k) \beta)^{[m-1-k,-\alpha]} . \end{gathered}(1+α+mβ)[m−1,−α](Qmα,βe1)(x)==∑k=1mkm(mk)x(x+α+kβ)[k−1;−α](1−x)(1−x+α+(m−k)β)[m−1−k,−α]==∑k=1m(m−1k−1)x(x+α+kβ)[k−1,−α](1−x)(1−x+α+(m−k)β)[m−1−k,−α].
If we change the index of summation k − 1 = j k − 1 = j k-1=jk-1=jk−1=j, we obtain
(5.8) ( 1 + α + m β ) [ m − 1 , − α ] ( Q m α , β e 1 ) ( x ) = (5.8) ( 1 + α + m β ) [ m − 1 , − α ] Q m α , β e 1 ( x ) = {:(5.8)(1+alpha+m beta)^([m-1,-alpha])(Q_(m)^(alpha,beta)e_(1))(x)=:}\begin{equation*} (1+\alpha+m \beta)^{[m-1,-\alpha]}\left(Q_{m}^{\alpha, \beta} e_{1}\right)(x)= \tag{5.8} \end{equation*}(5.8)(1+α+mβ)[m−1,−α](Qmα,βe1)(x)=
= x ∑ j = 0 m − 1 ( m − 1 j ) ( x + α + β + j β ) [ j , − α ] ( 1 − x ) ( 1 − x + ( m − 1 − j ) β ) [ m − 2 − j , − α ] = x ∑ j = 0 m − 1   ( m − 1 j ) ( x + α + β + j β ) [ j , − α ] ( 1 − x ) ( 1 − x + ( m − 1 − j ) β ) [ m − 2 − j , − α ] =xsum_(j=0)^(m-1)((m-1)/(j))(x+alpha+beta+j beta)^([j,-alpha])(1-x)(1-x+(m-1-j)beta)^([m-2-j,-alpha])=x \sum_{j=0}^{m-1}\binom{m-1}{j}(x+\alpha+\beta+j \beta)^{[j,-\alpha]}(1-x)(1-x+(m-1-j) \beta)^{[m-2-j,-\alpha]}=x∑j=0m−1(m−1j)(x+α+β+jβ)[j,−α](1−x)(1−x+(m−1−j)β)[m−2−j,−α].
Now we consider an extension to factorial powers, with increment h h hhh, of an Abel combinatorial formula
(5.9) ( u + v + n β ) [ n , h ] = ∑ k = 0 m ( n k ) ( u + k β ) [ k , h ] v ( v + ( n − k ) β − h ) [ n − 1 − k , h ] (5.9) ( u + v + n β ) [ n , h ] = ∑ k = 0 m   ( n k ) ( u + k β ) [ k , h ] v ( v + ( n − k ) β − h ) [ n − 1 − k , h ] {:(5.9)(u+v+n beta)^([n,h])=sum_(k=0)^(m)((n)/(k))(u+k beta)^([k,h])v(v+(n-k)beta-h)^([n-1-k,h]):}\begin{equation*} (u+v+n \beta)^{[n, h]}=\sum_{k=0}^{m}\binom{n}{k}(u+k \beta)^{[k, h]} v(v+(n-k) \beta-h)^{[n-1-k, h]} \tag{5.9} \end{equation*}(5.9)(u+v+nβ)[n,h]=∑k=0m(nk)(u+kβ)[k,h]v(v+(n−k)β−h)[n−1−k,h]
If we replace here n = m − 1 , h = − α , u = x + α + β n = m − 1 , h = − α , u = x + α + β n=m-1,h=-alpha,u=x+alpha+betan=m-1, h=-\alpha, u=x+\alpha+\betan=m−1,h=−α,u=x+α+β and v = 1 − x v = 1 − x v=1-xv=1-xv=1−x, we obtain
(5.10) ( 1 + α + m β ) [ m − 1 , − α ] = (5.10) ( 1 + α + m β ) [ m − 1 , − α ] = {:(5.10)(1+alpha+m beta)^([m-1,-alpha])=:}\begin{equation*} (1+\alpha+m \beta)^{[m-1,-\alpha]}= \tag{5.10} \end{equation*}(5.10)(1+α+mβ)[m−1,−α]=
= ∑ k = 0 m − 1 ( m − 1 k ) ( x + α + β + k β ) [ k , − α ] ( 1 − x ) ( 1 − x + ( m − 1 − k ) β ) | m − 2 − k , − α | . = ∑ k = 0 m − 1   ( m − 1 k ) ( x + α + β + k β ) [ k , − α ] ( 1 − x ) ( 1 − x + ( m − 1 − k ) β ) | m − 2 − k , − α | . =sum_(k=0)^(m-1)((m-1)/(k))(x+alpha+beta+k beta)^([k,-alpha])(1-x)(1-x+(m-1-k)beta)^(|m-2-k,-alpha|).=\sum_{k=0}^{m-1}\binom{m-1}{k}(x+\alpha+\beta+k \beta)^{[k,-\alpha]}(1-x)(1-x+(m-1-k) \beta)^{|m-2-k,-\alpha|} .=∑k=0m−1(m−1k)(x+α+β+kβ)[k,−α](1−x)(1−x+(m−1−k)β)|m−2−k,−α|.
