Representation formulae for

POLONICI MATHEMATICI LXIII.3 (1996)

Representation formulae for (C

0

) m -parameter operator semigroups

by Mi Zhou and George A. Anastassiou (Memphis, Tenn.)

Abstract. Some general representation formulae for (C

0

) m-parameter operator semi- groups with rates of convergence are obtained by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (C

0

) one- and m- parameter operator semigroups as special cases. When we consider special semigroups we recover well-known convergence theorems for multivariate approximation operators.

1. Introduction. Recently the study of representation formulae for (C

0

) operator semigroups has attracted much attention (Shaw [17, 18], Butzer–

Hahn [2], Pfeifer [13–15] and Chen–Zhou [3]). They gave some general formu- lae that include earlier (Post–Widder, Hille–Phillips [8] and Chung [4]) con- crete representation formulae. But most of the work done so far is confined to the one-parameter case, while Shaw’s method for the multi-parameter case is not an easy one to get new formulae and the results are without rates of convergence. In this article we try to give some general representation formulae for (C

0

) m-parameter operator semigroups. The main idea is the use of a probabilistic setting in the representation of operator semigroups, initiated by Chung [4] and developed by Butzer–Hahn [2] and Pfeifer [13], and the so-called multiplier enlargement method of Hsu–Wang [9, 19] and Shaw [17, 18]. At the same time, by introducing a modified second modu- lus of continuity of an operator semigroup and a Steklov-type element we establish quantitative estimates of the obtained formulae.

To the best of our knowledge, all existing representation formulae for (C

0

) one- and multi-parameter operator semigroups are special cases of our results. In particular, Shaw’s formulae [17, 18] for m-parameter operator

1991 Mathematics Subject Classification: Primary 47D03, 41A36, 41A63, 60G50; Sec- ondary 41A65, 60H30, 41A17, 41A80, 47A56.

Key words and phrases: multi-parameter operator semigroups, representation formu- lae, multivariate approximation, sum of random vectors, Banach space, multiplier enlarge- ment method, rate of convergence, second modulus of continuity, inequalities.

[247]

semigroups are special cases of our results when specifying the random vec- tors considered. Also with our method it is easier to obtain new formulae.

At the end we give examples to show the application of our results in multivariate operator approximation theory when we consider particular operator semigroups.

2. Preliminaries. Let X be a Banach space with norm k·k, and E (X ) be the Banach algebra of endomorphisms of X . If T ∈ E (X ), kT k also denotes the norm of T . Let R

^m

be the m-dimensional Euclidean space supplied with the usual definition of arithmetical operations and metric. We write t = (t

1

, . . . , t

m

) ∈ R

^m

, t = t

1

+ . . . + t

m

, t = |t

1

| + . . . + |t

_m

| and denote the unit vectors by e

1

, . . . , e

m

, where e

k

= (0, . . . , 1, . . . , 0) with 1 in the kth place and 0 elsewhere. Further, let

R

^m+

= {t ∈ R

^m

: t

k

≥ 0, k = 1, . . . , m},

the first closed 2

^m

-ant in R

^m

. Z

+

denotes the set of all non-negative integers and

Z

^m+

= {n = (n

1

, . . . , n

m

) : n

k

∈ Z

⁺

, k = 1, . . . , m}, while N is the set of all positive integers.

A family of bounded linear operators {T (t) : t ∈ R

^m₊

} on X is called a (C

0

) m-parameter operator semigroup in E (X ) when the following three conditions are satisfied:

T (t + s) = T (t)T (s), t, s ∈ R

^m+

; (1)

T (0) = I (identity operator);

(2)

s-lim

t∈R^m₊, t→0

T (t)f = f, f ∈ X . (3)

It is known that {T (t) : t ∈ R

^m+

} is then the direct product of m (C

₀

) one-parameter operator semigroups in E (X ):

(4) T (t) =

m

Y

k=1

T

k

(t

k

),

where T

k

(t

k

) = T (t

k

e

k

). The operators {T

k

(t

k

) : 0 ≤ t

k

< ∞} (k = 1, . . . , m) commute with each other.

Let A

k

be the infinitesimal generator of {T

k

(t

k

) : 0 ≤ t

k

< ∞} with domain D(A

k

), k = 1, . . . , m. Then if f is in D(A

k

) so is T (t)f for each t ∈ R

^m+

and

A

k

T (t)f = T (t)A

k

f.

Further, if f ∈ D(A

j

) and f ∈ D(A

j

A

k

) then f ∈ D(A

k

A

j

) and

A

k

A

j

f = A

j

A

k

f (j, k = 1, . . . , m). In the following we use the notation

D

²

:=

m

\

k,j=1

D(A

k

A

j

).

D

²

is a linear subspace of X .

To each k = 1, . . . , m, there correspond two numbers M

k

≥ 1 and ω

_k

≥ 0 such that

kT

k

(t

k

)k ≤ M

k

e

^ωktk

, 0 ≤ t

k

< ∞.

Hence we have the inequality

(5) kT (t)k ≤ M exp(ω(t

1

+ . . . + t

m

)) = M e

^ω¯t

, t ∈ R

^m+

, where M = M

1

. . . M

m

and ω = max{ω

k

: 1 ≤ k ≤ m}.

In the following it is always understood that {T (t) : t ∈ R

^m+

} satisfies (5), unless otherwise specified.

For the above definitions and properties of operator semigroups we refer to Butzer–Berens [1], Hille–Phillips [8] or W. K¨ ohnen [11].

Let (Ω, A, P ) be a probability space. For every real-valued random vari- able X defined on (Ω, A, P ), E(X) denotes its expectation. If ξ = E(X) exists then σ

²

= σ

²

(X) = E[(X − ξ)

²

] is the variance of X. Let fur- ther Ψ

X

(u) = E(u

^X

), u ≥ 0, and Ψ

_X^∗

(u) = E(e

^uX

), u ∈ R, denote the probability-generating function and the moment-generating function of X respectively.

We need to consider m-dimensional random vectors, also denoted by X, Y, . . . , on (Ω, A, P ). For an m-dimensional random vector X = (X

01

, . . . , X

0m

), we also use E(X) to denote its expectation:

E(X) := (E(X

01

), . . . , E(X

0m

)) and define

σ

²_i

(X) := σ

²

(X

0i

).

It is not difficult to extend the theory of the extended Pettis integral developed in [13] to the multivariate case.

