Publicacions Ma em`a iques, Vol 39 (1995), 263–271.
GLOBAL APPROXIMATION BY
MODIFIED BASKAKOV TYPE OPERATORS
Vijay Gup a
Abs ac
In he p esen pape , we p o e a global di ec heo em o he
modified Baskako ype ope a o s in e ms o so called Di zian-
To ik modulus o smoo hness.
1. In oduc ion
Mo i a ed by he in eg al modifica ion o Be ns ein polynomials by
Du meye [3], Sahai and P asad [6] fi s defined and s udied modified
Baskako ope a o s. Sinha e al. [7] imp o ed and co ec ed he e-
sul s o [6]. Recen ly he au ho [4], in oduced ano he modifica ion o
Baskako ope a o s by aking he weigh unc ion o Be a ope a o s on
L1[0,∞)as
(1.1) (Bn )(x)=
∞

k=0
pn,k(x)∞
0
bn,k( ) ( )d , x ∈[0,∞)
whe e
pn,k(x)=n+k−1
kxk(1 + x)−n−k
and
bn,k( )=[B(k+1,n)]−1 k(1 + )−n−k−1,
B(k+1,n) being he Be a unc ion gi en by k!(n−1)!/(n+k)!.
In [4], he au ho has ob ained only local di ec heo ems in simul a-
neous app oxima ion, as he ope a o s defined by (1.1) gi e be e ap-
p oxima ion han he ea lie in eg al modifica ion o Baskako ope a o s
Resea ch suppo ed by Council o Scien ific and Indus ial Resea ch, India unde
awa d no. 9/143(163)/91-EMR-1.
264 V. Gup a
s udied in [5], [6] and [7] e c., his mo i a ed us o ex end he esul s
o [4] o he whole in e al [0,∞) and we s udy a global esul o he
ope a o s (1.1).
By L
1[0,∞), we deno e he class o unc ions ggi en by
L
1[0,∞):={g:g( )∈L1[0,a] o e e y a∈(0,∞) and
|g( )( )|≤M(1 + )m,Mand ma e cons an s depending on g}.
We may ema k ha L
p[0,∞) is no con ained in L
1[0,∞).
Following [2], he modulus o smoo hness o is gi en by
ω2
φ( , )p= sup
0<h≤
∆2
hφ p,φ(x)=x(1 + x)
whe e
∆2
h (x)= (x−h)−2 (x)+ (x+h),i [x−h, x +h]⊂[0,∞)
0,o he wise.
This modulus o smoo hness is equi alen o he modified k- unc ional
(see e.g. [2]) gi en by
¯
K2
φ( , 2)p= in { −gp+ 2φ2gp+ 4gp;g∈¯
W2
p(φ, [0,∞))}
whe e
¯
W2
p(φ, [0,∞)) = {g∈Lp[0,∞):g∈ACloc[0,∞); φ2g ∈Lp[0,∞)}.
In [4] he au ho was no able o ob ain global esul s. In he p esen
pape , we p o e a global di ec heo em in simul aneous app oxima ion
o he ope a o s (Bn )(x) defined by (1.1) in e ms o Di zian-To ik
modulus o second o de .
Th oughou he pape we deno e by C he posi i e cons an s no nec-
essa ily he same a each occu ence.
2. Auxilia y esul s
In his sec ion, we shall gi e ce ain defini ions and lemmas which will
be used in he sequel.
Fo e e y n∈Nand n>( +1)weha e
(2.1)
∞

k=0
pn,k(x)=1,∞
0
bn,k( )d =1
k
npn,k(x)=xpn+1,k−1(x),∞
0
bn− ,k+ ( )d =k+ +1
n− −1.
Modi ied Baskako ype ope a o s 265
Lemma 2.1 [4]. Le m, ∈N0, we define
T ,n,m(x)=
∞

k=0
pn+ ,k(x)∞
0
bn− ,k+ ( )( −x)md
hen
T ,n,0(x)=1,T
,n,1(x)=1+ +x(1+2 )
(n− −1) ,
T ,n,2(x)=2(2 2+4 +n+1)x2+2(2 2+5 +2+n)x+( 2+3 +2)
(n− −1)(n− −2) ,
and he e holds he ecu ence ela ion:
(n−m− −1)T ,n,m+1(x)=φ2(x)[T(1)
,n,m(x)+2mT ,n,m−1(x)]
+[(m+ +1)(1+2x)−x]T ,n,m(x),n>m+ +1.
Consequen ly o each x∈[0,∞),T ,n,m(x)=0(n−[(m+1)/2]),[α]de-
no es he in eg al pa o α.
The p oo o his lemma easily ollows along he lines o [6], [7] using
φ2(x)p
n,k(x)=(k−nx)pn,k(x) and φ2( )b
n,k( )=[k−(n+1) ]bn,k( ).
F om he abo e lemma, we ha e
(2.2)
T ,n,2m(x)=
m

i=0
qi,m,n(x)φ2(x)
nm−i
n−2i
T ,n,2m+1(x)=(1+2x)
m

i=0
si,m,n(x)φ2(x)
nm−i
n−2i−1,
whe e qi,m,n(x) and si,m,n(x) a e polynomials in xo fixed deg ee wi h
coefficien s ha a e bounded uni o mly o all n.
Lemma 2.2. I ∈L
p[0,∞)∪L
1[0,∞),1≤p≤∞,n> (1 + m)
and x∈[0,∞), hen
(2.3) (Bn )( )(x)=α(n, )
∞

k=0
pn+ ,k(x)∞
0
bn− ,k+ ( ) ( )( )d
266 V. Gup a
whe e
α(n, )=(n+ −1)!(n− −1)!
((n−1)!)2=
−1