By taking into account the identities (5.8) and (5.10), we obtain Q m α , β e 1 = e 1 Q m α , β e 1 = e 1 Q_(m)^(alpha,beta)e_(1)=e_(1)Q_{m}^{\alpha, \beta} e_{1}=e_{1}Qmα,βe1=e1. Consequently, the degree of exactness of formula (5.5) is indeed one.
If we replace this result in (5.7), we find that
(5.11) ( Q m α , β , γ , δ e 1 ) ( x ) = m x + γ m + δ = x + γ − δ x m + δ (5.11) Q m α , β , γ , δ e 1 ( x ) = m x + γ m + δ = x + γ − δ x m + δ {:(5.11)(Q_(m)^(alpha,beta,gamma,delta)e_(1))(x)=(mx+gamma)/(m+delta)=x+(gamma-delta x)/(m+delta):}\begin{equation*} \left(Q_{m}^{\alpha, \beta, \gamma, \delta} e_{1}\right)(x)=\frac{m x+\gamma}{m+\delta}=x+\frac{\gamma-\delta x}{m+\delta} \tag{5.11} \end{equation*}(5.11)(Qmα,β,γ,δe1)(x)=mx+γm+δ=x+γ−δxm+δ
In the case of the monomial e 2 ( x ) = x 2 e 2 ( x ) = x 2 e_(2)(x)=x^(2)e_{2}(x)=x^{2}e2(x)=x2 we have
( Q m α , β , γ , δ e 2 ) ( x ) = 1 ( m + δ ) 2 [ γ 2 + 2 m γ x + m 2 ( Q m α , β e 2 ) ( x ) ] Q m α , β , γ , δ e 2 ( x ) = 1 ( m + δ ) 2 γ 2 + 2 m γ x + m 2 Q m α , β e 2 ( x ) (Q_(m)^(alpha,beta,gamma,delta)e_(2))(x)=(1)/((m+delta)^(2))[gamma^(2)+2m gamma x+m^(2)(Q_(m)^(alpha,beta)e_(2))(x)]\left(Q_{m}^{\alpha, \beta, \gamma, \delta} e_{2}\right)(x)=\frac{1}{(m+\delta)^{2}}\left[\gamma^{2}+2 m \gamma x+m^{2}\left(Q_{m}^{\alpha, \beta} e_{2}\right)(x)\right](Qmα,β,γ,δe2)(x)=1(m+δ)2[γ2+2mγx+m2(Qmα,βe2)(x)]
Next we can write
( Q m α , β , e 2 ) ( x ) = 1 m ∑ k = 1 m [ k m + k ( k − 1 ) m ] q m , k α , β ( x ) = 1 m ( Q m α , β e 1 ) ( x ) + + 1 m ∑ k = 2 m ( m − 1 k − 2 ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) [ m − 1 − k 1 − α ] = x m + + m − 1 m ∑ j = 0 m − 2 ( m − 2 j ) ( x + α + 2 β + j β ) [ j + 1 , − α ] ( 1 − x + α + ( m − 2 − j ) β ) [ m − 3 − j , − α ] Q m α , β , e 2 ( x ) = 1 m ∑ k = 1 m   k m + k ( k − 1 ) m q m , k α , β ( x ) = 1 m Q m α , β e 1 ( x ) + + 1 m ∑ k = 2 m   ( m − 1 k − 2 ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + α + ( m − k ) β ) m − 1 − k 1 − α = x m + + m − 1 m ∑ j = 0 m − 2   ( m − 2 j ) ( x + α + 2 β + j β ) [ j + 1 , − α ] ( 1 − x + α + ( m − 2 − j ) β ) [ m − 3 − j , − α ] {:[(Q_(m)^(alpha,beta,)e_(2))(x)=(1)/(m)sum_(k=1)^(m)[(k)/(m)+(k(k-1))/(m)]q_(m,k)^(alpha,beta)(x)=(1)/(m)(Q_(m)^(alpha,beta)e_(1))(x)+],[+(1)/(m)sum_(k=2)^(m)((m-1)/(k-2))x(x+alpha+k