Let {T (t) : t ∈ R

^m+

} be as above and X be an R

^m+

-valued random vector such that

Ψ

^∗

X

(ω) < ∞, X = X

01

+ . . . + X

0m

. Then for every f ∈ X define

E[T (X)f ] := R

Ω

T (X)f dP,

which exists in the Bochner sense in X by the strong continuity of {T (t) : t ∈ R

^m+

} and (5). Moreover, the map E[T (X)] : f → E[T (X)f ] on X defines a bounded linear operator E[T (X)] ∈ E (X ) with

kE[T (X)]k ≤ M Ψ

^∗

X

(ω).

E[T (X)] is called the expectation of T (X) and is understood as an extended Pettis integral following [13].

If X and Y are independent R

^m+

-valued random vectors such that Ψ

^∗

X

(ω)

< ∞ and Ψ

^∗

Y

(ω) < ∞ then E[T (X)], E[T (Y )] and E[T (X + Y )] exist in E(X ) and

E[T (X) ◦ T (Y )] = E[T (X + Y )] = E[T (X)] ◦ E[T (Y )], where “◦” denotes composition.

For the above see [13], [15], [16] and the references cited there.

3. Auxiliary results. We need a Taylor expansion integral formula for (C

0

) m-parameter operator semigroups.

Lemma 1. Assume {T (t) : t ∈ R

^m+

} is a (C

₀

) m-parameter operator semigroup satisfying (5). Then for every g ∈ D

²

and s, t ∈ R

^m+

,

(6) T (t)g − T (s)g

= T (s)[(t

1

− s

1

)A

1

g + . . . + (t

m

− s

m

)A

m

g]

+

1

R

0

(1 − u)T (s + u(t − s))((t

1

− s

1

)A

1

+ . . . + (t

m

− s

m

)A

m

)

²

g du.

P r o o f. Let G(u) = T (s + u(t − s))g ∈ X , u ∈ [0, 1]. Then G

⁰

(u) := dG(u)

du

= T (s + u(t − s))[(t

1

− s

₁

)A

1

+ . . . + (t

m

− s

_m

)A

m

]g and

G

⁰⁰

(u) = T (s + u(t − s))[(t

1

− s

₁

)A

1

+ . . . + (t

m

− s

_m

)A

m

]

²

g.

Now (6) follows from the Taylor formula with integral remainder for Banach space valued functions (see, e.g., [5, Theorem 8.14.3]).

For our purpose we need a second modulus of continuity ω

2

(T f, δ) and the Steklov operator J

h

(f ) (h > 0) for a (C

0

) m-parameter operator semi- group {T (t) : t ∈ R

^m+

} and f ∈ X .

Definition 1.

ω

2

(T f, δ) = sup

t=(t1,...,tm) 0≤ti, tj≤δ

{k(T (t) − I)

²

f k, k(T

i

(t

i

) − I)(T

j

(t

j

) − I)f k}.

We have ω

2

(T f, δ) → 0 as δ → 0, by the strong continuity of {T (t) : t ∈ R

^m+

}.

Definition 2.

J

h

(f ) =  2 h



2m h/2

R

0

. . .

h/2

R

0

| {z }

2m

[2T (ξ

1

+ η

1

, . . . , ξ

m

+ η

m

)

− T (2ξ

₁

+ 2η

1

, . . . , 2ξ

m

+ 2η

m

)]f dξ

1

dη

1

. . . dξ

m

dη

m

. The integral may be considered as a multi-X -valued Riemann integral.

We have the following lemma:

Lemma 2. (i) J

^h

(f ) ∈ D

²

for all f ∈ X ; (ii) kf − J

h

(f )k ≤ ω

2

(T f, h);

(iii) kA

i

A

j

J

h

(f )k ≤ 9M e

^2(m−1)hω

ω

2

(T f, h)/h

²

, 1 ≤ i, j ≤ m.

P r o o f. (i) Let J

1

=

h/2

R

0

. . .

h/2

R

0

| {z }

2m

T (ξ

1

+ η

1

, . . . , ξ

m

+ η

m

)f dξ

1

dη

1

. . . dξ

m

dη

m

, (7)

J

2

=

h/2

R

0

. . .

h/2

R

0

| {z }

2m

T (2ξ

1

+ 2η

1

, . . . , 2ξ

m

+ 2η

m

)f dξ

1

dη

1

. . . dξ

m

dη

m

(8)

=  1 2



2m h

R

0

. . .

h

R

0

| {z }

2m

T (ξ

1

+ η

1

, . . . , ξ

m

+ η

m

)f dξ

1

dη

1

. . . dξ

m

dη

m

.

It is not difficult to show that J

1

∈ D

²

, J

2

∈ D

²

(cf. [1, p. 10]) and hence (i) holds.

(ii) We have kf − J

_h

(f )k =

 2 h



2m h/2

R

0

. . .

h/2

R

0

| {z }

2m

[f − 2T (ξ

1

+ η

1

, . . . , ξ

m

+ η

m

)

+ T (2ξ

1

+ 2η

1

, . . . , 2ξ

m

+ 2η

m

)]f dξ

1

dη

1

. . . dξ

m

dη

m

=

 2 h



2m h/2

R

0

. . .

h/2

R

0

| {z }

2m

[T (ξ

1

+ η

1

, . . . , ξ

m

+ η

m

) − I]

²

f

× dξ

₁

dη

1

. . . dξ

m

dη

m

≤  2 h



2m h/2

R

0

. . .

h/2

R

0

| {z }

2m

k[T (ξ

₁

+ η

1

, . . . , ξ

m

+ η

m

) − I]

²

f k

× dξ

1

dη

1

. . . dξ

m

dη

m

≤ ω

2

(T f, h).

(iii) When i 6= j, similar to the one-parameter operator semigroup case (ibid.), we can show

A

i

A

j

J

1

=

h/2

R

0

. . .

h/2

R

0

| {z }

2m−2

Y

k6=i,j

T

k

(ξ

k

+ η

k

)T

i

(η

i

)T

j

(η

j

)

× (T

_i

(h/2) − I)(T

j

(h/2) − I)f Y

k6=i,j

dξ

k

dη

k

dη

i

dη

j

and

A

i

A

j

J

2

=  1 2



2m h

R

0

. . .

h

R

0

| {z }

2m−2

Y

k6=i,j

T

k

(ξ

k

+ η

k

)T

i

(η

i

)T

j

(η

j

)

× (T

_i

(h) − I)(T

j

(h) − I)f Y

k6=i,j

dξ

k

dη

k

dη

i

dη

j

. So

kA

_i

A

j

J

h

(f )k = k(2/h)

^2m

[2A

i

A

j

J

1

− A

_i

A

j

J

2

]k

≤  2 h



2m

 2

h/2

R

0

. . .

h/2

R

0

| {z }

2m−2

Y

k6=i,j

M

k

e

^ωk(ξk+ηk)