=0
n+
n−( +1).
P oo : By using Leibni z heo em, we ha e
(Bn )( )(x)=

i=0
∞

k=i
i(n+k+ −i−1)!
(n−1)!(k−i)!
×(−1) −ixk−i(1 + x)−n−k− +i
×∞
0
bn,k( ) ( )d
=(n+ −1)!
(n−1)!
∞

k=0
pn+ ,k(x)
×∞
0

i=0
(−1) −i
ibn,k+i( ) ( )d .
Again, by he use o Leibni z heo em, we ha e
b( )
n− ,k+ ( )= (n−1)!
(n− −1)!

i=0
(−1)i
ibn,k+i( ).
Hence,
(Bn )( )(x)=(n+ −1)!(n− −1)!
((n−1)!)2
∞

k=0
pn+ ,k(x)∞
0
(−1) b( )
n− ,k+ ( ) ( )d .
On in eg a ing imes by pa s, we ge he equi ed esul .
We see ha he ope a o s defined in (2.3) by B( )
n := (Bn )( ), ∈
L
p[0,∞)∪L
1[0,∞) a e no posi i e. To make he ope a o s posi i e we
in oduce he ope a o
Bn, ≡D BnI , ∈Lp[0,∞)∪L
1[0,∞),
whe e Dand Ia e diffe en ia ion and in eg a ion ope a o s espec i ely.
The e o e we define he ope a o by
(Bn, )(x)=α(n, )
∞

k=0
pn+ ,k(x)∞
0
bn− ,k+ ( ) ( )d ,
Modi ied Baskako ype ope a o s 267
∈Lp[0,∞)∪L
1[0,∞), n> (1 + m).
The ope a o s Bn, a e posi i e and he es ima ion (Bn )( )− ( )p.
∈L
p[0,∞) is equi alen o Bn, − p, ∈Lp[0,∞).
Using (2.1), we can easily p o e ha o n>( + 1), Bn, 1≤
C 1, o ∈L1[0,∞) and Bn, ≤C ∞ o ∈L∞[0,∞).
Making use o Riesz-Tho in heo em, we ge
(2.4) Bn, p≤C p, ∈Lp[0,∞),1≤p≤∞,n>( +1).
Co olla y 2.3. Fo e e y m∈N0,n>( +2m+1)and x∈[0,∞)
we ha e
(2.5) |Bn, (( −x)2m,x)|≤Cn−m(φ2(x)+n−1)m,
|Bn, (( −x)2m+1,x)|≤C(1+2x)n−m−1(φ2(x)+n−1)m
whe e he cons an Cis independen o n. Fo fixed x∈[0,∞)we ob ain
(2.6) |Bn, (( −x)m,x)|=0(n−[(m+1)/2]),n→∞.
P oo : Since Bn, (( −x)m,x)=α(n, )T ,n,m(x) he es ima e (2.5)
ollows om (2.2) along he lines o [5], (2.6) immedia ely ollows om
(2.5).
Lemma 2.4. Le ∈[0,∞)and n>( +m) hen
Bn, ((1 + )−m,x)≤C(1 + x)−m,x∈[0,∞)
whe e he cons an Cis independen o n.
P oo : I is easily e ified ha
(1 + )−mbn− ,k+ ( )=
m−1

=0
n− +
n+ +k+1bn− +m,k+ ( )
and
pn+ ,k(x) = (1 + x)−m
m

=1
n+ − +k
n+ − pn+ −m,k(x).

268 V. Gup a
Making use o hese wo iden i ies and (2.1) we ge
Bn, ((1 + )−m,x)=α(n, )
∞

k=0
pn+ ,k(x)∞
0
bn− ,k+ ( )(1 + )−md
=α(n, )
∞

k=0
pn+ ,k(x)
m−1

=0
n− +
n+ +k+1
×∞
0
bn− +m,k+ ( )d
=α(n, )
∞

k=0
(1+x)−mpn+ −m,k(x)
m

=1
(n+ − +k)
(n+ − )
×
m−1

=0
n+ −
n+ +k+1
≤C(1 + x)−m
∞

k=0
pn+ −m,k(x)
=C(1 + x)−m.
Fo he wo monomials e0,e1and x∈[0,∞), n→∞we ob ain by di ec
compu a ion
Bn, (e0,x)=1+0(n−1)(2.7)
Bn, (e1,x)=x(1+0(n−1)).(2.8)
Lemma 2.5. Fo Hn(u)gi en by
Hn(u)=∞
0u
0
−u
0∞
0∞

