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Approximation by a class of new type Bernstein polynomials of one and two variables.

1. Introduction

The more classical examples of linear positive operators throughout approximation process are the Bernstein polynomials, which are defined by Bernstein [3] as following:

[B.sub.n](f; X) = [n.summation over (k=0)] f(k/n)[[phi].sub.n,k](x), 0 [less than or equal to] x [less than or equal to] 1 (1)

for any f [member of] C[0,1], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Due to the importance of Bernstein polynomials, many of their generalizations and related topics has been intensive research. [2], [5], [13], [16], [14], [8] and [17]. And also, the Bernstein polynomials and their modifications have been used in many branches and computer (for example see [15], [11]).

Stancu [17] generalized (1) as following

[B.sub.n,([alpha],[beta])] (f; X) = [n.summation over (k=0)] f (k + [alpha]/n + [beta]) [[phi].sub.n,k] (X), 0 [less than or equal to] x [less than or equal to] 1 (2)

where [alpha], [beta] [member of] R and 0 [less than or equal to] [alpha] [less than or equal to] [beta], n = 1,2,....

Recently Deo and his collaborates [6] gave a modification of (1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The operators (3) convert in classical Bernstein polynomials which are given in (1) for n sufficiently large.

In this paper we introduce a new type Bernstein polynomials as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is clear that the operators (4) convert in classical Bernstein polynomials for a = b and convert in (3) for a = 0 and b = 1. Therefore the operators (4) include both (1) and (3). Although (2) and (4) may seem similar but they are different.

The first purpose of this article is to see approximation of the operators (4) in the space of continuous functions.

The second is to give definitions of operators for bivariate functions as (4) corresponding to the both square and triangle domain respectively. We also study approximation and rate of approximation of the operators which mentioned above in the space of two variables continuous functions.

The last is to give some plots and numerical examples corresponding to the obtained approximation results. We see that in plots and numerical values:

1) Approximation of [V.sub.n](f; x)to f is better than approximation of [B.sub.n](f; x) to f. Mean, [absolute value of [V.sub.n](f;x) - f(x)] [less than or equal to] [absolute value of [B.sub.n](f;x) - f(x)]

2) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For all these inequalities see figures and tables 1, 2 and 3.

2. Approximation of One Variable of New Type Bernstein Polynomials

The new type Bernstein polynomials given in (4) are sequences of linear positive operators in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where C(A) denotes the class of all continuous functions defined on A. Because of this reality we can use the study of approximation the classical Korovkin's theorem [12], (see also [1])

Firstly, we give the following Lemma.

Lemma 2.1. The following equalities are hold for the polynomials (4).

[F.sub.n,a,b] (1; x) = 1 (5)

[F.sub.n,a,b] (t; X) = x (6)

[F.sub.n,a,b]([t.sup.2]; x) = [x.sup.2] + [x(n + 1) - (n + b)x]/n((n + b)] (7)

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can write [k.sup.2] = k(k - 1) + k then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.2. If f e C[0,1], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. From Lemma 2.1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore the proof follows from Korovkin's theorem.

Lemma 2.3. Let p--th degree moment for the polynomials (4) defined by

T[n.sub.,p](X) = [F.sub.n,a,b] ([(t - x).sup.p]; x), p = 0,1,2, ... (8)

Then we have [T.sub.n,0] (x) = 1, [T.sub.n,1](x) = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Consequently, for each x [member of] [0, n + a/n + b]

(i) [T.sub.n,p](x) is a polynomial in x of degree [less than or equal to] p,

(ii) [T.sub.n,p](x) = O (n-[[absolute value of p + 1/2]]) where [[absolute value of a]] denotes the integral part of a.

Proof. First we get the following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

From the definition (9) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and from (9) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Now we will measure smoothness of approximation of continuous function f by the polynomial (4). In order to do that we use modulus of continuity of function f defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

for any positive numbers [degrees]. [omega](f; [delta]) has some useful properties which can be found, for instance [1, cf. 266-269].

Theorem 2.4. If f [member of] C ([0, n + a/n + b]), then the following inequality holds.

[absolute value of [F.sub.n,a,b](f;x) - f(x)] [less than or equal to] [1 + 1/2 (n + a/n + b)][omega](f; 1/[square root of n]).

Proof. From the well-known properties of modulus of continuity we have

[absolute value of f(t) - f(x)] [less than or equal to] [omega](f; [[delta].sub.n])[1 + [[absolute value of t - x]/[[delta].sub.n]]

where ([[delta].sub.n]) is any sequences of positive numbers. Since the polynomials [F.sub.n,a,b](f; x) are also linear positive operators, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Use Cauchy-Schwartz inequality, (5) and (11), then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Put [[delta].sub.n] = [square root of 1/n], then the we get desired result.

