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On the rate of convergence of the Szász-Mirakyan operator for functions of bounded variation
JOURNAL OF APPROXIMATION THEORY 40,226-241
(1984)
On the Rate of Convergence of the SzBsz-Mirakyan Operator for Functions of Bounded Variation FUHUA CHENG Department of Mathematics, Ohio State University, Columbus, Ohio 43210*, U.S.A. Communicated by R. Bojanic Received August 16, 1982
1. INTRODUCTION Let f be a function defined on the infinite Szisz-Mirakyan operator S, applied to f is
interval
[0, co). The
S,(f,x>= 2 f(X) PkW),
(1.1)
k=O
where
pk(t)= C$. In 1950 Szisz [l] proved’: THEOREM A. Zf f (t) is bounded in every finite subinterval of [0, co), is equal to O(tk) for some k > 0 as t -+ co, and is continuous at the point t = x, then S,(f, x) converges uniformly to f (x).
Later on, in 197 1, Gr6f [3] gave the following estimate for the rate of convergence of S,(f, x) when f (t) is continuous on 10, co): ’ Mirakyan
[2] considered the partial sum w,.,(L
and proved that lim,,, w,.,(A x) =f(x) uniformly We do not consider this matter in this paper. * Present address: Institute of Computer University, Hsinchu, Taiwan 300, R.O.C.
Copyright All rights
$3.00
0 1984 by Academic Press, Inc. of reproduction in any form reserved.
in 10, r’ 1, if lim,,,
and Decision
226 0021.9045/84
x) of s,(A x),
(m/n) = r > r’ > 0.
Sciences, National
Tsing Hua
221
SZASZ-MIRAKYAN
THEOREM B. If f is continuous on [0, co) and is equal to O(eax) for some a > 0 as x+ 00, then for all A > 0,
S”(f, x>-f(x) = 0(%4(f, n- 1’2)>, xE [&A], where w,(f, S) = sup{(f(x
(1.2)
It] < 6, x E ]O, A 1I.
+ t) -f(x):
It is easy to see, by considering the function f(t) = 1t - x / at the point t = x (x > 0), that this result is essentially the best possible. This result was improved by Hermann [4] in 1977. He showed that (1.2) holds iff(t) = O(tar) (a > 0). He proved in the same paper that the series in (1.1) does not converge if f(t) 2 tmcr)‘r, where 4(t) is any monotonically increasing function such that lim,,, 4(t) = co. In this paper, we shall study S,(f, x) for functions of bounded variation on every finite subinterval of [0, oo) and prove that S,(f, x) converges to f (f(x + 0) +f(x - 0)) under H ermann’s condition on the magnitude off by giving quantitative estimates of the rate of convergence. We shall also prove that our estimates are essentially best possible.
2. RESULTS AND REMARKS Our main result may be stated as follows THEOREM. Let f be a function of bounded variation on everyfinite subinterval of [0, a) and let f (t) = O(F) for some a > 0 as t + co. If x E (0, 00) is irrational, then for n suflciently large we have
Is,(f,X)-t(f(X+)+f(X-))l~
(3+;)xp’
C v:l:‘$(g,) k:l
+
0(x-‘/y
n1,2 If @+I -f @->I
+ 0(1)(4x)4ax(nx))1’2
where Vi< g) is total variation of g on [a, b] and ‘!m =f (t) -0x+),
x
= 0,
t = x,
=f (t) -f (x-),
O
t+jnx,
(2.1)
228
FUHUACHENG
The right-hand side of (2.1) converges to zero as n -+ co since the continuity of g,(t) at t = x implies that
Remarks.
If f is not constant in any neighborhood of x then (4x)4”“(m)
for n sufficiently
- “2
(+)
nx < &
v;y
g,)
(2.2)
large. So in that case (2.1) becomes
+ 0(x--“*) I/Z It
I.!++) -Ax-11.
(2.3)
If, in addition, f is continuous at x then (2.3) can be further simplified to
for sufficiently large n. If, however, f is constant in some neighborhood of x, then since cl:;$(f) = 0 f or all except a finite number of k’s, we have
for some positive constant c (depending on x), Using this and (2.2), (2.1) becomes
IS,(.ttx>-.m>l < “4n; x)
(2.5)
for n sufficiently large. As far as the precision of the above estimates is concerned, we can prove that (2.4) cannot be asymptotically improved. Consider the function f(t) = /t-xl (x > 0) at t = x. From (2.4) we have
IS,(.Lx>-f(x)1 < $2
k=l
229
SZikZ-MIRAKYAN
Since V:‘:(S)
= 26, it follows that
ILs,(f,X)-f(X)I< 2(4n+x)T- 1<4(4+x). GJit fi On the other hand, by a result of SZ~SZ [ 1, p. 2401,
~ (2x/7ce4)‘12 nw . Hence, for the function f(t) = 1t - xl (x > 0), we have (2x$;*4)“2
< (S,(f; x) -f(x)1
Therefore (2.4) cannot be asymptotically
< 4’4, I’,“‘.
improved.