beta)^([k-1,-alpha])(1-x)(1-x+alpha+(m-k)beta)^([m-1-k_(1)-alpha])=(x)/(m)+],[+(m-1)/(m)sum_(j=0)^(m-2)((m-2)/(j))(x+alpha+2beta+j beta)^([j+1,-alpha])(1-x+alpha+(m-2-j)beta)^([m-3-j,-alpha])]:}\begin{gathered} \left(Q_{m}^{\alpha, \beta,} e_{2}\right)(x)=\frac{1}{m} \sum_{k=1}^{m}\left[\frac{k}{m}+\frac{k(k-1)}{m}\right] q_{m, k}^{\alpha, \beta}(x)=\frac{1}{m}\left(Q_{m}^{\alpha, \beta} e_{1}\right)(x)+ \\ +\frac{1}{m} \sum_{k=2}^{m}\binom{m-1}{k-2} x(x+\alpha+k \beta)^{[k-1,-\alpha]}(1-x)(1-x+\alpha+(m-k) \beta)^{\left[m-1-k_{1}-\alpha\right]}=\frac{x}{m}+ \\ +\frac{m-1}{m} \sum_{j=0}^{m-2}\binom{m-2}{j}(x+\alpha+2 \beta+j \beta)^{[j+1,-\alpha]}(1-x+\alpha+(m-2-j) \beta)^{[m-3-j,-\alpha]} \end{gathered}(Qmα,β,e2)(x)=1m∑k=1m[km+k(k−1)m]qm,kα,β(x)=1m(Qmα,βe1)(x)++1m∑k=2m(m−1k−2)x(x+α+kβ)[k−1,−α](1−x)(1−x+α+(m−k)β)[m−1−k1−α]=xm++m−1m∑j=0m−2(m−2j)(x+α+2β+jβ)[j+1,−α](1−x+α+(m−2−j)β)[m−3−j,−α]
If we take into account (5.6) and (5.11) and if we use again the extension to factorial powers of the Abel combinatorial formula, we can apply the BohmanKorovkin convergence criterion and we can state
THEOREM 5.1. If f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] and the parameters α α alpha\alphaα and β β beta\betaβ are nonnegative and depend on m m mmm such that α = α ( m ) → 0 , m β ( m ) → 0 α = α ( m ) → 0 , m β ( m ) → 0 alpha=alpha(m)rarr0,m beta(m)rarr0\alpha=\alpha(m) \rightarrow 0, m \beta(m) \rightarrow 0α=α(m)→0,mβ(m)→0, when m → ∞ m → ∞ m rarr oom \rightarrow \inftym→∞, then we have
lim m → ∞ Q m α , β , γ , δ f = f lim m → ∞   Q m α , β , γ , δ f = f lim_(m rarr oo)Q_(m)^(alpha,beta,gamma,delta)f=f\lim _{m \rightarrow \infty} Q_{m}^{\alpha, \beta, \gamma, \delta} f=flimm→∞Qmα,β,γ,δf=f
uniformly on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].

6. AN INTEGRAL REPRESENTATION OF THE REMAINDER IN THE APPROXIMATION FORMULA BY Q m α , β Q m α , β Q_(m)^(alpha,beta)Q_{m}^{\alpha, \beta}Qmα,β

  1. Since the degree of exactness of the approximation formula (5.5) is one, we can apply the Peano theorem in order to find an integral representation of the remainder term.