3. Approximation of Bidimensional New Type Bernstein Polynomials

As it is known, there are two type bidimensional of the polynomials (1): First we give the following polynomials corresponding to the square of (4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second we give the following polynomials corresponding to the triangle of (4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

If f (x,y) = g(x) + h(y)or f (x,y) = g(x) x h(y) then the polynomial (13) reduce [F.sub.n,m,a,b](f;x,y) = [F.sub.n,a,b](g;x) + [F.sub.m,a,b](h;y)or [F.sub.n,m,a,b](f;x,y) = [F.sub.n,a,b](g;x) x [F.sub.m,a,b](h; y) respectively. Because of these facts , using well known analogous of Korovkin's theorem of Volkov [19] two dimensional and its modifications as in Theorem 2.2 we get that: If f is continuous function in the rectangle D' then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similar statements come true for the polynomials which are given (14) on triangle [DELTA].

Now we want to measure the smoothness of the approximation by using the partial moduli of continuity of f(x, y)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the complete moduli of continuity of f (x, y)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.1. If f [member of] C(D), then the following inequalities holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where [[alpha].sub.nm] = max{(n + a/n + b), (m + a/m + b)}.

Proof. First we prove (15). Because of linearity of [F.sub.n,m,a,b] properties of modulus of continuity, use Cauchy-Schwartz inequality and (4) , (11) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similar calculations are made to obtain [I.sub.2] [less than or equal to] [1 + [1/2](m + a/m + b)][[omega].sup.(2)](f; 1/[square root of m]).

Now we prove (16).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now use properties of modulus of continuity then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Use Cauchy-Schwartz inequality and (4), (11) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now we need the following property of complete modulus of continuity which proved in [9]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Theorem 3.2. If f [member of] C([DELTA])' then the following inequalities holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Proof. Because of (17) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And because of linearity of [F.sub.n,a,b](f; x y)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now use Cauchy-Schwartz inequality, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Put [[delta].sub.1] = [[delta].sub.2] = 1/[square root of n] then we get desired result.

4. Applications

We give some animated plots and numerical tables here to see the difference of approximation of [F.sub.n,a,b] than [B.sub.n] and [V.sub.n].

1) Figures.

It can be seem that if b - a< 1, then approximation of [V.sub.n] is better than approximation of both [B.sub.n] and [F.sub.n,a,b] (Fig1), if b - a> 1 then approximation of [F.sub.n,a,b] is better than approximation of both Bn and Vn (Fig2, Fig3). The function we use here is f (x) = x sin ([pi] x)/1 + [x.sup.2].

For the function f(x, y) = sin ((x + y)[pi])/[square root of 1 + [x.sup.4] + [y.sup.4]] we have the following figures: It can be seem that if b - a < 1, then approximation of [V.sub.n,m] is better than approximation of both [B.sub.n,m] and [F.sub.n,m,a,b] (Fig4), if b - a > 1 then approximation of [F.sub.n,m,a,b] is better than approximation of both [B.sub.n,m] and [V.sub.n,m] (Fig5). To get a better approximation is more appropriate to use [F.sub.n,m,a,b] instead of of [B.sub.n,m] and [V.sub.n,m].

2.Numeric tables: Let x = ([x.sub.n]) and y = ([y.sub.n]) be sequences with limits [x.sub.0] and [y.sub.0] respectively. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then it is said that y converges faster than x (See:[4]).

(Or equivalent [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. And if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], than it is said that x and y converges equivalent. That means that there is any [n.sub.0] such that for all n [greater than or equal to] [n.sub.0] the inequality [absolute value of [y.sub.n] - [y.sub.0]] < M [absolute value of [x.sub.n] - [x.sub.0]] holds. If 0 < M < 1, then it is sa,d that convergence of y is better than convergence of x. If 1 < M < [infinity] ,then it is said that convergence of x is better than convergence of y (See:[18]).

Our aim was to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in order to compare approximation of f(x) by [F.sub.n,a,b](f; x) and [B.sub.n](f; x), approximation of f(x) by [F.sub.n,a,b](f; x) and [V.sub.n](f; x). But we could not do that by our software programma which we have. That is why we give those comparisons for some, a, b, n and x.

The following numerical values are calculated for the function

Table 1: Comparison of Approximation of [V.sub.n](f; x) and [B.sub.n]
(f; x) to f(x)

f(x) = (1 + x + [x.sup.2]) sin ([pi], x).

[absolute value of [V.sub.n](f; x) - f(x)]/[absolute value of
[B.sub.n](f; x) - f(x)]

n \ x   0.1       0.2       0.3       0.4       0.5       0.6

10      0.93200   0.10324   0.83363   0.83358   0.81321   0.77465
100     0.98928   0.98968   0.98520   0.98326   0.98012   0.97528
500     0.99778   0.99755   0.99713   0.99667   0.99601   0.99501
900     0.99878   0.99872   0.99837   0.99813   0.99777   0.99722

n \ x   0.7       0.8

10      0.70426   0.55588
100     0.96713   0.95073
500     0.99335   0.99003
900     0.99629   0.99445

Table 2: Comparison of Approximation of [F.sub.n,1,3](f; x) and
[B.sub.n](f; x)to f(x)

