I ntt. J. Math. Math. Si.
Vol. 3 No. 4 (1980) 793-796
793
APPROXIMATION ON THE SEMI-INFINITE INTERVAL
A. McD. MERCER
Department of Mathematics and Statistics
University of Guelph
Guelph, Ontario, Canada
(Received February ii, 1980)
ABSTRACT.
The approximation of a function f E
is well-known.
C[a,b] by Bernsteln polynomials
It is based on the binomial distribution.
O. Szasz has shown that
there are analogous approximations on the interval [ O,) based on the Polsson
distribution.
Recently R. Mohapatra has generalized Szasz’ result to the case in
which the approximating function is
se
lux) I+-i
-ux
k=N
r (ks+8)
f
ka
(u)
The present note shows that these results are special cases of a Tauberlan theorem
for certain infinite series having positive coefficients.
KEYWORDS AND PHRASES. Szasz operators, Borel sure,ability, Taubian theorems.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
Primary A36, Secondary A10, A46.
INTRODUCTION.
Let us denote the class of functions
f
such that f E C[O,=) and for which
A. McD.MERCER
794
by
0 we shall denote
The subclass for which llm f(t)
CL
lim f(t) exists by
t-o
t-o
Coo.
It is known that if f
lim e
C
E
-xu
ufor each x
E
(0,oo).
>
Here
then
L
(xu) k-i
F (ke+)
k--N
f()
Z
f(x)
(i)
is a real number and N is a positive integer
O,
This result was proved in [i] and is a generalization of e
exceeding-/.
result due to 0.
Szsz [2]
i, N
which was the special case
O.
The proof of (i) depends heavily on a result due to D. Borweln [3], namely
that
(2)
Z
k=N
e
lira
u-o
F (ke+8)
and it is the purpose of the present note to show that the deduction of (i) from
(2) is a special case of a general theorem about infinite series.
This theorem
is of the Tauberlan type and the method of proof which we give is of rather wide
applicability.
THEOREM.
{vk} be
Our result is
CL,oo.
Suppose that f
lira e
implies
>
0, let K be a constant and let
lira e
-xu
Z a k uVk
-u
Z
k=0
2.
k
strictly increasing sequence of positive numbers.
,a
for each
Let a
X
a
k(xu)
Then
(3)
i
f
u
f(x)
-’)
(O,,x,).
PROOF OF THE THEOREM
Since the result is trivially true if f is a constant
function there is no loss of generality in supposing
f
E
APPROXIMATION ON THE SEMI-INFINITE INTERVAL
space
C,
.
CL, As
namely llfll
instead of f
E
lfll
usual we will denote by
If(x) l.
sup
[0,)
lim
e
-xu
u
Now for each x
Z
k:O
defines a linear functional on
C
ak(xu
Vk
795
the norm of f in
he
(0, )
E
Vk+K
f(----)
which we will denote by
X
And if llm is
replaced by llm the corresponding linear functional will be denoted by
First we consider
Since
X
(vk+K)
Vk
functional on
C
u-o, that
I(f)
X
(e -%t)
that-x
lfll.
<
(f)
BV[O, =o)
for some function e
see
k:0
H==
X
is a bounded linear
and so we will have
x
in the usual way.
k
u
k:0
we see, on letting
v
f(t)
and we shall take e
-%x
as having been normalized
X
Now if we take f(t)
e
dex(t)
e
-At (%
>
O)
it is a simple matter to
In this calculation the hypothesis (3) is used
in the form
lira e
-xu
u
7.
k:O
ak(xU
Vk
(x > 0).
1
Henc e
g-(-xt)
o
e
-xt
d(Xx(t)
e
-x
(,I. > 0).
By a well known theorem [4] this determines the normalized function e uniquely
X
and by inspection it is seen to be
796
A. McD. MERCER
(0-<t <x)
0
(t)
(t
x)
(X < t)
Hence for f
C
we have
x(f)
fCt) dc (t)
X
Now all of the above analysis involving
The same function e
X
X
f(x)
could be repeated with
instead.
would be obtained and so we have
--X
(f)
X
(f)
f(x)
That is to say, if x > 0
then
-xu
llm e
u-o
and equals f(x).
7.
a
k=0
k(xu)
Vk
f(
Vk+K-)
u
exists
This concludes the proof of the theorem.
We conclude with two remarks.
The above theorem is about point-wise
convergence whereas in [i] and [2] the uniform convergence of a set of
functions P (x) to f(x) at each point
U
x
[0,=o) was considered
O
For the
definition of this type of convergence we refer the reader to either of these
sources but, when f
C
to go from pointwise convergence to this other
L oo,
type of convergence is, any way, a simple matter.
in [i]
Ne+8
the result (i) was stated for x
i the point x
Secondly, we mention that
[0,) but except
in the case
0 should be omitted.
REFERENCES
i.
Mohapatra, R.N., A Note on Approximation of Continuous Functions by
Szsz Ope’r’ators, .Na.nta. Mathematica, i0 (1977), 181-184.
Szsz, O., G.eneralization of. S.. Ber.nste.in’s P.!/n.mia! to the Infi,ni..te
Interval, J. Nat. Bur. Standards, 45 (1950), 239-245.
Generalized
2.
3.
Borwein, D., Rel, atlons Between Borel-type Methods of Summability,
J. London Math. Soc., 35 (1960), 65-70.
4
Widder, D V
"The Laplace Transform, " Princeton (1941).