M

i

e

^ωiηi

M

j

e

^ωjηj

×

 T

i

 h 2



− I



T

j

 h 2



− I

 f

Y

k6=i,j

dξ

k

dη

k

dη

i

dη

j

+  1 2



2m h

R

0

. . .

h

R

0

| {z }

2m−2

Y

k6=i,j

M

k

e

^ωk(ξk+ηk)

M

i

e

^ωiηi

M

j

e

^ωjηj

× k(T

i

(h) − I)(T

j

(h) − I)f k Y

k6=i,j

dξ

k

dη

k

dη

i

dη

j



≤ (2/h)

^2m

M {2e

^{(2m−2)ωh/2}

(h/2)

^2m−2

+ (1/2)

^2m

e

^(2m−2)ωh

h

^2m−2

}

× ω

₂

(T f, h)

≤ M e

^(2m−2)ωh

{2(h/2)

⁻²

+ 1/h

²

}ω

₂

(T f, h)

= 9M e

^2(m−1)ωh

ω

2

(T f, h)/h

²

. When i = j, the same estimate holds.

Lemma 3. For any R

^m+

-valued random vector Y = (Y

01

, . . . , Y

0m

) with E(Y ) = x = (x

1

, . . . , x

m

) and f ∈ X ,

(9) kE[T (Y )]f − T (x)f k

= kE[T (Y )f ] − T (x)f k

≤ M ω

₂

(T f, h)



2E(e

^ωY

)

+ 9

2 mM e

^{2ω ¯x}

e

^2(m−1)hω

[E(e

^pωY

)]

^1/p

h X

^m

i=1

(E((Y

0i

− x

i

)

^2q

))

^1/q

i h

²

 , where p > 0, q > 0, 1/p + 1/q = 1, h > 0. If ω = 0, we have

(10) kE[T (Y )]f − T (x)f k ≤ 2M ω

₂

(T f, h)



1 + 9mM 4h

²

m

X

i=1

σ

²

(Y

0i

)

 . P r o o f. We have

kE[T (Y )]f − T (x)f k = kE[T (Y )f ] − T (x)f k (11)

≤ kE[T (Y )f ] − E[T (Y )J

_h

f ]k + kE[T (Y )J

h

f ] − T (x)J

h

f k + kT (x)J

h

f − T (x)f k

=: I

1

+ I

2

+ I

3

. Now,

I

1

≤ E[kT (Y )(J

h

f − f )k]

(12)

≤ E[M e

^ωY

kJ

_h

f − f k] ≤ M E(e

^ωY

)ω

2

(T f, h) by Lemma 2, and

(13) I

3

≤ M e

^{ω ¯x}

ω

2

(T f, h) ≤ M E(e

^ωY

)ω

2

(T f, h) by Jensen’s inequality.

Note that g := J

h

(f ) ∈ D

²

, by Lemma 2. Apply Lemma 1 to get I

2

=

E n

T (x)[(Y

01

− x

₁

)A

1

+ . . . + (Y

0m

− x

_m

)A

m

]g (14)

+

1

R

0

(1 − u)T (x + u(Y − x))[(Y

01

− x

₁

)A

1

+ . . . . . . + (Y

0m

− x

_m

)A

m

]

²

g du

o

= E

n R

¹

0

(1 − u)T (x + u(Y − x))[(Y

01

− x

1

)A

1

+ . . . . . . + (Y

0m

− x

m

)A

m

]

²

g du o

≤ E n R

¹

0

(1 − u)kT (x + u(Y − x))k

× k[(Y

₀₁

− x

₁

)A

1

+ . . . + (Y

0m

− x

_m

)A

m

]

²

gk du o

≤ M E n R

¹

0

(1 − u) exp(ω(x + u(Y − x))

× k[(Y

₀₁

− x

₁

)A

1

+ . . . + (Y

0m

− x

_m

)A

m

]

²

gk du o

≤ 1

2 M e

^{2ω ¯x}

E n e

^ωY

m

X

i=1 m

X

j=1

|Y

_0i

− x

_i

| · |Y

_0j

− x

_j

| · kA

_i

A

j

gk o

≤ 1

2 M e

^{2ω ¯x}

E

 e

^ωY

m

X

i=1 m

X

j=1

1

2 [(Y

0i

− x

_i

)

²

+ (Y

0j

− x

_j

)

²

]



× 9M e

^2(m−1)hω

ω

2

(T f, h)/h

²

(by Lemma 2(iii))

= 9

2 M

²

e

^{2ω ¯x}

e

^2(m−1)hω

m

m

X

i=1

E[e

^ωY

(Y

0i

− x

_i

)

²

]ω

2

(T f, h)/h

²

≤ 9

2 mM

²

e

^{2ω ¯x}

e

^2(m−1)hω

[E(e

^pωY

)]

^1/p

h X

^m

i=1

(E((Y

0i

− x

_i

)

^2q

))

^1/q

i

× ω

₂

(T f, h)/h

²

by H¨ older’s inequality.

Therefore by (11)–(14) we get (9).

If ω = 0 we have I

1

≤ M ω

₂

(T f, h), I

3

≤ M ω

₂

(T f, h), and I

2

≤

⁹₂

M

²

m × P

m

i=1

σ

²

(Y

0i

)ω

2

(T f, h)/h

²

so (10) follows.

4. Main results. Here comes our first main result:

Theorem 1. Let X = (X

⁰¹

, . . . , X

0m

) be an R

^m+

-valued random vec- tor with E(X) = x = (x

1

, . . . , x

m

) and suppose that there exists a δ > 0 such that Ψ

^∗

X

(δ) < ∞. Then for any (C

0

) m-parameter operator semigroup satisfying (5), and all n > max(pω/δ, 1/δ

²

),

k{E[T (X/n)]}

ⁿ

f − T (x)f k (15)

≤ 2M ω

₂

(T f, 1/ √ n)



e

^{ω ¯x}

exp

 2nω

²

e

²

(nδ − ω)

²

Ψ

_X^∗

(δ)



+ 2

^1/q

9M m

²

q

²

e

²

e

^{3ω ¯x}

e

^2(m−1)ω/

√n

× exp

 2npω

²

e

²

(nδ − pω)

²

+ 2ne

^{δ ¯x}

e

²

q(δ √

n − 1)

²

 Ψ

_X^∗

(δ)



,

where p, q > 0, 1/p + 1/q = 1, is an arbitrary conjugate pair. If ω = 0, then (16) k{E[T (X/n)]}

ⁿ

f − T (x)f k

≤ 2M ω

₂

(T f, 1/ √ n)

 1 + 9

4 mM

m

X

i=1

σ

²

(X

0i

)

 .