We can state
THEOREM 6.1. Let x x xxx be any fixed point in [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]. If f ∈ C 2 [ 0 , 1 ] f ∈ C 2 [ 0 , 1 ] f inC^(2)[0,1]f \in C^{2}[0,1]f∈C2[0,1], then the remainder of formula (5.5) can be represented under the following integral form
(6.1) ( R m α , β f ) ( x ) = ∫ 0 1 G m α , β ( t ; x ) f ′ ′ ( t ) d t (6.1) R m α , β f ( x ) = ∫ 0 1   G m α , β ( t ; x ) f ′ ′ ( t ) d t {:(6.1)(R_(m)^(alpha,beta)f)(x)=int_(0)^(1)G_(m)^(alpha,beta)(t;x)f^('')(t)dt:}\begin{equation*} \left(R_{m}^{\alpha, \beta} f\right)(x)=\int_{0}^{1} G_{m}^{\alpha, \beta}(t ; x) f^{\prime \prime}(t) \mathrm{d} t \tag{6.1} \end{equation*}(6.1)(Rmα,βf)(x)=∫01Gmα,β(t;x)f′′(t)dt
where the Peano kernel G m α , β G m α , β G_(m)^(alpha,beta)G_{m}^{\alpha, \beta}Gmα,β is defined by the formula
G m α , β ( t ; x ) = ( R m α , β φ x ) ( t ) , φ x ( t ) = 1 2 [ x − t + | x − t | ] , G m α , β ( t ; x ) = R m α , β φ x ( t ) , φ x ( t ) = 1 2 [ x − t + | x − t | ] , G_(m)^(alpha,beta)(t;x)=(R_(m)^(alpha,beta)varphi_(x))(t),quadvarphi_(x)(t)=(1)/(2)[x-t+|x-t|],G_{m}^{\alpha, \beta}(t ; x)=\left(R_{m}^{\alpha, \beta} \varphi_{x}\right)(t), \quad \varphi_{x}(t)=\frac{1}{2}[x-t+|x-t|],Gmα,β(t;x)=(Rmα,βφx)(t),φx(t)=12[x−t+|x−t|],
the operator acting on φ x ( t ) φ x ( t ) varphi_(x)(t)\varphi_{x}(t)φx(t) as a function of x x xxx.
We shall derive an explicit formula for the Peano kernel.
By starting from the equation
(6.2) ( R m α , β φ x ) ( t ) = ( x − t ) + − ∑ k = 0 ∞ q m , k α , β ( x ) ( k m − t ) + , (6.2) R m α , β φ x ( t ) = ( x − t ) + − ∑ k = 0 ∞   q m , k α , β ( x ) k m − t + , {:(6.2)(R_(m)^(alpha,beta)varphi_(x))(t)=(x-t)_(+)-sum_(k=0)^(oo)q_(m,k)^(alpha,beta)(x)((k)/(m)-t)_(+)",":}\begin{equation*} \left(R_{m}^{\alpha, \beta} \varphi_{x}\right)(t)=(x-t)_{+}-\sum_{k=0}^{\infty} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right)_{+}, \tag{6.2} \end{equation*}(6.2)(Rmα,βφx)(t)=(x−t)+−∑k=0∞qm,kα,β(x)(km−t)+,
we shall be able to deduce an explicit expression for G m α , β G m α , β G_(m)^(alpha,beta)G_{m}^{\alpha, \beta}Gmα,β.
