View of An approximation operator of Stancu type

Rev. Anal. Num´er. Th´eor. Approx., vol. 34 (2005) no. 1, pp. 115–121 ictp.acad.ro/jnaat

AN APPROXIMATION OPERATOR OF STANCU TYPE

SILVIA TOADER^∗

Abstract. We study the behavior of Stancu-Goldman’s operator on the second degree functions.

MSC 2000. 41A10, 60E05.

Keywords. Stancu’s approximation operator, Friedman’s distribution.

1. INTRODUCTION

Using the P`olya-Eggenberger-Markov distribution, D. D. Stancu has defined in [4] a polynomial approximation operator, which today is bearing his name (see [1]). The above distribution was generalized by B. Friedman in [2]. Using this new distribution, R. N. Goldman has defined in [3] an approximation operator of Stancu type for which has proved that it preserves the first degree functions. In this paper, we begin the study of the behavior of this operator on the second degree functions. This is necessary for the application of Popoviciu- Bohman-Korovkin approximation theorem (see [1]).

2. FRIEDMAN’S DISTRIBUTION

In [2] it is considered the following probabilistic model: an urn contains N balls,aof which are white andbblack. A ball is draw out at random, its color noted and it is returned together with c balls of the same color and dballs of the opposite color. This procedure is repeated n times. We denote with P(n, k) the probability that the total number of white balls chosen be k.

This model was studied in detail in [3] where the following recurrence rela- tion

P(n, k) =P(n−1, k−1)·a+(k−1)·c+(n−k)·d

N+(n−1)·(c+d) +P(n−1, k)·b+(n−k−1)·c+k·d N+(n−1)·(c+d)

is established. If we denote

a

N =x, _N^c =α, _N^d =β

∗Department of Mathematics, Technical University of Cluj-Napoca, Romania, e-mail:

[email protected].

and P(n, k) =P(n, k, x) the recurrence relation becomes P(n, k, x) =s(n−1, k−1, x)·P(n−1, k−1, x)

+t(n−1, k, x)·P(n−1, k, x), where

s(n, k, x) = [x+k·α+ (n−k)·β]·ρn

and

t(n, k, x) = 1−s(n, k, x) with

ρ_n= _1+n·(α+β)¹ .

For β = 0 we obtain the P`olya-Eggenberger-Markov distribution with the explicit expression

P(n, k, x) = ⁿ_k^{x(k,−α)(1−x)}₁(n,−α)^{(n−k,−α)}, where

x^(k,−α)=x(x+α)...[x+ (k−1)α].

As we have shown in [5], such an expression seems to be impossible to obtain in the case β6= 0.

3. THE DETERMINATION OF THE MOMENTS

Though the expressions of the distribution are missing, as it is shown in [3]

we can calculate its moments of order r Mr(n, x) =

n

X

k=0

k^r·P(n, k, x).

Of course

M0(n, x) = 1, Mr(1, x) =x.

For the calculation of other moments, the following recurrence relation M_r(n+ 1, x) =

r

X

i=0

M_i(n, x)·Γ(r, n, x, i) is used, where

Γ(r, n, x, i) = Φ(r, n, i)·x+ Ψ(r, n, i) with

Φ(r, n, r) = 0, Ψ(r, n, r) = 1 +r·(α−β)·ρn

and

Φ(r, n, i) = ^r_i·ρ_n, Ψ(r, n, i) =^h _i−1^r ·(α−β) +β·n· ^r_iⁱ·ρ_n, i < r.

In the special case r = 1 we obtain:

M₁(n+ 1, x) = [1 + (α−β)·ρ_n]·M₁(n, x) + (x+β·n)·ρ_n which leads at

Lemma 1. There exist the constants pn>0, qn≥0, such that (1) M1(n, x) =pn·x+qn, n≥1.

The proof is done by mathematical induction, starting with the initial values p₁ = 1, q₁ = 0

obtaining the recurrence relations

pn+1 =µn·pn+ρn>0 and

q_n+1 =µ_n·q_n+β·n·ρ_n≥0, where

µn= [1 +α−β+ (α+β)·n]·ρn. To get explicit expressions we use the following

Lemma 2. If the sequence (xn)_n≥l verifies the recurrence relation xn+1=An·xn+Bn, n≥l

then it has the expression xn=

n−1

X

i=l

Bi

n−1

Y

j=i+1

Aj

+xl

n−1

Y

i=l

Ai, n > l with the convention

n−1

Y

j=n

· · ·= 1.

We obtain

Theorem 1. The coefficients of the mean values(1) have the expressions p_n=

n−1

X

i=0

ρ_i

n−1

Y

j=i+1

µ_j

and

q_n=β·

n−1

X

i=0

i·ρ_i

n−1

Y

j=i+1

µ_j

. Analogously, for the moments of order two we have

Theorem 2. There exist the positive constants u_n, v_n, w_n such that M2(1, x) =x

and

(2) M₂(n, x) =u_n·x²+v_n·x+w_n, n≥2,

where

u_n=

n−1

X

i=1

2·ρ_i·p_i·

n−1

Y

j=i+1

ϕ_j

,

vn=

n−1

X

i=1

(2·ρi·qi+ψi·pi+ρi)·

n−1

Y

j=i+1

ϕj

+

n−1

Y

i=1

ϕi,

w_n=

n−1

X

i=1

(ψ_i·q_i+β·i·ρ_i)·

n−1

Y

j=i+1

ϕ_j

, with

ϕi = [1 + 2·(α−β) + (α+β)·i]·ρi

and

ψ_i = (α−β+ 2·β·i)·ρ_i. Proof. Forr= 2 the recurrence relation is

M₂(n+ 1, x) =ϕn·M₂(n, x) + (2·ρn·x+ψn)·M(n, x) +ρn·(x+β·n).