[absolute value of [F.sub.n,1,3](f; x) - f(x)]/[absolute value of
[B.sub.n](f;x) - f(x)]

n \ x   0.1       0.2       0.3       0.4       0.5       0.6

10      0.88039   0.40223   0.72437   0.72108   0.68483   0.61768
100     0.97897   0.97972   0.97099   0.96717   0.96101   0.95152
500     0.99562   0.99527   0.99423   0.99334   0.99203   0.99005
900     0.99758   0.99744   0.99676   0.99628   0.99555   0.99445

n \ x   0.7       0.8

10      0.49562   0.23818
100     0.93552   0.90335
500     0.98675   0.98013
900     0.99261   0.98892

Table 3: Comparison of Approximation of [F.sub.n,1,3](f; x) and
[V.sub.n](f; x) to f(x)

[absolute value of [F.sub.n,1,3](f; x) - f(x)]/[absolute value of
[V.sub.n](f;x) - f(x)]

n \ x   0.1       0.2       0.3       0.4       0.5       0.6

10      0.94462   3.8959    0.86894   0.86503   0.84213   0.79737
100     0.98958   0.98993   0.98558   0.98364   0.98050   0.97563
500     0.99783   0.99772   0.99709   0.99666   0.99601   0.99502
900     0.99879   0.99871   0.99838   0.99814   0.99778   0.99722

n \ x   0.7       0.8

10      0.70374   0.42847
100     0.96732   0.95016
500     0.99335   0.99000
900     0.99630   0.99444


Remark 4.1. We have aims to discuss the following operators (19) and (20) which are Kantorovich [10] and Durrmeyer [7] type respectively elsewhere.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

References

[1] F. Altomare and M.Campiti. Korovkin-type Approximation Theory and its Applications. Berlin: New York: de Gruyter, (1994)

[2] A. Attalanti and M. Campiti, Bernstein type operators on the half line, Czec. Math. J., 52(127)(2002), no.4, 851-860.

[3] S.N. Bernstein, Demonstration dutheor'emedeWeierstrass baseesurlec alcul des probabilities, Commun. Soc. Math. Kharkow (2)13(1912-1913)12.

[4] C. Brezinski. Convergence acceleration during 20-th century. J. Comput. Appl. Math. 122(2000), 1-21.

[5] J. D. Cao, A Generalization of the Bernstein Polynomials, J. Math. Analy. and Apll. 209, (1997), 140-146.

[6] N. Deo, M. A. Noor and M. A. Siddiqui. On Approximation by A Class Of New Bernstein Type Operators, Applied Math. and Comp. 201, (2008), 604-612.

[7] J.L. Durrmeyer, une formule d inversion de la ransformee de Laplace:Applicationsa la theorie de moments , these de 3e cycle, Faculte des sciences de 1 universite de Paris, 197.

[8] A. Ilinskii, S. Ostrovska, Convergence of generalized Bernstein polynomials, J. Approx. Theory 116, (2002), 100-112.

[9] A. F. Ipatov, Estimation of the error and order of approximation of two variables by Bernstein polynomials(Russian), Uc. Zap. Petrozavodsk. Cos. Univ. 4(4). (1955), 3-48 (1957).

[10] I. V. Kantorovich, Sur certains developments suviant les polynomes de la forme de S.Bernstein, I.H.C.R. Acad. URSS (1930), 563-568, 595-600.

[11] A. Karaci ,I. Buyukyazici and M. Aktumen, Recognition of human speech using q-Bernstein polynomials international Journal of Computer Applications (0975 8887) 2(5), (2010), 22-28.

[12] P. P. Korovkin, Linear Operators and Approximation Theory, Delhi, 1960. Translated from the russian ed. (1959).

[13] S. Lewanowicz and P. Wonzy, Generalized Bernstein polynomials, BIT Numerical Mathematics 44 (2004), 6378.

[14] G. G. Lorentz, Bernstein polynomials, Chelsea, New York, 1986.

[15] P. S. V. Nataraj and M. Arounassalame, A New Subdivision Algorithm for the Bernstein Polynomial Approach to Global Optimization. International Journal of Automation and Computing, 4(4), (2007), 342-352.

[16] G.M. Phillips, Bernstein polynomials based on the q-integers, Annals of Numerical Mathematics 4 (1997), 511-518.

[17] D. D. Stancu. Approximation of functions by new class of linear positive operators, Rev. Roum. Math. Pure Apply. 13(1968), 1173-1194.

[18] B. S. Theodore Ho Hsu. A non-linear transformation for sequences and integrals. Thesis master science' Graduate Faculty of Texas Thechnological College, 1968.

[19] V. I. Volkov. On the convergence of sequences of linear positive operators in the space of continuous functions of two variable, Math. Sb. N. S. 43(85) (1957) 504 (Russian).

Aydin izgi

Harran University, Sciences and arts Faculty, Department of Mathematics, Osmanbey Kampusu, S.Urfa/TURKEY

E-mail: aydinizgi@yahoo.com
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