We can state
THEOREM 6.2. Assuming that x ∈ [ s − 1 m , s m ] ( 1 ≤ s ≤ m ) x ∈ s − 1 m , s m ( 1 ≤ s ≤ m ) x in[(s-1)/(m),(s)/(m)](1 <= s <= m)x \in\left[\frac{s-1}{m}, \frac{s}{m}\right](1 \leq s \leq m)x∈[s−1m,sm](1≤s≤m), the Peano kernel G m α , β G m α , β G_(m)^(alpha,beta)G_{m}^{\alpha, \beta}Gmα,β, corresponding to the operator Q m α , β Q m α , β Q_(m)^(alpha,beta)Q_{m}^{\alpha, \beta}Qmα,β, can be represented as follows:
[ − ∑ k = 0 j − 1 q m , k α , β ( x ) ( x − k m ) if t ∈ [ j − 1 m , j m ] ( 1 ≤ j ≤ s − 1 ) (6.3) G m α , β ( t ; x ) = { − ∑ k = 0 s − 1 q m , k α , β ( x ) ( x − k m ) if − ∑ k = s m q m , k α , β ( x ) ( k m − t ) if − ∑ k = j m q m , k α , β ( x ) t ∈ ( x , s m ] ] . − ∑ k = 0 j − 1   q m , k α , β ( x ) x − k m  if  t ∈ j − 1 m , j m ( 1 ≤ j ≤ s − 1 ) (6.3) G m α , β ( t ; x ) = − ∑ k = 0 s − 1   q m , k α , β ( x ) x − k m  if  − ∑ k = s m   q m , k α , β ( x ) k m − t  if  − ∑ k = j m   q m , k α , β ( x ) t ∈ x , s m . {:[[-sum_(k=0)^(j-1)q_(m,k)^(alpha,beta)(x)quad(x-(k)/(m))quad" if "quad t in[(j-1)/(m),(j)/(m)]quad(1 <= j <= s-1):}],[(6.3)G_(m)^(alpha,beta)(t;x)={[-sum_(k=0)^(s-1)q_(m,k)^(alpha,beta)(x),(x-(k)/(m))," if "],[-sum_(k=s)^(m)q_(m,k)^(alpha,beta)(x),((k)/(m)-t)," if "],[-sum_(k=j)^(m)q_(m,k)^(alpha,beta)(x),t in(x,(s)/(m)]]].]:}\begin{align*} & {\left[-\sum_{k=0}^{j-1} q_{m, k}^{\alpha, \beta}(x) \quad\left(x-\frac{k}{m}\right) \quad \text { if } \quad t \in\left[\frac{j-1}{m}, \frac{j}{m}\right] \quad(1 \leq j \leq s-1)\right.} \\ & G_{m}^{\alpha, \beta}(t ; x)=\left\{\begin{array}{lll} -\sum_{k=0}^{s-1} q_{m, k}^{\alpha, \beta}(x) & \left(x-\frac{k}{m}\right) & \text { if } \\ -\sum_{k=s}^{m} q_{m, k}^{\alpha, \beta}(x) & \left(\frac{k}{m}-t\right) & \text { if } \\ -\sum_{k=j}^{m} q_{m, k}^{\alpha, \beta}(x) & t \in\left(x, \frac{s}{m}\right] \end{array}\right] . \tag{6.3} \end{align*}[−∑k=0j−1qm,kα,β(x)(x−km) if t∈[j−1m,jm](1≤j≤s−1)(6.3)Gmα,β(t;x)={−∑k=0s−1qm,kα,β(x)(x−km) if −∑k=smqm,kα,β(x)(km−t) if −∑k=jmqm,kα,β(x)t∈(x,sm]].
Proof. Since x ∈ [ s − 1 m , s m ] x ∈ s − 1 m , s m x in[(s-1)/(m),(s)/(m)]x \in\left[\frac{s-1}{m}, \frac{s}{m}\right]x∈[s−1m,sm], if we assume that t ∈ [ j − 1 m , j m ] ( 1 ≤ j ≤ s − 1 ) t ∈ j − 1 m , j m ( 1 ≤ j ≤ s − 1 ) t in[(j-1)/(m),(j)/(m)]quad(1 <= j <= s-1)t \in\left[\frac{j-1}{m}, \frac{j}{m}\right] \quad(1 \leq j \leq s-1)t∈[j−1m,jm](1≤j≤s−1), we can write
(6.4) G m α , β ( t ; x ) = x − t − ∑ k ≥ j q m , k α , β ( x ) ( k m − t ) (6.4) G m α , β ( t ; x ) = x − t − ∑ k ≥ j   q m , k α , β ( x ) k m − t {:(6.4)G_(m)^(alpha,beta)(t;x)=x-t-sum_(k >= j)q_(m,k)^(alpha,beta)(x)((k)/(m)-t):}\begin{equation*} G_{m}^{\alpha, \beta}(t ; x)=x-t-\sum_{k \geq j} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right) \tag{6.4} \end{equation*}(6.4)Gmα,β(t;x)=x−t−∑k≥jqm,kα,β(x)(km−t)