N o t e. All the right hand sides of (15) and (16) are finite.

P r o o f o f T h e o r e m 1. Let X

k

be a sequence of independent random vectors identically distributed as X, and Y = (1/n) P

n

k=1

X

k

. Then E(Y ) = 1

n

n

X

k=1

E(X

k

) = x, E[T (Y )f ] = {E[T (X/n)]}

ⁿ

f.

For u > 0 we have Ψ

^∗

Y

(u) = E(e

^{(u/n)Σnk=1Xk}

) = (E(e

^(u/n)X

))

ⁿ

≤

 1 + u

n E(X) + E  u

²

X

²

2n

²

e

^(u/n)X



n

≤

 1 + u

n x + u

²

2n

²

 2

δ − u/n



2

e

⁻²

E(e

^δX

)



n

≤ e

^u¯x

exp

 2nu

²

e

²

(nδ − u)

²

Ψ

^∗

X

(δ)



whenever u/n < δ.

Above we made use of the inequalities (see also Pfeifer [14, p. 275]) (17) r

^α

e

^ηr

≤

 α δ − η



α

e

^−α

e

^δr

(for η < δ, r > 0, α > 0) and

(18) (1 + r)

ⁿ

≤ e

^nr

.

So for n > pω/δ ≥ ω/δ,

E(e

^ωY

) ≤ e

^{ω ¯x}

exp

 2nω

²

e

²

(nδ − ω)

²

Ψ

_X^∗

(δ)



and

[E(e

^pωY

)]

^1/p

≤



e

^{pω ¯x}

exp

 2np

²

ω

²

e

²

(nδ − pω)

²

Ψ

_X^∗

(δ)



1/p

= e

^{ω ¯x}

exp

 2npω

²

e

²

(nδ − pω)

²

Ψ

^∗

X

(δ)

 . Observe that for Y = (Y

01

, . . . , Y

0m

) we have

E((Y

0i

− x

_i

)

^2q

)

= E  1 n

n

X

k=1

X

ki

− x

i



2q



≤  2q

√ n



2q

e

^−2q

E(e

√n|(1/n)Σⁿ_k=1Xki−xi|

) (by (17))

≤  2q

√ n



2q

e

^−2q

[E(e

^(1/

√n)Σⁿ_k=1(Xki−xi)

) + E(e

^(1/

√n)Σⁿ_k=1(xi−Xki)

)]

≤  2q

√ n



2q

e

^−2q

2 exp

 E  1

2 (X

0i

− x

_i

)

²

e

^(1/

√n)|X0i−xi|



(by Taylor’s expansion and (18))

≤  2q

√ n



2q

e

^−2q

2 exp

 2

δ − 1/ √ n



2

e

⁻²

E  1

2 e

^δ|X0i−xi|



(by (17) for 1/ √ n < δ)

≤  2q

√ n



2q

e

^−2q

2 exp

 2n

(δ √

n − 1)

²

e

⁻²

e

^{δ ¯x}

Ψ

_X^∗

(δ)



, 1 ≤ i ≤ m.

Hence we established that [E((Y

0i

− x

_i

)

^2q

)]

^1/q

≤ 4q

²

n e

⁻²

2

^1/q

exp

 2n

q(δ √

n − 1)

²

e

⁻²

e

^{δ ¯x}

Ψ

^∗

X

(δ)

 . Now apply Lemma 3 and take h = 1/ √

n:

k{E[T (X/n)]}

ⁿ

f − T (x)f k = kE[T (Y )]f − T (x)f k

≤ M ω

₂

(T f, 1/ √ n)



2e

^{ω ¯x}

exp

 2nω

²

e

²

(nδ − ω)

²

Ψ

^∗

X

(δ)



+ 9

2 mM e

^{2ω ¯x}

e

^2(m−1)ω/

√n

e

^{ω ¯x}

exp

 2npω

²

e

²

(nδ − pω)

²

Ψ

_X^∗

(δ)



× m 4q

²

n e

⁻²

2

^1/q

exp

 2n

q(δ √

n − 1)

²

e

⁻²

e

^{δ ¯x}

Ψ

_X^∗

(δ)

 n



= 2M ω

2

(T f, 1/ √ n)



e

^{ω ¯x}

exp

 2nω

²

e

²

(nδ − ω)

²

Ψ

^∗

X

(δ)



+ 2

^1/q

9M m

²

q

²

e

²

e

^{3ω ¯x}

e

^2(m−1)ω/

√n

× exp

 2npω

²

e

²

(nδ − pω)

²

+ 2ne

^{δ ¯x}

e

²

q(δ √

n − 1)

²

 Ψ

^∗

X

(δ)



. When ω = 0, noting that σ

²

(Y

0i

) = σ

²

(X

0i

)/n by (10), we get (16).

A ramification of Theorem 1 follows:

Theorem 2. Let N be a Z

+

-valued random variable with E(N ) = η, η > 0, and let Y = (Y

01

, . . . , Y

0m

) be an R

^m+

-valued random vector indepen- dent of N with E(Y ) = γ = (γ

1

, . . . , γ

m

). Assume that there exists a δ > 0 such that

Ψ

N

(Ψ

_Y^∗

(δ)) < ∞.

Then for n > max(pω/δ, 1/δ

²

),

k{Ψ

_N

[E(T (Y /n))]}

ⁿ

f − T (ηγ)f k (19)

≤ 2M ω

₂

(T f, 1/ √ n)



e

^ωη¯γ

exp

 2nω

²

e

²

(nδ − ω)

²

Ψ

N

(Ψ

_Y^∗

(δ))



+ 2

^1/q

9M m

²

q

²

e

²

e

^3ωη¯γ

e

^2(m−1)ω/

√n

× exp

 2npω

²

e

²

(nδ − pω)

²

+ 2ne

^δη¯γ

e

²

q(δ √

n − 1)

²

 Ψ

N

(Ψ

^∗

X

(δ))



, where p, q > 0, 1/p + 1/q = 1, is an arbitrary conjugate pair. If ω = 0, then (20) k{Ψ

_N

[E(T (Y /n))]}

ⁿ

f − T (ηγ)f k

≤ 2M ω

₂

(T f, 1/ √ n)

 1 + 9

4 mM

m

X

i=1

[ησ

²

(Y

0i

) + σ

²

(N )γ

_i²

]

 .

P r o o f. Consider Y

k i.i.d.

