A REMARK ON A MODIFIED SZ ´ ASZ–MIRAKJAN OPERATOR

C O L L O Q U I U M M A T H E M A T I C U M

VOL. 79 1999 NO. 2

A REMARK ON A MODIFIED SZ ´ ASZ–MIRAKJAN OPERATOR

BY

GUANZHEN Z H O U (NINGBO

HANGZHOU)

SONGPING Z H O U (NANGCHONG

HANGZHOU)

Abstract. We prove that, for a sequence of positive numbers δ(n), if n

δ(n) 6→ ∞ as n → ∞, to guarantee that the modified Sz´ asz–Mirakjan operators S

(f, x) converge to f (x) at every point, f must be identically zero.

1. Introduction. Let C

be the set of all continuous functions on [0, ∞) satisfying |f (t)| ≤ M t

for some real numbers M > 0 and α > 0. For f ∈ C

and x ∈ [0, ∞), the well-known Sz´ asz–Mirakjan operator is defined by S

(f, x) =

∞

X

k=0

f  k n



e

(nx)

k! =:

∞

X

k=0

f  k n



p

(nx).

From Hermann [2], we know that for f ∈ C

, S

(f, x) converges to f (x) uniformly on any closed subset of [0, ∞), hence in particular at every point x in [0, ∞). At the same time, Hermann also pointed out that C

, for all α > 0, are the largest sets, in the usual sense, which guarantee S

(f, x) to exist.

For computational reasons, Gr´ of [1] and Lehnhoff [3] suggested using a partial sum of S

(f, x) (which only has a finite number of terms depending upon n and x) to approximate f (x). Let δ = δ(n) be a sequence of positive numbers. Lehnhoff examined the operator

S

(f, x) =

[n(x+δ)]

X

k=0

f  k n



p

(nx),

and he proved that, for all f in C

satisfying |f (t)| ≤ M

+ M

t

for some positive numbers M

, M

and some natural number m, S

(f, x) converges to f (x) at every point on [0, ∞) if

(1) lim

n→∞

n

δ(n) = ∞.

1991 Mathematics Subject Classification: Primary 41A36.

Key words and phrases: modified Sz´ asz–Mirakjan operator.

Research supported in part by National and Provincial Natural Science Foundations, and by State Key Laboratory of Southwest Institute of Petroleum.

[157]

158

Recently, Sun [4] showed that the condition (1) is a sharp necessary and sufficient condition for S

(f, x) to converge pointwise to f (x) for f in C

. More precisely, he showed that for f ∈ C

the condition (1) is sufficient for S

(f, x) to converge to f (x) uniformly on any closed subset of [0, ∞), and he also proved that if (1) does not hold, then for the function f

(x) = x

∈ C

, S

(f

, x) does not converge to f

(x) at some point x.

A natural question is whether (1) can be weakened if we consider a subset of C

(for example, the subset that Lehnhoff studied). Our result exhibits a surprising phenomenon that if (1) does not hold, then to guarantee that S

(f, x) converges to f (x) at every point, f must be identically zero.

2. Result and proof. In what follows, we always use C to indicate a positive constant, whose value may be different in different situations.

Theorem. Let δ = δ(n), n = 1, 2, . . . , be a sequence of positive numbers such that n

δ(n) 6→ ∞ as n → ∞, and assume that f ∈ C

. If f (x

) 6= 0 for some x

∈ [0, ∞), then S

(f, x

) 6→ f (x

) as n → ∞.

P r o o f. Suppose n

δ(n) 6→ ∞ as n → ∞. Without loss of generality, there exists a constant A > 0 and a sequence {n

} of positive integers such that n

δ(n

) ≤ A, and f (x

) > 0 for x

∈ (0, ∞), say. There are M

> 0 and ε

> 0 such that f (x) > M

for all x ∈ (x

− ε

, x

+ ε

) ⊂ (0, ∞).

Write

f

(x) =

(f (x) + |f (x)|), f

(x) =

(f (x) − |f (x)|).