In the case t ∈ [ s − 1 m , x ] t ∈ s − 1 m , x t in[(s-1)/(m),x]t \in\left[\frac{s-1}{m}, x\right]t∈[s−1m,x] we obtain
(6.5) G m α , β ( t ; x ) = x − t − ∑ k ≥ s q m , k α , β ( x ) ( k m − t ) (6.5) G m α , β ( t ; x ) = x − t − ∑ k ≥ s   q m , k α , β ( x ) k m − t {:(6.5)G_(m)^(alpha,beta)(t;x)=x-t-sum_(k >= s)q_(m,k)^(alpha,beta)(x)((k)/(m)-t):}\begin{equation*} G_{m}^{\alpha, \beta}(t ; x)=x-t-\sum_{k \geq s} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right) \tag{6.5} \end{equation*}(6.5)Gmα,β(t;x)=x−t−∑k≥sqm,kα,β(x)(km−t)
and when t ∈ [ x , s m ] t ∈ x , s m t in[x,(s)/(m)]t \in\left[x, \frac{s}{m}\right]t∈[x,sm] we get
G m α , β ( t ; x ) = − ∑ k = s m q m , k α , β ( x ) ( k m − t ) . G m α , β ( t ; x ) = − ∑ k = s m   q m , k α , β ( x ) k m − t . G_(m)^(alpha,beta)(t;x)=-sum_(k=s)^(m)q_(m,k)^(alpha,beta)(x)((k)/(m)-t).G_{m}^{\alpha, \beta}(t ; x)=-\sum_{k=s}^{m} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right) .Gmα,β(t;x)=−∑k=smqm,kα,β(x)(km−t).
When t ∈ [ j − 1 m , j m ] ( j > s ) t ∈ j − 1 m , j m ( j > s ) t in[(j-1)/(m),(j)/(m)](j > s)t \in\left[\frac{j-1}{m}, \frac{j}{m}\right](j>s)t∈[j−1m,jm](j>s) we have
G m α , β ( t ; x ) = − ∑ k = j m q m , k α , β ( x ) ( k m − t ) G m α , β ( t ; x ) = − ∑ k = j m   q m , k α , β ( x ) k m − t G_(m)^(alpha,beta)(t;x)=-sum_(k=j)^(m)q_(m,k)^(alpha,beta)(x)((k)/(m)-t)G_{m}^{\alpha, \beta}(t ; x)=-\sum_{k=j}^{m} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right)Gmα,β(t;x)=−∑k=jmqm,kα,β(x)(km−t)
Since the degree of exactness of formula (5.5) is one, by replacing in it f ( x ) = x − t f ( x ) = x − t f(x)=x-tf(x)=x-tf(x)=x−t, the corresponding remainder vanishes and we obtain
x − t = ∑ k = 0 m q m , k α , β ( x ) ( k m − t ) = ∑ k = 0 j − 1 q m , k α , β ( x ) ( k m − t ) + ∑ k = j m q m , k α , β ( x ) ( k m − t ) . x − t = ∑ k = 0 m   q m , k α , β ( x ) k m − t = ∑ k = 0 j − 1   q m , k α , β ( x ) k m − t + ∑ k = j m   q m , k α , β ( x ) k m − t . x-t=sum_(k=0)^(m)q_(m,k)^(alpha,beta)(x)((k)/(m)-t)=sum_(k=0)^(j-1)q_(m,k)^(alpha,beta)(x)((k)/(m)-t)+sum_(k=j)^(m)q_(m,k)^(alpha,beta)(x)((k)/(m)-t).x-t=\sum_{k=0}^{m} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right)=\sum_{k=0}^{j-1} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right)+\sum_{k=j}^{m} q_{m, k}^{\alpha, \beta}(x)\left(\frac{k}{m}-t\right) .x−t=∑k=0mqm,kα,β(x)(km−t)=∑k=0j−1qm,kα,β(x)(km−t)+∑k=jmqm,kα,β(x)(km−t).
