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n n n +1


and comparing coe cients on both sides of this equation proves the claim.
We now deÿne, for = 1; : : : ; k, coe cient functions c (x) by
˜
1
c (x):=d ; (x) +
˜ (x)D f(x): (21)
;
!
∈Ns
0
| |=2


We now claim: For m = 1; : : : ; k, it is
m’1
c (x)
˜
m
lim n Bn (f; x) ’ f(x) ’ = cm (x):
˜ (22)
n
n’∞
=1

For m = 1, this was established in [3] as a corollary to Theorem 3.
Now let 26m6k. From (19) in connection with Theorem 4, we get
d1; m (x) dm; m (x)
lim nm Bn (f; x) ’ f(x) ’ + ··· +
n nm
n’∞
« 
m s
1 i
 ÿ
j
= j (v ’ x) D f(x):
!
∈Ns ÿ1 ;:::;ÿm ∈Ns i=1 j=0
0 0
ÿ1 +···+ÿm =
| |=2m
|ÿi |¿2; i=1;:::; m


Together with (21), (20) and (18), this gives
m’1
c (x) dm; m (x)
˜ 1
m
lim n Bn (f; x) ’ f(x) ’ ’ = (x)D f(x);
;
n nm !
n’∞
∈Ns
=1 0
| |=2m


and so, using (21) once more, (22) is proved.
This also completes the proof of the existence of the asymptotic expansion, as stated in (10).
To verify (11) (i.e., to prove that c = c ), we once again analyse the sum in (19). Using (16)
˜
gives
« 
||
s
2
1 i; i
 ’ x)ÿ  D f(x):
j
j (v
! n| |’i
∈Ns =1 i=1 i=1 j=0
ÿ1 ;:::;ÿ ∈Ns
0 0
ÿ1 +···+ÿ =
| |62 ’1
|ÿi |¿2; i=1;:::;
G. Walz / Journal of Computational and Applied Mathematics 122 (2000) 317“328 325

Table 1a
Errors in approximating f1

0:2083e(00)
0:1042e(’1)
0:9896e(’1) 0:5208e(’2)
0:6510e(’2) 0:0000e(1)
0:4622e(’1) 0:6510e(’3)
0:2116e(’2) 0:0000e(1)
0:2205e(’1) 0:8138e(’4)
0:5900e(’3) 0:0000e(1)
0:1073e(’1) 0:1017e(’4)
0:1551e(’3) 0:0000e(1)
0:5288e(’2) 0:1272e(’5)
0:3974e(’4)
0:2624e(’2)

Table 1b
Quotients of the entries of Table 1a

2:105
1:600
2:141 8:000
3:077
2:096 8:000
3:586
2:055 8:000
3:803
2:029 8:000
3:904
2:015 8:000
3:953
2:008


Collecting in this expression all terms containing 1=n shows that the coe cient of this power of
n is
« 
||
s
2
1 i
 ’ x)ÿ  D f(x):
j
j (v
| |’ ;
!
∈Ns =1 i=1 j=0
ÿ1 ;:::;ÿ ∈Ns
0 0
ÿ1 +···+ÿ =
| |62 ’1
|ÿi |¿2; i=1;:::;

Since = 0 for i60 and i ¿ , this is equal to
i;
« 
||
s
2
1 i
 ’ x)ÿ  D f(x):
j
j (v
| |’ ;
! i=1 j=0
ÿ1 ;:::;ÿ ∈Ns
=| |’
∈Ns 0
0
ÿ1 +···+ÿ =
+16| |62 ’1 |ÿi |¿2; i=1;:::;

Now using once more relation (18) completes the proof of Theorem 5.
326 G. Walz / Journal of Computational and Applied Mathematics 122 (2000) 317“328