Then

R

(f, x) := S

(f, x) − S

(f, x) = R

(f

, x) + R

(f

, x)

since R

is a linear operator. At the same time, noting that R

is also a positive operator, we calculate that

R

(f

, x

) =

∞

X

k=[nj(x0+δ)]+1

f (k/n

)p

(n

x

)

≥ X

njx0+An^1/2_j +1≤k≤njx0+2An^1/2_j +2

f (k/n

)p

(n

x

).

For n

x

+ An

+ 1 ≤ k ≤ n

x

+ 2An

+ 2 and sufficiently large j, k/n

∈ (x

− ε

, x

+ ε

), so that

R

(f

, x

) ≥ M

X

njx0+An^1/2_j +1≤k≤njx0+2An^1/2_j +2

p

(n

x

)

≥ AM

n

p

jx0+2An^1/2_j ]

(n

x

)

MODIFIED SZ ´ASZ–MIRAKJAN OPERATOR

159

by the monotonicity of {p

(nx

)} for k ≥ nx

. It is easy to obtain p

jx0+2An^1/2_j ]

(n

x

) ≥ C(n

x

)

e

from Stirling’s formula, hence

R

(f

, x

) ≥ CAM

x

e

> 0, that is,

(2) R

(f

, x

) 6→ 0 as j → ∞.

For R

(f

, x

), we see that

R

(f

, x

) = X

k/n−x0≥ε0

f (k/n)p

(nx

)

in view of f

(x) = 0 for x ∈ (x

− ε

, x

+ ε

). Thus for any given ε > 0, there is an N > 0 such that

∞

X

k=N +1

f (k/n)p

(nx

)    < ε.

Similarly to the standard proof of the Korovkin theorem, we have 

N

X

k=[nx0+nε0]+1

f (k/n)p

(nx

)  

 ≤ max

0≤t≤N/n

|f (t)| X

k/n−x0≥ε0

p

(nx

)

≤ M N

ε

S

((t − x

)

, x

) → 0 as n → ∞, or

R

(f

, x

) → 0 as n → ∞, therefore, with (2),

R

(f, x

) ≥ R

(f

, x

) − |R

(f

, x

)| ≥ C > 0,

or R

(f, x

) 6→ 0 as n → ∞. Consequently, f (x

)−S

(f, x

) = R

(f, x

)+

f (x

) − S

(f, x

) 6→ 0 as n → ∞. The theorem is proved.

Corollary. Let δ = δ(n), n = 1, 2, . . . , be a sequence of positive num- bers such that n

δ(n) 6→ ∞ as n → ∞, and assume that f ∈ C

. Then lim

S

(f, x) = f (x) holds for every x ∈ [0, ∞) if and only if f ≡ 0.

REFERENCES

[1] J. G r ´ o f, ¨ Uber Approximation durch Polynome mit Belegungsfunktion, Acta Math.

Acad. Sci. Hungar. 35 (1980), 109–116.

[2] T. H e r m a n n, Approximation of unbounded functions on unbounded interval , ibid.

29 (1977), 393–398.

[3] H. G. L e h n h o f f, On a modified Sz´ asz–Mirakjan-operator , J. Approx. Theory 42

(1984), 278–282.

160

[3] X. H. S u n, On the convergence of the modified Sz´ asz–Mirakjan operator , Approx.

Theory Appl. 10 (1994), no. 1, 20–25.

Guanzhen Zhou

Department of Mathematics Ningbo University

Ningbo, Zhejiang 315211 China

Department of Mathematics Hangzhou University Hangzhou, Zhejiang 310028 China

Songping Zhou State Key Laboratory of Oil/Gas Reservoir Geology and Exploitation Southwest Institute of Petroleum Nangchong, Sichuan 637001 China Department of Mathematics Hangzhou University Hangzhou, Zhejiang 310028 China E-mail: spz@public.hz.zj.cn

Received 9 March 98