3. LEMMAS The proof of our theorems is based on a number of lemmas. The first lemma originated in one of the questions posed by Ramanujan in a letter of January 16, 1913, to G. H. Hardy. A complete proof of this lemma was given by Watson [ 5 1, Szego 161, and Karamata [ 7 1. LEMMA
1.
If x is a positive integer, then 1+++$+ .
.
. . . + s e(x) = + e.‘,
where O(x) lies between f and f . For an arbitrary positive number x, we have LEMMA 2.
If x is a positive number, then
230
FUHUACHENG
Proof of Lemma 2. Set n = [x). Define a function v(t) on [n, n + 1) as
y(t)=e-’ <’ t”
t E [n, n + 1).
kfko k! ’
Since y’(t) = -e-‘;
< 0,
if n < t < n + 1 we have
w((n + 1) -I< v/(t) G w(n). In particular iy((n+
l)-)
t’
g
kto k!
By Lemma 1,
1 =y+ew((n+
l)-)=e-‘“tl’
Ix1[Xl[Xl(l Ix]!
- 8((x])), TIx1 ([xl kr+ qk
2 (“;,l)k=e-Ilrl+l~ k=O -
1 z---e
2
-([x1+1) ([xl + P+’
kY0
q[x]
+ 1)
(1x1 + 1Y
also
Lemma 2 now follows immediately from the fact that O(x) is bounded. Lemma 2 has an evident application in probability. The function S,(f, x)
231
SZtiSZ-MIRAKYAN
corresponds to the Poisson distribution, theory; the distribution function is
using the terminology of probability
with parameter a > 0. If a = nx for some x > 0, then, by Lemma 2,
This result cannot be proved directly by applying the Central Limit Theorem; there is a similar result when x is a positive integer (see, e.g., 18, p. 3021). The following lemma was proved by Szisz (see 11, p. 239 1). LEMMA 3.
‘If x is a positive number, then
Lemma 4 is a Ramanujan-type result. The second part will not be needed in the proof of our theorems. It is given here merely because of its own interest. LEMMA 4.
(i)
If 2x is a positive integer then
where 6(x) lies between 2 & (ii)
- 3 and 1.
If x is a positive integer, then
X+1 a(x)c
x!
+
2X-I
(xx+ l)! + .** + p-
2r
l)! + ch,! m) =+
where a(x) lies between $ and 3 and p(x) lies between 2(& Proof of Lemma 4. 6(x)
=
- 1) and 2.
It is easy to see that, when 2x is a positive integer,
ex - Cpzo xk/k! =&T x27(2x)!
2
+ (2x + 2;(2x + 1) 3
+ (2x + 3)(2xX+ 2)(2x + 1) + -“’
232
FUHUACHENG
We shall adopt this equality as the definition of 6(x) for all x > i. It is obvious that s(j) = 2 & - 3 and lim x-m 6(x) = 1. Therefore (i) will be proved if we can show that 6(x) is an increasing function. However, this follows immediately from the fact that
x’ n;=I
ui
(2x+9
j = 1) 2, 3)...,
G n::=, (2y+i)’
if x
5. If x > f , then
where cl = (2 fi
- 3)/2 6
and c, = \/e/47c.
Proof of Lemma 5. First, let us assumethat n < x < n + i, where n is a positive integer. Define a function v(t) on [n, n + i) as t E [n, n + +). Since 2n
ifn -1. In particular
By Lemma 4 and Stirling’s formula,
233
SZhZ-MIRAKYAN
Therefore, when II < x < n + $, as 6(x) lies between 2 fi have
- 3 and 1, we
(3.1) if n + i < x < n + 1 for some nonnegative integer n, we define y(t) on [n+i,n+ 1) as
Next,
tE
In+&n+
1).
Along the same lines, we can prove that, when n + f < x < n + 1 for some nonnegative integer n, e-” Ck > 2.rxk/k! satisfies again (3.1). This completes the proof. The last lemma of this section is similar to a result in Hermann’s paper [4], but has a more precise estimate. LEMMA
6.