∼ Y , which are also independent of N . In Theo- rem 1, take X = P

N

k=1

Y

k

(as usual, an empty sum equals 0). Then E

 T  1

n X



= E

 T  1

n

N

X

k=1

Y

k



=

∞

X

l=0

P (N = l)E

 T  1

n

l

X

k=1

Y

k



=

∞

X

l=0

P (N = l)

 E

 T  1

n Y



l

= Ψ

N

 E

 T  1

n Y



,

E(X) =

∞

X

l=0

P (N = l)E



^l

X

k=1

Y

k



= E(N )E(Y ) = ηγ.

Also

Ψ

_X^∗

(δ) = E(e

^δX

) = E(e

^δΣNk=1Yk

) =

∞

X

l=0

P (N = l)E(e

^δΣlk=1Yk

)

=

∞

X

l=0

P (N = l)(E(e

^δY

))

^l

= Ψ

N

(Ψ

_Y^∗

(δ)).

If X = (X

01

, . . . , X

0m

) we have σ

²

(X

0i

) = σ

²

 X

^N

k=1

Y

ki



=

∞

X

l=0

P (N = l)E  X

^l

k=1

Y

ki



2



− η

²

γ

_i²

=

∞

X

l=0

P (N = l)(lE(Y

_0i²

) + l(l − 1)γ

_i²

) − η

²

γ

_i²

= ησ

²

(Y

0i

) + σ

²

(N )γ

_i²

. Then (19), (20) follow by (15), (16).

An application of Lemma 3 comes next:

Theorem 3. For each positive real number τ , let N

τ

be a Z

+

-valued random variable with E(N

τ

) = τ η, where η ∈ R

+

is fixed. Let X be an R

^m+

-valued random vector with E(X) = γ = (γ

1

, . . . , γ

m

), independent of N

τ

. Assume that there exists a δ > 0 such that Ψ

^∗

X

(δ) < ∞ and further there are p, q > 0 with 1/p + 1/q = 1 such that

lim sup

τ →∞

Ψ

Nτ



Ψ

_X^∗

 pω τ



= d

1

< ∞, (21)

lim sup

τ →∞

τ

 E  1

τ N

τ

− η



2q



1/q

= d

2

< ∞ (22)

and

(23) lim sup

τ →∞

Ψ

_N^∗_τ

 2

e

²

( √

τ δ − 1)

²

e

^δ¯γ

Ψ

_X^∗

(δ)



= d

3

< ∞.

Then for τ > 1/δ

²

,

(24) kΨ

_Nτ

(E[T (X/τ )])f − T (ηγ)f k

≤ M ω

2

(T f, 1/ √ τ )

 2d

1

+ 9mM e

^2ωη¯γ

e

^2(m−1)ω/

√τ

d

^1/p₁



m2

^1/q

 2q e



2

d

^1/q₃

+ d

2 m

X

i=1

γ

_i²



.

If ω = 0, then

(25) kΨ

_Nτ

(E[T (X/τ )])f − T (ηγ)f k

≤ 2M ω

₂

(T f, 1/ √ τ )

 1 + 9

4 mM

m

X

i=1



ησ

²

(X

0i

) + γ

²_i

1

τ σ

²

(N

τ

)



.

P r o o f. Take random vectors X

k i.i.d.

∼ X, which are also independent of N

τ

. Consider Y

τ

= (1/τ ) P

Nτ

k=1

X

k

, where Y

τ

= (Y

01

, . . . , Y

0m

), then apply Lemma 3 with h = 1/ √

τ . We have E[T (Y

τ

)]f =

∞

X

l=0

P (N

τ

= l)E

 T  1

τ

l

X

k=1

X

k



f = Ψ

Nτ

(E[T (X/τ )]f ),

E(Y

τ

) =

∞

X

l=0

P (N

τ

= l) 1 τ E

 X

^l

k=1

X

k



= 1

τ E(N

τ

)E(X) = ηγ, E(e

^ωYτ

) ≤ E(e

^pωYτ

) = Ψ

Nτ

(Ψ

^∗

X

(pω/τ )) ≤ d

1

. Furthermore,

(E((Y

0i

− ηγ

_i

)

^2q

))

^1/q

=

 E  1

τ

Nτ

X

k=1

X

ki

− 1

τ N

τ

γ

i

+ 1

τ N

τ

γ

i

− ηγ

_i



2q



1/q

≤ 2

 E  1

τ

Nτ

X

k=1

(X

ki

− γ

_i

)



2q



1/q

+ 2γ

_i²

 E  1

τ N

τ

− η



2q



1/q

=: 2I

1

+ 2I

2

. We observe that I

₁^q

= E  1

τ

Nτ

X

k=1

(X

ki

− γ

i

)



2q



≤  2q

√ τ



2q

e

^−2q

E[e

^(1/

√τ )|Σ^Nτ_k=1(Xki−γi)|

] (by (17))

≤

 2q e √ τ



2q

{E[e

^(1/

√τ )Σ^Nτ_k=1(Xki−γi)

] + E[e

^(1/

√τ )Σ^Nτ_k=1(γi−Xki)

]}

≤

 2q e √ τ



2q

{E[(E(e

^{(1/√τ )(X0i−γi)}

))

^Nτ

] + E[(E(e

^(1/

√τ )(γi−X0i)

))

^Nτ

]}

≤

 2q e √ τ



2q

 E



E

 1 + 1

√ τ (X

0i

− γ

_i

)

+ 1

2τ (X

0i

− γ

_i

)

²

e

^(1/

√τ )|X0i−γi|



Nτ



+ E



E

 1 + 1

√ τ (γ

i

− X

_0i

) + 1

2τ (X

0i

− γ

_i

)

²

e

^(1/

√τ )|X0i−γi|



Nτ



≤ 2

 2q e √ τ



2q

E{(exp[E(X

0i

− γ

_i

)

²

e

^(1/

√τ )|X0i−γi|

])

^{Nτ/(2τ )}

} (1/ √ τ < δ)

≤ 2

 2q e √ τ



2q

E



exp

 2

δ − 1/ √ τ



2

e

⁻²

E(e

^{δ|X0i−γi|}

)



Nτ/(2τ )



≤ 2

 2q e √ τ



2q

E



exp

 2 √ τ e( √

τ δ − 1)



2

e

^δ¯γ

Ψ

_X^∗

(δ)



Nτ/(2τ )



≤ 2

 2q e √ τ



2q

d

3

. So

I

1

≤ 2

^1/q

4q

²

e

²

τ d

^1/q₃

and

I

2

= γ

_i²

 E  1

τ N

τ

− η



2q



1/q

≤ γ

_i²

d

2

/τ.