Consequently, if x ∈ [ s − 1 m , s m ] x ∈ s − 1 m , s m x in[(s-1)/(m),(s)/(m)]x \in\left[\frac{s-1}{m}, \frac{s}{m}\right]x∈[s−1m,sm] and t ∈ [ j − 1 m , j m ] t ∈ j − 1 m , j m t in[(j-1)/(m),(j)/(m)]t \in\left[\frac{j-1}{m}, \frac{j}{m}\right]t∈[j−1m,jm] we have
G m α , β ( t ; x ) = − ∑ k = 0 j − 1 q m , k α , β ( x ) ( t − k m ) G m α , β ( t ; x ) = − ∑ k = 0 j − 1   q m , k α , β ( x ) t − k m G_(m)^(alpha,beta)(t;x)=-sum_(k=0)^(j-1)q_(m,k)^(alpha,beta)(x)(t-(k)/(m))G_{m}^{\alpha, \beta}(t ; x)=-\sum_{k=0}^{j-1} q_{m, k}^{\alpha, \beta}(x)\left(t-\frac{k}{m}\right)Gmα,β(t;x)=−∑k=0j−1qm,kα,β(x)(t−km)
while for t ∈ ( s − 1 m , x ] t ∈ s − 1 m , x t in((s-1)/(m),x]t \in\left(\frac{s-1}{m}, x\right]t∈(s−1m,x] we obtain
G m α , β ( t ; x ) = − ∑ k = 0 s − 1 q m , k α , β ( x ) ( t − k m ) . G m α , β ( t ; x ) = − ∑ k = 0 s − 1   q m , k α , β ( x ) t − k m . G_(m)^(alpha,beta)(t;x)=-sum_(k=0)^(s-1)q_(m,k)^(alpha,beta)(x)(t-(k)/(m)).G_{m}^{\alpha, \beta}(t ; x)=-\sum_{k=0}^{s-1} q_{m, k}^{\alpha, \beta}(x)\left(t-\frac{k}{m}\right) .Gmα,β(t;x)=−∑k=0s−1qm,kα,β(x)(t−km).
  1. We can also deduce a Cauchy type form for the remainder of the approximation formula (5.5).
THEOREM 6.3. If f ∈ C 2 [ 0 , 1 ] f ∈ C 2 [ 0 , 1 ] f inC^(2)[0,1]f \in C^{2}[0,1]f∈C2[0,1], the remainder of the approximation formula (5.5) can be represented under the form
(6.6) ( R m α , β f ) ( x ) = 1 2 ( R m α , β e 2 ) ( x ) f ′ ′ ( ξ ) , 0 < ξ < 1 . (6.6) R m α , β f ( x ) = 1 2 R m α , β e 2 ( x ) f ′ ′ ( ξ ) , 0 < ξ < 1 . {:(6.6)(R_(m)^(alpha,beta)f)(x)=(1)/(2)(R_(m)^(alpha,beta)e_(2))(x)f^('')(xi)","quad0 < xi < 1.:}\begin{equation*} \left(R_{m}^{\alpha, \beta} f\right)(x)=\frac{1}{2}\left(R_{m}^{\alpha, \beta} e_{2}\right)(x) f^{\prime \prime}(\xi), \quad 0<\xi<1 . \tag{6.6} \end{equation*}(6.6)(Rmα,βf)(x)=12(Rmα,βe2)(x)f′′(ξ),0<ξ<1.