For any fixed positive numbers a and x, a(kln)
if n is sufficiently
large.
FUHUA
CHENG
Let a(kln) zz
By a simple calculation
if n is sufficiently
k > 2x.
Pk(X),
we can show that
large. Hence, if k > 2x, dl2xltI)ln
ocx’n’pk(x) < 3b,,,,+, = 3
x i”
n“‘1
< \ 3c n]1a(2xtl)ln
Pl2xj+
I@>
2x +
P12x]+
(3.2)
l(X).
By Stirling’s formula (x/(l2xlt
IHtlog(x/(l2*ltl))
12xltI >
Since the function g(y) = 1 - y + logy g(i)-i-log2<0, we have
is increasing
.
on
(0, i 1 and
(3.3) Lemma 6 follows immediately from (3.2) and (3.3). If, in Lemma 6, we replace x by nx, then, when n is suffkiently get the inequality atkIn)
p&x)
< -+ (4x)4ryx
which is what we really need in the proof of our theorem.
large, we
SZ.bZ-MIRAKYAN
235
4. PROOF OF THE THEOREM For any fixed x E (0, co), define g, as
g,(t) =fW -“ox+>3 = 0, =f(t)
x
(4.1)
O
-./-(x-)3
g, is continuous at t = x and inherits all the properties ofJ Using (4.1), (1.1) can be written as
+f(x+)-.0x-) S,(.Lx>= S,(%!?A.) x>+j-(x+)+.I-(-) 2 2 x @?I@> -B,(x))* where A,(X)
=
x
p,(nx)
= emnx \’
k>nx
B,(x)=
k;x
\’
pk(nx)= emnx \‘
kzinx
k3-.r
~(4” k!
’
(nx)” -. k!
Hence
IS,(s,x>- xm+> +f(x->>I,
(4.2)
By Lemma 2,
B,(x)=++ 0i-k ) and
A,(x)= 1- B,(x)= + + 0 ~ i A. 1 Therefore, for the second summand on the right-hand side of (4.2) we have
$(x+)
-“I-(x-)I IA,(x) - B,(x)1= 0 (-g=,
P-(x+)-J-(x-)I
(4.3)
236
FUHUACHENG
and our theorem will be proved if we establish that
for suffciently large n. To do this we first observe that Lebesgue-Stieltjes integral
S,(g,, x)
can be written
S,(g,,x>= !‘ag,(t)d,K,ht>,
as a
(4.5)
0
where the kernel K,(x, t) is defined by
K”(x, ‘>= y p&x),
O
k
0,
t =
0,
the so-called Poisson distribution of probability. We decompose the integral on the right-hand side of (4.5) into three parts, as
d,K,(x, 0=L,(.L xl+M,,(f, xl+R&L x>, Img,(t)
(4.6)
0
where L,(f, x) = j;-x’y;
g,(t) d,K,(x, t>,
_g,(t) d,K,(x, th c.TiXlfi Rn(f, x>=jx;,,fig,(t) 4&(x, t>. We shall evaluate consecutively t E [x - XlJii, x + x/&l],
M,,(f, x), L,(f, x),
I g,(t)l = I g,@>- g,(x)l G V’I x :j$hJ. Hence
and R,,(f, x).
For
231
SZkW-MIRAKYAN
b
i a qfqx,
t)
<
1
for any [a, b] G [O, oo),
therefore
Next, we evaluate L,,(f, x). The method used here is similar approach used by Bojanic and Vuilleumier [9]. Using partial integration with y = x - x/&, we have
to the
Ll(.Lx>=g,Q+)Kn(x,Y+) - JYk(x,t>4 s,(t), 0
where k,(x, t) is the normalized K,,(x, Y+> = K,,(x, v) and
form of K,(x, t). If 0 < y < 00, then
I gh+)l = I g,(y+) - g,(x)1< q+(g,)T where Vz+ (g,) = lim,,,,
Vc+ ,( g,). Therefore
IUf, XI G q+ (g,) K,(x,Y)+ 1; Qx, t) d,(- q g,)). Since k,(x, t) < K,(x, t) on (0, co) and since by Lemma 3,
we have
Since, for nx > 0,
h40i40/3-4
238
F’UHUA CHENG
and -x
y
1 4-~X&))
n I ot (t-.g2 x
+ $
vi+(&)
1
y
=-i n o (t-x)’
4(-W
&>>Y
it follows that
Using partial integration again, we have Y i
1
C+k> cx&) g,)) = - (x _ y)’ + -g--
4-w
0 (x-O2
+ qy0 Wgx) (x $3 . Hence
Replacing the variable t in the last integral by x -x/d,
x-xl&Y I
Wg,)
1
dt (x-
t)3 =Tg
0
we find that
n I , Cx,&L)
dt
Thus
(4.8) Finally, we evaluate R,(f, x). Let z =x +x/v’% [0,2x] as
Q,(x, 0 = 1 - P,(x, t-h = 0,
and define Q,(x, t) on
0 < t < 2x, t=2x.