Therefore by inequality (9) of Lemma 3 for Y = Y

τ

and h = 1/ √ τ , we get

kΨ

_Nτ

[E(T (X/τ ))]f − T (ηγ)f k

≤ M ω

₂

(T f, 1/ √ τ )



2d

1

+ 9

2 mM e

^2ωη¯γ

e

^2(m−1)ω/

√τ

d

^1/p₁

× 2

m

X

i=1



2

^1/q

 4q

²

e

²

τ d

^1/q₃

 + d

2

γ

_i²

τ

 τ



= M ω

2

(T f, 1/ √ τ )



2d

1

+ 9mM e

^2ωη¯γ

e

^2(m−1)ω/

√τ

d

^1/p₁

×



m2

^1/q

 2q e



2

d

^1/q₃

+ d

2 m

X

i=1

γ

_i²



. If ω = 0, we apply inequality (10) of Lemma 3. Note that σ

²

(Y

0i

) = σ

²

 1

τ

Nτ

X

k=1

X

ki



= 1 τ

²

∞

X

l=0

P (N

τ

= l)E  X

^l

k=1

X

ki



2



− η

²

γ

²_i

= 1 τ

²

∞

X

l=0

P (N

τ

= l)(lE(X

_0i²

) + l(l − 1)γ

_i²

) − η

²

γ

_i²

= 1

τ

²

E(N

τ

)E(X

_0i²

) + 1

τ

²

(E(N

_τ²

))γ

_i²

− 1

τ

²

(E(N

τ

))γ

_i²

− η

²

γ

_i²

= 1 τ



ησ

²

(X

0i

) + γ

²_i

1

τ σ

²

(N

τ

)

 . By (10), when h = 1/ √

τ , we obtain (25).

Another generalization of Theorem 1 is presented next.

Theorem 4. Let N = (N

1

, . . . , N

m

) be a Z

^m+

-valued random vector with E(N ) = η = (η

1

, . . . , η

m

). For each i (1 ≤ i ≤ m), let {Y

ki

}

^∞_k=1

be a sequence of i.i.d. real-valued random variables distributed as Y , a fixed random vari- able with E(Y ) = γ. N and Y

ki

are assumed to be independent. Also assume that there exists a δ > 0 such that

Ψ

_N

(Ψ

_Y^∗

(δ)) < ∞.

Then for n > max(pω/δ, 1/δ

²

), (26)

 E

 T



^N1

X

k1=1

1

n Y

k11

, . . . ,

Nm

X

km=1

1 n Y

kmm



n

f − T (γη)f

≤ 2M ω

₂

(T f, 1/ √ n)



e

^{ωγ ¯η}

exp

 2nω

²

e

²

(nδ − ω)

²

Ψ

_N

(Ψ

_Y^∗

(δ))



+ 2

^1/q

9M m

²

q

²

e

²

e

^{3ωγ ¯η}

e

^2(m−1)ω/

√n

× exp

 2npω

²

e

²

(nδ − pω)

²

+ 2ne

^{δγ ¯η}

e

²

q(δ √

n − 1)

²



Ψ

_N

(Ψ

_Y^∗

(δ))



, where p, q > 0, 1/p + 1/q = 1, is an arbitrary conjugate pair. If ω = 0, then (27)

 E

 T



^N1

X

k1=1

1

n Y

k11

, . . . ,

Nm

X

km=1

1 n Y

kmm



n

f − T (γη)f

≤ 2M ω

₂

(T f, 1/ √ n)

 1 + 9

4 mM

m

X

i=1

[η

i

σ

²

(Y ) + σ

²

(N

i

)γ

²

]

 .

P r o o f. In Theorem 1, take X := ( P

N1

k1=1

Y

k11

, . . . , P

Nm

km=1

Y

kmm

) and let X

k

i.i.d.

∼ X. Then E(X) = 

E h X

^N1

k1=1

Y

k11

i

, . . . , E h X

^Nm

km=1

Y

kmm

i

= (EN

1

EY, . . . , EN

m

EY ) = γη.

We observe that Ψ

_X^∗

(δ) = E(e

^δX

)

= E(e

^{δ(ΣN1k1=1Yk11+...+ΣNmkm=1Ykmm)}

)

=

∞

X

l1=0

. . .

∞

X

lm=0

P (N = (l

1

, . . . , l

m

))E(e

^{δΣl1k1=1Yk11+...+δΣlmkm=1Ykmm}

)

=

∞

X

l1=0

. . .

∞

X

lm=0

P (N = (l

1

, . . . , l

m

))E(e

^{δΣl1k1=1Yk11}

) . . . E(e

^{δΣlmkm=1Ykmm}

)

=

∞

X

l1=0

. . .

∞

X

lm=0

P (N = (l

1

, . . . , l

m

))(E(e

^δY

))

^l1

. . . (E(e

^δY

))

^lm

=

∞

X

l1=0

. . .

∞

X

lm=0

P (N = (l

1

, . . . , l

m

))(E(e

^δY

))

^l1+...+lm

= E((E(e

^δY

))

^N1+...+Nm

) = Ψ

_N

(Ψ

_Y^∗

(δ)).

Thus (26) follows from (15).

If ω = 0 we see that σ

²

(X

0i

) = σ

²

 X

^Ni

ki=1

Y

kii



= η

i

σ

²

(Y ) + σ

²

(N

i

)γ

²

,

established similarly to the fact at the end of the proof of Theorem 2. Now (27) follows from (16).

5. Further results: multiplier enlargement formulae. In this sec- tion we modify the formulae obtained in the previous section by the so-called multiplier enlargement method (see [6]) initiated by Hsu–Wang [9, 19] in the 60’s and also used by Shaw [17, 18] in the representation of operator semi- groups. The modified representation formulae have a larger range of appli- cations and when we specify the random vectors (variables) considered, we recover the representation formulae for m-parameter operator semigroups of Shaw [18]. For simplicity we only consider equibounded operator semi- groups, i.e.,

kT (t)k ≤ M for all t ∈ R

^m+

.

Here we only need to give two versions related to Theorems 1 and 4. Others can be similarly obtained.

Theorem 5. Suppose kT (t)k ≤ M for all t ∈ R

^m+

, and α

n

is a se-

quence of positive real numbers with lim inf

n→∞

α

n

> 0. For each n ∈ N

let X(n) be an R

^m+

-valued random vector with E[X(n)] = x/α

n

. Assume

lim sup

_n→∞

α

n

σ

²_i

(X(n)) < ∞, i = 1, . . . , m. Then (28)

 E

 T  α

n

n X(n)



n

f − T (x)f

≤ 2M ω

₂

(T f, (α

n

/n)

^1/2

)

 1 + 9

4 mM α

n m

X

i=1

σ

_i²

(X(n))

 .