So

M2(2, x) = 2·ρ1·x²+ (ϕ1+ψ1+ρ1)·x+β·ρ1,

thus (2) is verified for n = 2. If we assume it to be valid for a given n, we deduce

M₂(n+ 1, x) = (u_n·x²+v_n·x+w_n)·ϕ_n

+ (pn·x+qn)·(2·ρn·x+ψn) +ρn·(x+β·n).

We have so (2) for n+ 1,with

u_n+1 =ϕ_n·u_n+ 2·ρ_n·p_n,

v_n+1 =ϕ_n·v_n+ 2·ρ_n·q_n+ψ_n·p_n+ρ_n and

w_n+1 =ϕ_n·w_n+ψ_n·q_n+β·n·ρ_n.

For ending the proof it is sufficient to apply the last lemma with the initial values

u₁ = 0, v₁ = 1 and w₁ = 0.

4. THE DEFINITION OF THE APPROXIMATION OPERATOR

Using Friedman’s distribution, the following operator Un:C[a, b]→C[0,1]

was defined in [3] by

(3) (U_nf)(x) =

n

X

k=0

P(n, k, x)·f(x_n,k),

where

a≤xn,0< xn,1 < ... < xn,n≤b.

This operator is linear, positive, of polynomial type, and with the property that

Une0 =e0, where

e_k(x) =x^k, k = 0,1,2, ...

To apply Korovkin’s approximation theorem (see [1]) the knots x_n,kmust be determined so that

U_ne_k→e_k, n→ ∞, k= 1,2.

As it is stated in [3], Ch. Micchelli had the idea of choosing the knots as follows.

Theorem 3. If the interval of definition of the functions verifies the con- dition

[a, b]⊇^h−_p^qn

n,^n−q_p ⁿ

n

i, where p_n, q_n are from(1), then choosing the knots

x_n,k = ^k−q_p ⁿ

n , k= 0,1, ..., n,

the operator (3)reproduces the linear functions such as it has the property U_ne₁ =e₁.

The proof is done by direct computation. In a similar way we obtain Theorem 4. In the conditions of the above theorem, we have

U_ne₂ = _p¹2

n ·^hu_n·e₂+ (v_n−2·q_n·p_n)·e₁+ (w_n−q_n²)·e₀ⁱ. Proof. Step by step we have

(Une₂)(x) =

n

X

k=0

_k−q

n

pn

2

·P(n, k, x)

= _p¹₂

n ·^hM₂(n, x)−2·q_n·M₁(n, x) +q_n²ⁱ

= _p¹₂

n ·^hu_n·x²+v_n·x+w_n−2·q_n·(p_n·x+q_n) +q_n²ⁱ

which gives the desired result.

5. A SPECIAL CASE

Choosing β= 0 we obtain Stancu’s operator. Let us study the case α= 0.

We have first of all

ρn= _1+β·n¹ andµn= ^{1+β·(n−1)}_1+β·n . As

n−1

Y

j=i+1

µj = _{1+β·(n−1)}^1+β·i it follows that

pn= _{1+β·(n−1)}ⁿ andqn= 2·[1+β·(n−1)]^{β·n·(n−1)} . Then

ϕn= ^{1+β·(n−2)}_1+β·n andψn= ^{β·(2·n−1)}_1+β·n . So

n−1

Y

j=i+1

ϕ_j = [1+β·(i−1)]·(1+β·i) [1+β·(n−2)]·[1+β·(n−1)], which gives

u_n= [1+β·(n−2)]·[1+β·(n−1)]^n·(n−1) , v_n= n·[β·n·(n−2)+1]

[1+β·(n−2)]·[1+β·(n−1)]

and

w_n= β·n·(n−1)·[β·(3·n²−5·n−2)+6]

12·[1+β·(n−2)]·[1+β·(n−1)] . We deduce that

u_n

p²_n →1, n→ ∞, and

v_n−2·p_n·q_n

p²_n →0, n→ ∞, but

w_n−q_n²

p²_n → ∞, n→ ∞, soUne2 does not converge.

It remains as an open problem that of determination of the parameters αand β such that

U_ne₂ →e₂, n→ ∞.

REFERENCES

[1] Altomare, F. andCampiti, M.,Korovkin-type approximation theory and its application, Walter de Gruyter, Berlin-New York, 1994.

[2] Friedman, B.,A simple urn model, Comm. Pure Appl. Math.,2, pp. 59–70, 1949.

[3] Goldman, R. N., Urn models, approximations, and splines, J. Approx. Theory, 54, pp. 1–66, 1988.

[4] Stancu, D. D.,Approximation of functions by a new class of linear polynomial operators, Revue Roum. Math. Pures Appl.,13, pp. 1173–1194, 1969.

[5] Toader, S., A generalization of Polya distribution, Bull. Applied & Comput. Math.

(Budapest), 86A, pp. 477–484, 1998.

Received by the editors: November 19, 1999.