239
SZkSZ-MIRAKYAN
Then
R,(f,x)= - jzx 4 - g,P) c P&X) z g,(f)4 Q,2nx
+
j*y
bm
4K,(X~
t>
(4.9)
=R,,,+R,,+R3,,.
Using partial integration for the first term on the right-hand side of (4.9), we get R,, = g,(z->
Q,
4 4 &r(f),
where 0,(x, t) is the normalized form of Q,(x, t). Since Q,(x, z-) = Q,(x, z), 0 < z < 1, and Ig,(z-)/ < V:-( g,), we have lR,,I G Vi-(g,>
Q,(x, z) + jzr z
&(x, 4 4 V:(g,).
Since by Lemma 3
Q,(x, t>= c p&x)< k>nl
and since &(x,
n(t
'
-
x < x)'
'
t < 2x,
t) < Q,(x, t) on [0,2x), we have
Next, the inequality 1 z
which follows from Lemma 5 and the identity -
2x-
X n
I*
(t
' -x)'
4
V:(gx)
+
j$
Vi;-(g,)
=
jz2x
(t
:x,z
4
V:C
g,>
240
FUHUACHENG
imply
IRAGC(&> n(z:x)2++j2x * (x;o2 4C(g,). Integrating by parts the last integral, we get 1 I L (t-x)2 2x
d, pH(g,) = WgJ
_ yy Z
X2 2x
$2
jz
dt ~XkL)
([
_
x)’
.
Hence (R,,, <+
(vi:$J
+ 2j
22*vdgx)
dt (t
-x)3
Replacing the variable in the last integral by x + x/fi, dt
I
2X v;(g,) z
(t-X)3
=‘jn
. i
2x2 1
we find that
v;+“i‘/‘(g,)dt
( &)*
Therefore
(4.10) The evaluation of R,, is relatively easy. By Lemma 5, we have IR2nl
G I g,W)l &(4,e)“x
But
I gx(2xI ,< i k=l
and
V,“‘“‘%9
*
241
SZASZ-MIRAKYAN
Consequently (4.11) Finally, by Lemma 6 and the assumption that f(t) = O(P’) t+ co, we see that for TZsuffkiently large,
(a > 0) as
(4.12) for some positive constant M. Hence, from (4. lo), (4.1 l), and (4.12), we obtain for n sufficiently
IR,(f, XII< $
$, K+x’fi
(g,) + o(l)(4x)4”‘(nl)~‘-‘(~~fl*.
Equation (4.4) now folows from (4.5)-(4.8),
large, (4.13)
(4.13), and the fact that
REFERENCES 1. 0. SzAsz,, Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Standards, Sect. B 45 (1950), 239-245. 2. G. MIRAKYAN, Approximation des fonctions continues au moyen de polynomes de la forme emnxCrmO Ck,“.xk, C. R. (D&lady) Acad. Sci. URSS 31 (1941), 201-205. 3. J. GR~F, A S&z Otto-fele operator approximacios tulajdonsagairol, Mat III, Oszf. Kiizl. 20 (1971), 3544. [Hungarian] 4. T. HERMANN,Approximation of unbounded functions on unbounded interval, Acta Math. Acad. Sci. Hungar. 29 (34)
(1977), 393-398.
5. G. N. WATSON,Theorems stated by Ramanujan. V. Approximations connected with e”, Proc. London Math. Sot. (2) 29 (1928), 293-308.
6. G. SZEGG, Ueber einige von S. Ramanujan gestellte Aufgaben, J. London Math. Sot. 3 (1928), 225-232. 7. J. KARAMATA, Sur quelques problemes poses par Ramanujan, J. Indian Math. Sot. (N.S.) 24 (1960) 343-365. 8. B. V. GNEDENKO,“The Theory of Probability,” Chelsea, New York, 1962. 9. R. BOJANIC AND M. VUILLEUMIER, On the rate of convergence of Fourier-Legendre series of functions of bounded variation, J. Approx. Theory 31 (1981) 67-79.