Proof. From (6.3) it is easy to see that the function y = G m α , β ( x ) y = G m α , β ( x ) y=G_(m)^(alpha,beta)(x)y=G_{m}^{\alpha, \beta}(x)y=Gmα,β(x) represents a polygonal continuous line situated beneath the x x xxx-axis. Consequently, we can apply the mean value theorem to the integral occurring in (6.1) and we obtain
(6.7) ( R m α , β f ) ( x ) = f ′ ′ ( ξ ) ∫ 0 1 G m α , β ( t ; x ) d t , ξ ∈ 0 , 1 ) (6.7) R m α , β f ( x ) = f ′ ′ ( ξ ) ∫ 0 1   G m α , β ( t ; x ) d t , ξ ∈ 0 , 1 {:(6.7){:(R_(m)^(alpha,beta)f)(x)=f^('')(xi)int_(0)^(1)G_(m)^(alpha,beta)(t;x)dt,quad xi in0,1):}\begin{equation*} \left.\left(R_{m}^{\alpha, \beta} f\right)(x)=f^{\prime \prime}(\xi) \int_{0}^{1} G_{m}^{\alpha, \beta}(t ; x) \mathrm{d} t, \quad \xi \in 0,1\right) \tag{6.7} \end{equation*}(6.7)(Rmα,βf)(x)=f′′(ξ)∫01Gmα,β(t;x)dt,ξ∈0,1)
By inserting f = e 2 f = e 2 f=e_(2)f=e_{2}f=e2 in the approximation formula
f ( x ) = ( Q m α , β f ) ( x ) + f ′ ′ ( ξ ) ∫ 0 1 G m α , β ( t ; x ) d t f ( x ) = Q m α , β f ( x ) + f ′ ′ ( ξ ) ∫ 0 1   G m α , β ( t ; x ) d t f(x)=(Q_(m)^(alpha,beta)f)(x)+f^('')(xi)int_(0)^(1)G_(m)^(alpha,beta)(t;x)dtf(x)=\left(Q_{m}^{\alpha, \beta} f\right)(x)+f^{\prime \prime}(\xi) \int_{0}^{1} G_{m}^{\alpha, \beta}(t ; x) \mathrm{d} tf(x)=(Qmα,βf)(x)+f′′(ξ)∫01Gmα,β(t;x)dt
we obtain
∫ 0 1 G m α , β ( t ; x ) d t = 1 2 [ x 2 − ( Q m α , β e 2 ) ( x ) ] = 1 2 ( R m α , β e 2 ) ( x ) ∫ 0 1   G m α , β ( t ; x ) d t = 1 2 x 2 − Q m α , β e 2 ( x ) = 1 2 R m α , β e 2 ( x ) int_(0)^(1)G_(m)^(alpha,beta)(t;x)dt=(1)/(2)[x^(2)-(Q_(m)^(alpha,beta)e_(2))(x)]=(1)/(2)(R_(m)^(alpha,beta)e_(2))(x)\int_{0}^{1} G_{m}^{\alpha, \beta}(t ; x) \mathrm{d} t=\frac{1}{2}\left[x^{2}-\left(Q_{m}^{\alpha, \beta} e_{2}\right)(x)\right]=\frac{1}{2}\left(R_{m}^{\alpha, \beta} e_{2}\right)(x)∫01Gmα,β(t;x)dt=12[x2−(Qmα,βe2)(x)]=12(Rmα,βe2)(x)
If we replace this result in (6.7), we arrive just to formula (6.6).
Since the polynomial Q m α , β f Q m α , β f Q_(m)^(alpha,beta)fQ_{m}^{\alpha, \beta} fQmα,βf is interpolatory at both ends of [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], it is clear that ( R m α , β e 2 ) ( x ) R m α , β e 2 ( x ) (R_(m)^(alpha,beta)e_(2))(x)\left(R_{m}^{\alpha, \beta} e_{2}\right)(x)(Rmα,βe2)(x) contains the factor x ( x − 1 ) x ( x − 1 ) x(x-1)x(x-1)x(x−1).
In the case β = 0 β = 0 beta=0\beta=0β=0 we find
( R m α f ) ( x ) = x ( x − 1 ) 2 m ⋅ 1 + α m 1 + α f ′ ′ ( ξ ) R m α f ( x ) = x ( x − 1 ) 2 m ⋅ 1 + α m 1 + α f ′ ′ ( ξ ) (R_(m)^(alpha)f)(x)=(x(x-1))/(2m)*(1+alpha m)/(1+alpha)f^('')(xi)\left(R_{m}^{\alpha} f\right)(x)=\frac{x(x-1)}{2 m} \cdot \frac{1+\alpha m}{1+\alpha} f^{\prime \prime}(\xi)(Rmαf)(x)=x(x−1)2m⋅1+αm1+αf′′(ξ)
given in [20] for the remainder of the approximation formula by the operator S α = Q m α , 0 S α = Q m α , 0 S^(alpha)=Q_(m)^(alpha,0)S^{\alpha}=Q_{m}^{\alpha, 0}Sα=Qmα,0, introduced in 1968 in the papers [18] and [19] of the first author.

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Received September 20, 1997.

D. D. StancuFaculty of Mathematics and Computer Science"Babeş-Bolyai" UniversityStr. Kogălniceanu, nr. I3400 Cluj-Napoca, Romania

M. R. Occorsio
Istituto per Applicazioni della Matematica, CNR
Via P. Castellino, 111
80131 Napoli , Italia

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