P r o o f. For each fixed n, let X

k i.i.d.

∼ X(n), k = 1, . . . , n, and consider Y := 1

n

n

X

k=1

α

n

X

k

. Then

E(Y ) = E  1 n

n

X

k=1

α

n

X

k



= α

n

E(X(n)) = α

n

x/α

n

= x and

E[T (Y )]f = E

 T  1

n

n

X

k=1

α

n

X

k



f =

 E

 T  α

n

n X(n)



n

f.

Furthermore,

σ

_i²

(Y ) = σ

_i²

 1 n

n

X

k=1

α

n

X

k



= α

²_n

n

²

nσ

²_i

(X(n)) = α

²_n

n σ

_i²

(X(n)).

Now take h = (α

n

/n)

^1/2

; then by (10), we get (28).

Theorem 6. Let α

n

be a sequence of positive real numbers satisfying

n→∞

lim α

n

/n = 0 and lim inf

n→∞

α

n

> 0.

For each n ∈ N, let N (n) := (N

1

(n), . . . , N

m

(n)) be a Z

^m+

-valued random vector with E(N (n)) = (1/α

n

)η = (1/α

n

)(η

1

, . . . , η

m

) and

lim sup

n→∞

α

n

σ

_i²

(N (n)) < ∞ for all i = 1, . . . , m.

For each i (1 ≤ i ≤ m), let {Y

ki

(n)}

^∞_k=1

be a sequence of i.i.d. random variables, distributed as Y (n), where Y (n) is a fixed real-valued random variable for each n ∈ N and E(Y (n)) = α

n

γ. Suppose that Y

0i

(n) (i = 1, . . . , m) and N (n) are all independent. Assume also that

lim sup

n→∞

σ

²

(Y (n))/α

²_n

< ∞.

Consider the equibounded operator semigroup {T (t) : t ∈ R

^m+

} with

kT (t)k ≤ M for all t ∈ R

^m+

.

Then (29)

 E

 T



^N1(n)

X

k1=1

1

n Y

k11

(n), . . . ,

Nm(n)

X

km=1

1

n Y

kmm

(n)



n

f − T (γη)f k

≤ 2M ω

₂

(T f, (α

n

/n)

^1/2

)

 1 + 9

4 mM

m

X

i=1

 η

i

α

²_n

σ

²

(Y (n)) + γ

²

α

n

σ

_i²

(N (n))



. P r o o f. We want to apply Lemma 3. Let

X

k i.i.d.

∼ X := 

N1(n)

X

k1=1

Y

k11

(n), . . . ,

Nm(n)

X

km=1

Y

kmm

(n)  and

Y = 1 n

n

X

k=1

X

k

. Then

E(Y ) = E(X) =  E 

N1(n)

X

k1=1

Y

k11

(n) 

, . . . , E 

Nm(n)

X

km=1

Y

kmm

(n) 

=

 1 α

n

η

1

α

n

γ, . . . , 1 α

n

η

m

α

n

γ



= (η

1

γ, . . . , η

m

γ) = γη.

Moreover,

E[T (Y )]f =

 E

 T



^N1(n)

X

k1=1

1

n Y

k11

(n), . . . ,

Nm(n)

X

km=1

1

n Y

kmm

(n)



n

f.

Furthermore,

σ

_i²

(Y ) = σ

_i²

 1 n

n

X

k=1

X

k



= 1 n σ

²



Ni(n)

X

ki=1

Y

kii

(n)



= 1

n [E(N

i

(n))σ

²

(Y

0i

(n)) + σ

²

(N

i

(n))(E(Y

0i

(n)))

²

]

= 1 n

 1 α

n

η

i

σ

²

(Y (n)) + σ

²

(N

i

(n))α

²_n

γ

²

 . Pick h := (α

n

/n)

^1/2

; then by (10) of Lemma 3 we have

 E

 T



^N1(n)

X

k1=1

1

n Y

k11

(n), . . . ,

Nm(n)

X

km=1

1

n Y

kmm

(n)



n

f − T (γη)f

≤ 2M ω

₂

(T f, (α

n

/n)

^1/2

)

×



1 + 9mM 4

n α

n

m

X

i=1

1 n

 1 α

n

η

i

σ

²

(Y

0i

(n)) + σ

²

(N

i

(n))α

²_n

γ

²



≤ 2M ω

₂

(T f, (α

n

/n)

^1/2

)

×

 1 + 9

4 mM

m

X

i=1

 η

i

α

²_n

σ

²

(Y (n)) + γ

²

α

n

σ

²_i

(N (n))



.

6. Applications. In this section we specify the random vectors (vari- ables) and α

n

of Theorems 1–6 to derive some concrete representation for- mulae for (C

0

) m-parameter operator semigroups. We also illustrate how to get the results on multivariate approximation operators from the corre- sponding ones on operator semigroups. Unless otherwise mentioned all (C

0

) m-parameter operator semigroups considered satisfy (5).

Example 1. Take X = (X

01

, . . . , X

0m

) that follows the multi-point dis- tribution with EX = x = (x

1

, . . . , x

m

):

P (X = e

i

) = x

i

(e

i

= (0, . . . , 1, . . . , 0)) and

P (X = 0) = 1 − x, where 0 < x < 1 (x = x

1

+ . . . + x

m

).

Then

Ψ

_X^∗

(δ) = E(e

^δX

) = P (X = 0) + P (X = 1)e

^δ

= 1 − x + xe

^δ

< ∞.

Furthermore, we have

E[T (X/n)] = I +

m

X

i=1

x

i

(T

i

(1/n) − I).

Hence by Theorem 1 there is a constant K = K(ω, M, x, δ, m) such that (30) k(I +

m

X

i=1

x

i

(T

i

(1/n) − I))

ⁿ

f − T (x)f k

≤ Kω

₂

(T f, 1/ √

n) → 0 (n → ∞).

From the above result on operator semigroups we are able to recover the approximation theorem for multivariate Bernstein operators as follows.

Choose

X := BUC(R

^m

)

:= {f : f is a bounded uniformly continuous function from R

^m

into R}

and define

T (t)f (x) := f (x + t) = f (x

1

+ t

1

, . . . , x

m

+ t

m

)

for each f ∈ X and x ∈ R

^m

. Then {T (t) : t ∈ R

^m+

} is a (C

₀

) m-parameter operator semigroup in E (X ).

Now let x = 0, t = (t

1

, . . . , t

m

), 0 < t < 1, 0 < t

i

< 1, i = 1, . . . , m.