(1984)
On the Rate of Convergence of the SzBsz-Mirakyan Operator for Functions of Bounded Variation FUHUA CHENG Department of Mathematics, Ohio State University, Columbus, Ohio 43210*, U.S.A. Communicated by R. Bojanic Received August 16, 1982
1. INTRODUCTION Let f be a function defined on the infinite Szisz-Mirakyan operator S, applied to f is
interval
[0, co). The
S,(f,x>= 2 f(X) PkW),
(1.1)
k=O
where
pk(t)= C$. In 1950 Szisz [l] proved’: THEOREM A. Zf f (t) is bounded in every finite subinterval of [0, co), is equal to O(tk) for some k > 0 as t -+ co, and is continuous at the point t = x, then S,(f, x) converges uniformly to f (x).
Later on, in 197 1, Gr6f [3] gave the following estimate for the rate of convergence of S,(f, x) when f (t) is continuous on 10, co): ’ Mirakyan
[2] considered the partial sum w,.,(L
and proved that lim,,, w,.,(A x) =f(x) uniformly We do not consider this matter in this paper. * Present address: Institute of Computer University, Hsinchu, Taiwan 300, R.O.C.
Copyright All rights
$3.00
0 1984 by Academic Press, Inc. of reproduction in any form reserved.
in 10, r’ 1, if lim,,,
and Decision
226 0021.9045/84
x) of s,(A x),
(m/n) = r > r’ > 0.
Sciences, National
Tsing Hua
221
SZASZ-MIRAKYAN
THEOREM B. If f is continuous on [0, co) and is equal to O(eax) for some a > 0 as x+ 00, then for all A > 0,
S”(f, x>-f(x) = 0(%4(f, n- 1’2)>, xE [&A], where w,(f, S) = sup{(f(x
(1.2)
It] < 6, x E ]O, A 1I.
+ t) -f(x):
It is easy to see, by considering the function f(t) = 1t - x / at the point t = x (x > 0), that this result is essentially the best possible. This result was improved by Hermann [4] in 1977. He showed that (1.2) holds iff(t) = O(tar) (a > 0). He proved in the same paper that the series in (1.1) does not converge if f(t) 2 tmcr)‘r, where 4(t) is any monotonically increasing function such that lim,,, 4(t) = co. In this paper, we shall study S,(f, x) for functions of bounded variation on every finite subinterval of [0, oo) and prove that S,(f, x) converges to f (f(x + 0) +f(x - 0)) under H ermann’s condition on the magnitude off by giving quantitative estimates of the rate of convergence. We shall also prove that our estimates are essentially best possible.
2. RESULTS AND REMARKS Our main result may be stated as follows THEOREM. Let f be a function of bounded variation on everyfinite subinterval of [0, a) and let f (t) = O(F) for some a > 0 as t + co. If x E (0, 00) is irrational, then for n suflciently large we have
Is,(f,X)-t(f(X+)+f(X-))l~
(3+;)xp’
C v:l:‘$(g,) k:l
+
0(x-‘/y
n1,2 If @+I -f @->I
+ 0(1)(4x)4ax(nx))1’2
where Vi< g) is total variation of g on [a, b] and ‘!m =f (t) -0x+),
x
= 0,
t = x,
=f (t) -f (x-),
O
t+jnx,
(2.1)
228
FUHUACHENG
The right-hand side of (2.1) converges to zero as n -+ co since the continuity of g,(t) at t = x implies that
Remarks.
If f is not constant in any neighborhood of x then (4x)4”“(m)
for n sufficiently
- “2
(+)
nx < &
v;y
g,)
(2.2)
large. So in that case (2.1) becomes
+ 0(x--“*) I/Z It
I.!++) -Ax-11.
(2.3)
If, in addition, f is continuous at x then (2.3) can be further simplified to
for sufficiently large n. If, however, f is constant in some neighborhood of x, then since cl:;$(f) = 0 f or all except a finite number of k’s, we have
for some positive constant c (depending on x), Using this and (2.2), (2.1) becomes
IS,(.ttx>-.m>l < “4n; x)
(2.5)
for n sufficiently large. As far as the precision of the above estimates is concerned, we can prove that (2.4) cannot be asymptotically improved. Consider the function f(t) = /t-xl (x > 0) at t = x. From (2.4) we have
IS,(.Lx>-f(x)1 < $2
k=l
229
SZikZ-MIRAKYAN
Since V:‘:(S)
= 26, it follows that
ILs,(f,X)-f(X)I< 2(4n+x)T- 1<4(4+x). GJit fi On the other hand, by a result of SZ~SZ [ 1, p. 2401,
~ (2x/7ce4)‘12 nw . Hence, for the function f(t) = 1t - xl (x > 0), we have (2x$;*4)“2
< (S,(f; x) -f(x)1
Therefore (2.4) cannot be asymptotically
< 4’4, I’,“‘.
improved.