k=0
pn+ ,k(x)bn− ,k+ ( )(u− )d dx
we ha e Hn(u)≤Cn−1φ2(u), whe e Cis independen o nand u.
The p oo o he abo e lemma easily ollows by using (2.1) along he
lines o [1, Lemma 5.2].
3. Di ec esul
Theo em 3.1. Suppose ∈Lp[0,∞),1≤p<∞,n>( +5) hen
we ha e
Bn, − p≤C{ω2
φ( ,n−1/2)+n−1 p}
Modi ied Baskako ype ope a o s 269
whe e he cons an Cis independen o n.
P oo : By Taylo ’s expansion o g,weha e
(3.1) g( )=g(x)+( −x)g(x)+
x
( −u)g(u)du.
Nex , since Bn, ( ,x) a e uni o mly bounded ope a o s so o e e y g∈
¯
W2
p(φ, [0,∞)), we ha e
(3.2) Bn, − p≤C −gp+Bn, g−gp.
Using (2.5), (2.8) and (3.1) and ollowing [2], we ob ain
Bn, g−gp≤C{gp+gLp[0,1]}+(1+2x)gLp[1,∞)
+Bn, (R(g, , x),x)p
≤Cn−1[gp+φ2gp]+Bn, (R(g, , x),x)p
(3.3)
whe e R(g, , x)=
x( −u)g(u)du.
Now, we shall p o e ha
(3.4) Bn, (R(g, , x),x)p≤Cn−1(φ2+n−1)gp.
We p o e his o p= 1 and p=∞. The cases 1 <p<∞ ollows again
by Riesz-Tho in heo em.
Using (2.5) o he case m= 1 and Lemma 2.4, he case p=∞easily
ollows (see e.g. [5]).
Fo p= 1, we de i e (3.4) by applying Fubini’s heo em wice, he
defini ion o Hn(u) and Lemma 2.5 as
∞
0
|Bn, (R(g, , x),x)|dx
≤α(n, )∞
0
∞

k=0
pn+ ,k(x)∞
0
bn− ,k+ ( )|
x
( −u)g(u)|d dx
=α(n, )∞
0
|g(u)|∞
0u
0
−u
0∞
0(u− )
×
∞

k=0
pn+ ,k(x)bn− ,k+ ( )d dx du
=α(n, )∞
0
|g(u)|Hn(u)du
≤Cn−1φ2g1
≤Cn−1(φ2+n−1)g1,
270 V. Gup a
whe e Cis independen o n. Hence (3.4) holds by Riesz-Tho in heo em
o 1 ≤p≤∞. Combining he es ima es o (3.2), (3.3) and (3.4) we ge
Bn, − p=C −gp+Cn−1{ −gp+ p+φ2gp
+(φ2+n−1)gp}
≤C{ −gp+n−1φ2gp+n−2gp+n−1 p}.
Nex aking he infimum o e all g∈¯
W2
p(φ, [0,∞)) on he igh hand
side, we ge
Bn, − p≤C{¯
K2
φ( ,n−1)+n−1 p},
his comple es he p oo o Theo em 3.1.
Rema k. The conclusion o Theo em 3.1 is ue on he space
Lp[0,∞), 1 ≤p<∞(i.e. lim
n→∞ Bn, − p= 0 o e e y ∈Lp[0,∞)),
since he mos basic ac abou ω2
φ( ,n−1) is ha
lim
n→∞ ω2
φ( ,n−1) = 0 o all ∈Lp[0,∞),1≤p<∞,
o o all bounded unc ions ∈C[0,∞) which sa is y
lim
x→∞ (x)=L∞<∞,i p=∞(c . [2, p. 36]).
Acknowledgemen . The au ho is g a e ul o he e e ee o many
sugges ions ha g a ely imp o ed his pape .
Re e ences
1. Z. Di zian and K. I ano , Be ns ein ype ope a o s and hei
de i a i es, J. App ox. Theo y 56 (1989), 72–90.
2. Z. Di zian and V. To ik,“Moduli o smoo hness,” Sp inge Se ies
in Compu a ional Ma hema ics 9, Sp inge -Ve lag, Be lin, Heidel-
be g, New Yo k, 1987.
3. J. L. Du meye , Une o mule d’in e sion, de la ans o m´ee de
Laplace: Applica ion ´a la Theo ie des Momen s, Th´ese de 3e Cycle,
Facul ´e des Sciences de l’Uni e si e de Pa is, 1967.
4. V. Gup a, A no e on modified Baskako ype ope a o s, App ox.
Theo y and i s Appl. 10(3) (1994), 74–78.
Modi ied Baskako ype ope a o s 271
5. M. Heilmann, Di ec and con e se esul s o ope a o s o
Baskako -Du meye ype, App ox. Theo y and i s Appl. 5(1)
(1989), 105–127.
6. A. Sahai and G. P asad, On simul aneous app oxima ion by
modified Lupas ope a o s, J. App ox. Theo y 45 (1985), 122–128.
7. R. P. Sinha, P. N. Ag awal and V. Gup a, On simul aneous
app oxima ion by modified Baskako ope a o s, Bull. Soc. Ma h.
Belg. Se . B 42(2) (1991), 217–231.
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