Then n

I +

m

X

i=1

t

i

[T

i

(1/n) − I] o

n

f (0)

= X

k∈Z^m+, ¯k≤n

f (k

1

/n, . . . , k

m

/n) n!

k

1

! . . . k

m

!(n − k

1

− . . . − k

m

)!

× t

^k₁¹

. . . t

^k_m^m

(1 − t)

^n−¯k

= B

_n^f

(t

1

, . . . , t

m

),

where B

_n^f

(t

1

, . . . , t

m

) is the m-variate Bernstein operator over a simplex (cf.

[12]). So by (30), we obtain

n→∞

lim B

_n^f

(t

1

, . . . , t

m

) = T (t)f (0) = f (t

1

, . . . , t

m

), uniformly.

R e m a r k. The fact that the approximation theorem for the Bernstein operator can be derived from simple operator semigroup considerations has been observed by many authors (see, e.g., [1, p. 28], [10] and [14]). When considering other representation formulae for m-parameter operator semi- groups in the following examples we may derive other known convergence theorems for multivariate approximation operators, but we avoid to go into details here.

Example 2. Let α

n

be a sequence of positive real numbers with lim inf

n→∞

α

n

> 0 and lim

n→∞

α

n

/n = 0. For each n ∈ N take X(n) = (X

01

(n), . . . , X

0m

(n)) to have a modified multi-point distribution:

P (X(n) = e

i

) = x

i

/α

n

, 1 ≤ i ≤ m,

P (X(n) = 0) = 1 − x/α

n

(0 < x/α

n

< 1 and x

i

> 0).

Then

E[X(n)] = x α

n

(x = (x

1

, . . . , x

m

)) and

σ

_i²

(X(n)) = E(X

_i²

(n)) − (E(X

i

(n)))

²

= x

i

α

n

− x

²_i

α

_n²

.

For an equibounded (C

0

) m-parameter operator semigroup {T (t) : t ∈

R

^m+

} with kT (t)k ≤ M for all t ∈ R

^m+

, we have

E

 T  α

n

n X(n)



= T (0)P (X(n) = 0) +

m

X

i=1

T  α

n

n e

i



P (X(n) = e

i

)

= I +

m

X

i=1

x

i

α

n

 T

i

 α

n

n



− I

 . Thus by Theorem 5, we obtain

(31)

 I +

m

X

i=1

x

i

α

n

 T

i

 α

n

n



− I



n

f − T (x)f

≤ 2M ω

₂

(T f, (α

n

/n)

^1/2

)

 1 + 9

4 mM α

n m

X

i=1

 x

i

α

n

− x

²_i

α

²_n



= 2M ω

2

(T f, (α

n

/n)

^1/2

)

 1 + 9

4 mM

m

X

i=1



x

i

− x

²_i

α

n



→ 0 (n → ∞).

R e m a r k. (30) is the special case of (31) when α

n

≡ 1, but (30) is true for an arbitrary (C

0

) m-parameter operator semigroup.

Inequalities (31) and the following (32)–(34) are Shaw’s formulae [17, 18]

supplied with rates of convergence.

Example 3. Assume α

n

is as in Example 2. For each n, let X(n) :=

(X

01

(n), . . . , X

0m

(n)) follow the negative multi-point distribution:

P (X(n) = (k

1

, . . . , k

m

)) = k k



1 + 1 α

n

x



−1 m

Y

i=1

 x

i

α

n

+ x



ki

for all k = (k

1

, . . . , k

m

) ∈ Z

^m+

= {(n

1

, . . . , n

m

) : n

i

∈ Z

+

, 1 ≤ i ≤ m}, where x = (x

1

, . . . , x

m

) ∈ R

^m+

is fixed and

n k



= n(n − 1) . . . (n − k + 1) k

1

! . . . k

m

! . Then

P (X

0i

(n) = k

i

) =

 1 + 1

α

n

x

i



−1

 x

i

α

n

+ x

i



ki

(see [7, p. 165, (8.4)]), E(X

0i

(n)) =

∞

X

ki=0

k

i

 1 + 1

α

n

x

i



−1

 x

i

α

n

+ x

i



ki

= x

i

α

n

and

σ

_i²

(X(n)) = x

²_i

α

²_n

+ x

i

α

n

.

For an equibounded (C

0

) m-parameter operator semigroup {T (t) : t ∈ R

^m+

} with kT (t)k ≤ M for all t ∈ R

^m+

, we have

E

 T  α

_n

n X(n)



=

∞

X

k1=0

. . .

∞

X

km=0

T  α

n

n (k

1

, . . . , k

m

) k k



1 + 1 α

n

x



−1 m

Y

i=1

 x

i

α

n

+ x



ki

=

∞

X

k1=0

. . .

∞

X

km=0

k k



1 + 1 α

n

x



−1 m

Y

i=1

 x

i

T

i

(α

n

/n) α

n

+ x



ki

=

 1 + 1

α

n

x



−1



I − x

1

T

1

(α

n

/n) + . . . + x

m

T

m

(α

n

/n) α

n

+ x



−1

=

 I + I

α

n

x − x

1

α

n

T

1

(α

n

/n) − . . . − x

m

α

n

T

m

(α

n

/n)



−1

=

 I −

m

X

i=1

x

i

α

n

(T

i

(α

n

/n) − I)



−1

. By Theorem 5 we obtain

(32)

 I −

m

X

i=1

x

i

α

n

(T

i

(α

n

/n) − I)



−n

f − T (x)f

≤ 2M ω

₂

(T f, (α

n

/n)

^1/2

)

 1 + 9

4 mM α

n m

X

i=1

 x

_i

α

n

+ x

²_i

α

²_n



= 2M ω

2

(T f, (α

n

/n)

^1/2

)

 1 + 9

4 mM

m

X

i=1



x

i

+ x

²_i

α

n



→ 0 (n → ∞).

Example 4. In Theorem 6 take N (n) that follows the multi-point dis- tribution:

P (N (n) = e

i

) = x

i

/α

n

, 1 ≤ i ≤ m,

P (N (n) = 0) = 1 − x/α

n

, where x = (x

1

, . . . , x

m

) ∈ R

^m+

, is fixed.

Here α

n

is as in Example 2. Let Y

0i

(n), 1 ≤ i ≤ m, be exponentially dis- tributed with density (1/α

n

)e

^−v/αn

, v ∈ R

+

. Then

E(N (n)) = 1 α

n

x =

 1 α

n

x

1

, . . . , 1 α

n

x

m



, E(Y

0i

(n)) = α

n

, σ

²

(N

i

(n)) = x

i

α

n

− x

²_i

α

²_n

and σ

²

(Y

0i

(n)) = α

²_n

.