3. LEMMAS The proof of our theorems is based on a number of lemmas. The first lemma originated in one of the questions posed by Ramanujan in a letter of January 16, 1913, to G. H. Hardy. A complete proof of this lemma was given by Watson [ 5 1, Szego 161, and Karamata [ 7 1. LEMMA
1.
If x is a positive integer, then 1+++$+ .
.
. . . + s e(x) = + e.‘,
where O(x) lies between f and f . For an arbitrary positive number x, we have LEMMA 2.
If x is a positive number, then
230
FUHUACHENG
Proof of Lemma 2. Set n = [x). Define a function v(t) on [n, n + 1) as
y(t)=e-’ <’ t”
t E [n, n + 1).
kfko k! ’
Since y’(t) = -e-‘;
< 0,
if n < t < n + 1 we have
w((n + 1) -I< v/(t) G w(n). In particular iy((n+
l)-)
t’
g
kto k!
By Lemma 1,
1 =y+ew((n+
l)-)=e-‘“tl’
Ix1[Xl[Xl(l Ix]!
- 8((x])), TIx1 ([xl kr+ qk
2 (“;,l)k=e-Ilrl+l~ k=O -
1 z---e
2
-([x1+1) ([xl + P+’
kY0
q[x]
+ 1)
(1x1 + 1Y
also
Lemma 2 now follows immediately from the fact that O(x) is bounded. Lemma 2 has an evident application in probability. The function S,(f, x)
231
SZtiSZ-MIRAKYAN
corresponds to the Poisson distribution, theory; the distribution function is
using the terminology of probability
with parameter a > 0. If a = nx for some x > 0, then, by Lemma 2,
This result cannot be proved directly by applying the Central Limit Theorem; there is a similar result when x is a positive integer (see, e.g., 18, p. 3021). The following lemma was proved by Szisz (see 11, p. 239 1). LEMMA 3.
‘If x is a positive number, then
Lemma 4 is a Ramanujan-type result. The second part will not be needed in the proof of our theorems. It is given here merely because of its own interest. LEMMA 4.
(i)
If 2x is a positive integer then
where 6(x) lies between 2 & (ii)
- 3 and 1.
If x is a positive integer, then
X+1 a(x)c
x!
+
2X-I
(xx+ l)! + .** + p-
2r
l)! + ch,! m) =+
where a(x) lies between $ and 3 and p(x) lies between 2(& Proof of Lemma 4. 6(x)
=
- 1) and 2.
It is easy to see that, when 2x is a positive integer,
ex - Cpzo xk/k! =&T x27(2x)!
2
+ (2x + 2;(2x + 1) 3
+ (2x + 3)(2xX+ 2)(2x + 1) + -“’
232
FUHUACHENG
We shall adopt this equality as the definition of 6(x) for all x > i. It is obvious that s(j) = 2 & - 3 and lim x-m 6(x) = 1. Therefore (i) will be proved if we can show that 6(x) is an increasing function. However, this follows immediately from the fact that
x’ n;=I
ui
(2x+9
j = 1) 2, 3)...,
G n::=, (2y+i)’
if x
5. If x > f , then
where cl = (2 fi
- 3)/2 6
and c, = \/e/47c.
Proof of Lemma 5. First, let us assumethat n < x < n + i, where n is a positive integer. Define a function v(t) on [n, n + i) as t E [n, n + +). Since 2n
ifn
By Lemma 4 and Stirling’s formula,
233
SZhZ-MIRAKYAN
Therefore, when II < x < n + $, as 6(x) lies between 2 fi have
- 3 and 1, we
(3.1) if n + i < x < n + 1 for some nonnegative integer n, we define y(t) on [n+i,n+ 1) as
Next,
tE
In+&n+
1).
Along the same lines, we can prove that, when n + f < x < n + 1 for some nonnegative integer n, e-” Ck > 2.rxk/k! satisfies again (3.1). This completes the proof. The last lemma of this section is similar to a result in Hermann’s paper [4], but has a more precise estimate. LEMMA
6.