Table 2a
Errors in approximating f2

0:1097e(00)
0:2113e(’2)
0:5378e(’1) 0:1500e(’4)
0:5396e(’3) 0:2869e(’6)
0:2662e(’1) 0:1624e(’5) 0:1575e(’9)
0:1361e(’3) 0:1779e(’7)
0:1324e(’1) 0:1875e(’6) 0:7047e(’11)
0:3417e(’4) 0:1105e(’8)
0:6603e(’2) 0:2247e(’7) 0:2520e(’12)
0:8559e(’5) 0:6883e(’10)
0:3297e(’2) 0:2748e(’8) 0:8359e(’14)
0:2142e(’5) 0:4294e(’11)
0:1648e(’2) 0:3398e(’9) 0:2687e(’15)
0:5357e(’6) 0:2681e(’12)
0:8235e(’3) 0:4224e(’10)
0:1340e(’6)
0:4117e(’3)



Table 2b
Quotients of the entries of Table 2a

2:039
3:917
2:020 9:237
3:964 16:133
2:010 8:664 22:345
3:984 16:096
2:005 8:344 27:966
3:992 16:055
2:003 8:175 30:146
3:996 16:029
2:001 8:088 31:111
3:998 16:015
2:001 8:044
3:999
2:000




3. Numerical results

Having proved the existence of the asymptotic expansion (10), we can now apply the extrapolation
process (4) to the sequence of Bernstein approximants. It follows from (10) that k = k for all k.
In order to illustrate the numerical e ect of extrapolation, we show in this section a small selection
of a number of numerical tests that have been examined, and all of which showed the asymptotic
behaviour that was predicted.
G. Walz / Journal of Computational and Applied Mathematics 122 (2000) 317“328 327


The results shown below were obtained for s = 2 on the triangle T with vertices
1 0 0
v0 = v1 = v2 =
; and
0 1 2
in euclidean coordinates. We computed the absolute values of the error functions in the barycenter
of T , the point
ÿ = 1 (v0 + v1 + v2 ):
3

As a ÿrst test, we applied the method to the bivariate polynomial
f1 (x; y):=xy3 :
The errors of the approximations yi(k) of the true value f1 (ÿ) = 1 , computed by extrapolation with
3
K =3, n0 =2, and i =0; : : : ; 6, are shown in Table 1a. As expected, the entries of the third column are
identically zero, since f1 is a polynomial of total degree 4, and therefore the third extrapolation step
already gives the exact result. Note in this connection that the Bernstein approximants themselves
do not reproduce the polynomial f1 exactly, however high their degree might be.
As a second example, we consider approximation of the function
f2 (x; y):=exp(x + y)
and again compare our numerical approximations with the true value of f2 in ÿ, which is exp( 4 ). This
3
time, the errors (in absolute value) of the approximations computed by our method with K =4; n0 =4,
and i = 0; : : : ; 8 are shown, see Table 2a.
In Tables 1b and 2b, ÿnally, we have the quotients of two subsequent values in the columns of
Table 1a (resp. Table 2a). As predicted, the entries of the kth column (starting to count with k = 0)
converge to 2k+1 .


4. Conclusion

In contrast to the univariate case, the approximation of multivariate functions by polynomials is
still a very di cult task, and many problems are open. Up to now, there exist very few numeri-
cal methods for the computation of good polynomial approximations. Therefore, we are convinced
that the approach developed in this paper provides a very e cient new method for polynomial
approximation of multivariate functions.


References

[1] C. Brezinski, M. Redivo Zaglia, Extrapolation Methods, Theory and Practice, North-Holland, Amsterdam, 1992.
[2] F. Costabile, M.I. Gualtieri, S. Serra, Asymptotic expansion and extrapolation for Bernstein polynomials with
applications, BIT 36 (1996) 676“687.
[3] M.-J. Lai, Asymptotic formulae of multivariate Bernstein approximation, J. Approx. Theory 70 (1992) 229“242.
[4] G.G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953.
328 G. Walz / Journal of Computational and Applied Mathematics 122 (2000) 317“328


[5] G. Meinardus, G. Merz, Praktische Mathematik II, Bibl. Institut, Mannheim, 1982.
[6] E. Voronowskaja, DÃ termination de la forme asymptotique d™approximation des fonctions par les polynomes de M.
e
Bernstein, C. R. Acad. Sci. URSS (1932) 79“85.
[7] G. Walz, Asymptotics and Extrapolation, Akademie Verlag, Berlin, 1996.
Journal of Computational and Applied Mathematics 122 (2000) 329“356
www.elsevier.nl/locate/cam

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