For any fixed positive numbers a and x, a(kln)
if n is sufficiently
large.
FUHUA
CHENG
Let a(kln) zz
By a simple calculation
if n is sufficiently
k > 2x.
Pk(X),
we can show that
large. Hence, if k > 2x, dl2xltI)ln
ocx’n’pk(x) < 3b,,,,+, = 3
x i”
n“‘1
< \ 3c n]1a(2xtl)ln
Pl2xj+
I@>
2x +
P12x]+
(3.2)
l(X).
By Stirling’s formula (x/(l2xlt
IHtlog(x/(l2*ltl))
12xltI >
Since the function g(y) = 1 - y + logy g(i)-i-log2<0, we have
is increasing
.
on
(0, i 1 and
(3.3) Lemma 6 follows immediately from (3.2) and (3.3). If, in Lemma 6, we replace x by nx, then, when n is suffkiently get the inequality atkIn)
p&x)
< -+ (4x)4ryx
which is what we really need in the proof of our theorem.
large, we
SZ.bZ-MIRAKYAN
235
4. PROOF OF THE THEOREM For any fixed x E (0, co), define g, as
g,(t) =fW -“ox+>3 = 0, =f(t)
x
(4.1)
O
-./-(x-)3
g, is continuous at t = x and inherits all the properties ofJ Using (4.1), (1.1) can be written as
+f(x+)-.0x-) S,(.Lx>= S,(%!?A.) x>+j-(x+)+.I-(-) 2 2 x @?I@> -B,(x))* where A,(X)
=
x
p,(nx)
= emnx \’
k>nx
B,(x)=
k;x
\’
pk(nx)= emnx \‘
kzinx
k3-.r
~(4” k!
’
(nx)” -. k!
Hence
IS,(s,x>- xm+> +f(x->>I,
(4.2)
By Lemma 2,
B,(x)=++ 0i-k ) and
A,(x)= 1- B,(x)= + + 0 ~ i A. 1 Therefore, for the second summand on the right-hand side of (4.2) we have
$(x+)
-“I-(x-)I IA,(x) - B,(x)1= 0 (-g=,
P-(x+)-J-(x-)I
(4.3)
236
FUHUACHENG
and our theorem will be proved if we establish that
for suffciently large n. To do this we first observe that Lebesgue-Stieltjes integral
S,(g,, x)
can be written
S,(g,,x>= !‘ag,(t)d,K,ht>,
as a
(4.5)
0
where the kernel K,(x, t) is defined by
K”(x, ‘>= y p&x),
O
k
0,
t =
0,
the so-called Poisson distribution of probability. We decompose the integral on the right-hand side of (4.5) into three parts, as
d,K,(x, 0=L,(.L xl+M,,(f, xl+R&L x>, Img,(t)
(4.6)
0
where L,(f, x) = j;-x’y;
g,(t) d,K,(x, t>,
_g,(t) d,K,(x, th c.TiXlfi Rn(f, x>=jx;,,fig,(t) 4&(x, t>. We shall evaluate consecutively t E [x - XlJii, x + x/&l],
M,,(f, x), L,(f, x),
I g,(t)l = I g,@>- g,(x)l G V’I x :j$hJ. Hence
and R,,(f, x).
For
231
SZkW-MIRAKYAN
b
i a qfqx,
t)
<
1
for any [a, b] G [O, oo),
therefore
Next, we evaluate L,,(f, x). The method used here is similar approach used by Bojanic and Vuilleumier [9]. Using partial integration with y = x - x/&, we have
to the
Ll(.Lx>=g,Q+)Kn(x,Y+) - JYk(x,t>4 s,(t), 0
where k,(x, t) is the normalized K,,(x, Y+> = K,,(x, v) and
form of K,(x, t). If 0 < y < 00, then
I gh+)l = I g,(y+) - g,(x)1< q+(g,)T where Vz+ (g,) = lim,,,,
Vc+ ,( g,). Therefore
IUf, XI G q+ (g,) K,(x,Y)+ 1; Qx, t) d,(- q g,)). Since k,(x, t) < K,(x, t) on (0, co) and since by Lemma 3,
we have
Since, for nx > 0,
h40i40/3-4
238
F’UHUA CHENG
and -x
y
1 4-~X&))
n I ot (t-.g2 x
+ $
vi+(&)
1
y
=-i n o (t-x)’
4(-W
&>>Y
it follows that
Using partial integration again, we have Y i
1
C+k> cx&) g,)) = - (x _ y)’ + -g--
4-w
0 (x-O2
+ qy0 Wgx) (x $3 . Hence
Replacing the variable t in the last integral by x -x/d,
x-xl&Y I
Wg,)
1
dt (x-
t)3 =Tg
0
we find that
n I , Cx,&L)
dt
Thus
(4.8) Finally, we evaluate R,(f, x). Let z =x +x/v’% [0,2x] as
Q,(x, 0 = 1 - P,(x, t-h = 0,
and define Q,(x, t) on
0 < t < 2x, t=2x.
239
SZkSZ-MIRAKYAN
Then
R,(f,x)= - jzx 4 - g,P) c P&X) z g,(f)4 Q,
+
j*y
bm
4K,(X~
t>
(4.9)
=R,,,+R,,+R3,,.
Using partial integration for the first term on the right-hand side of (4.9), we get R,, = g,(z->
Q,
4 4 &r(f),
where 0,(x, t) is the normalized form of Q,(x, t). Since Q,(x, z-) = Q,(x, z), 0 < z < 1, and Ig,(z-)/ < V:-( g,), we have lR,,I G Vi-(g,>
Q,(x, z) + jzr z
&(x, 4 4 V:(g,).
Since by Lemma 3
Q,(x, t>= c p&x)< k>nl
and since &(x,
n(t
'
-
x < x)'
'
t < 2x,
t) < Q,(x, t) on [0,2x), we have
Next, the inequality 1 z
which follows from Lemma 5 and the identity -
2x-
X n
I*
(t
' -x)'
4
V:(gx)
+
j$
Vi;-(g,)
=
jz2x
(t
:x,z
4
V:C
g,>
240
FUHUACHENG
imply
IRAGC(&> n(z:x)2++j2x * (x;o2 4C(g,). Integrating by parts the last integral, we get 1 I L (t-x)2 2x
d, pH(g,) = WgJ
_ yy Z
X2 2x
$2
jz
dt ~XkL)
([
_
x)’
.
Hence (R,,, <+
(vi:$J
+ 2j
22*vdgx)
dt (t
-x)3
Replacing the variable in the last integral by x + x/fi, dt
I
2X v;(g,) z
(t-X)3
=‘jn
. i
2x2 1
we find that
v;+“i‘/‘(g,)dt
( &)*
Therefore
(4.10) The evaluation of R,, is relatively easy. By Lemma 5, we have IR2nl
G I g,W)l &(4,e)“x
But
I gx(2xI ,< i k=l
and
V,“‘“‘%9
*
241
SZASZ-MIRAKYAN
Consequently (4.11) Finally, by Lemma 6 and the assumption that f(t) = O(P’) t+ co, we see that for TZsuffkiently large,
(a > 0) as
(4.12) for some positive constant M. Hence, from (4. lo), (4.1 l), and (4.12), we obtain for n sufficiently
IR,(f, XII< $
$, K+x’fi
(g,) + o(l)(4x)4”‘(nl)~‘-‘(~~fl*.
Equation (4.4) now folows from (4.5)-(4.8),
large, (4.13)
(4.13), and the fact that
REFERENCES 1. 0. SzAsz,, Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Standards, Sect. B 45 (1950), 239-245. 2. G. MIRAKYAN, Approximation des fonctions continues au moyen de polynomes de la forme emnxCrmO Ck,“.xk, C. R. (D&lady) Acad. Sci. URSS 31 (1941), 201-205. 3. J. GR~F, A S&z Otto-fele operator approximacios tulajdonsagairol, Mat III, Oszf. Kiizl. 20 (1971), 3544. [Hungarian] 4. T. HERMANN,Approximation of unbounded functions on unbounded interval, Acta Math. Acad. Sci. Hungar. 29 (34)
(1977), 393-398.
5. G. N. WATSON,Theorems stated by Ramanujan. V. Approximations connected with e”, Proc. London Math. Sot. (2) 29 (1928), 293-308.
6. G. SZEGG, Ueber einige von S. Ramanujan gestellte Aufgaben, J. London Math. Sot. 3 (1928), 225-232. 7. J. KARAMATA, Sur quelques problemes poses par Ramanujan, J. Indian Math. Sot. (N.S.) 24 (1960) 343-365. 8. B. V. GNEDENKO,“The Theory of Probability,” Chelsea, New York, 1962. 9. R. BOJANIC AND M. VUILLEUMIER, On the rate of convergence of Fourier-Legendre series of functions of bounded variation, J. Approx. Theory 31 (1981) 67-79.