1. Introduction. C. R. May in [3], M. E. Ismail and C. R. May in [2] studied family of exponential operators. They considered the integral S

Seria I: PRACE MATEMATYCZNE XLV (1) (2005), 57-70

Agnieszka Tyliba, Eugeniusz Wachnicki

On some class of exponential type operators

Abstract. Starting from a differential equation _∂t^∂W (λ, t, u) = ^λ(u−t)_p(t) W (λ, t, u) − βW (λ, t, u) for the kernel of an operator Sλ(f, t) = RB

AW (λ, t, u)f (u)du with the normalization condition RB

AW (λ, t, u)du = 1 we prove some properties which are similar to properties proved by Ismail and May for the exponential operators. In particular, we show that all these operators are approximation operators. Moreover, a method of determining Sλfor a given function p is introduced.

2000 Mathematics Subject Classification: 41A36, 41A25, 35K15.

Key words and phrases: exponential operators, Voronovskaya type theorem, rate of convergence, limit value problems.

1. Introduction. C. R. May in [3], M. E. Ismail and C. R. May in [2] studied family of exponential operators. They considered the integral S

defined by (1) S

(f, t) =

Z

W (λ, t, u)f (u) du, −∞ ≤ A < B ≤ +∞

under the following assumptions: a kernel W is a positive function, satisfying the following homogenous partial differential equation

(2) ∂

∂t W (λ, t, u) = λ

p(t) W (λ, t, u)(u − t), λ ∈ R, u, t ∈ (A, B), p is analytic and positive for t ∈ (A, B) and

(3) S

(e

, t) = 1, t ∈ (A, B),

where e

(u) = 1 for u ∈ (A, B). They presented some well-known operators satis-

fying (2). For example, the Bernstein polynominals and operators of Szasz, Post-

Widder, Gauss-Weierstrass and Baskakov. The above operators satisfy the condition

S

(e

, t) = e

(t), where e

(u) = u for u ∈ (A, B).

Our purpose is to extend the results of May and Ismail to a family of operators, in which there are operators S

such that S

(e

, t) 6= e

(t). We investigate a similar family of operators. Instead of the equation (2), however, we consider the following

(4) ∂

∂t W (λ, t, u) = λ

p(t) W (λ, t, u)(u − t) − βW (λ, t, u),

where β is a non-negative real number, λ ∈ R, u, t ∈ (A, B), p is analitic and positive for t ∈ (A, B). For these operators we obtain similar results as May and Ismail for exponential operators. We state some estimates of the rate of convergence of S

. We also prove the Voronovskaya type theorem for these operators. In section 4 we give examples of operators satisfying (4) and on the strength of one of these we define the function U which is a solution of some limit problem.

Let C(A, B) be a set of all continuous, real-valued functions on (A, B). We define the space (C

, || ||

) as follows

C

= n

f ∈ C(A, B) : ∃ M > 0 ∀ t ∈ (A, B) |f (t)| ≤ M e

o

and ||f ||

= sup n

e

|f (t)| : t ∈ (A, B) o .

2. Preliminaries. In this section we give some properties of the above opera- tors, which we use in the proofs of the main theorems. The proofs of these lemmas follow by the same method as the proofs of the appropriate properties of exponential operators in [2].

Lemma 2.1 For each λ ∈ R and t ∈ (A, B) we have S

(e

, t) = e

(t) + βp(t)

λ and

S

(e

, t) = e

(t) + p(t) + 2βtp(t)

λ + βp(t)p

(t) + β

p

(t)

λ

,

where e

(u) = u

for u ∈ (A, B).

Lemma 2.2 For every n ∈ N

we get S

(e

, t) = e

(t) +

n

X

k=1

ϕ

(t)

λ

, t ∈ (A, B), where e

(u) = u

for u ∈ (A, B), the function ϕ

is defined by

ϕ

(t) = tϕ

(t) + p(t)



βϕ

(t) + ∂

∂t ϕ

(t)



for 1 ≤ k ≤ n and we put

ϕ

= e

, ϕ

≡ 0.

Let us denote

(5) A

(λ, t) = λ

S

((u − t)

, t).

Lemma 2.3

A

(λ, t) = 1, A

(λ, t) = βp(t) and

(6) A

(λ, t) = p(t) d

dt A

(λ, t) + βp(t)A

(λ, t) + mλp(t)A

(λ, t) for m ≥ 1.

Lemma 2.4 For all m ∈ N the operators A

and A

are polynomials in λ, whose the degree is m. Moreover, the coefficient of the term λ

is

c

p

(t) for A

and

c

p

(t) + c

p

(t)p

(t) for A

, where c

, c

and c

are constants.

Lemma 2.5 If

q(t) = Z

c

dv

p(v) , c ∈ (A, B) and

g(q(t)) = q(g(t)) = t, then the following equality holds

∞

X

m=0

S

((u − t)

, t) x

m! = exp

(

−xt + λ

Z

(

)

t

vdv p(v) + β 

g  q(t) + x

λ

 − t  )

.

3. Main results.

Theorem 3.1 Let N > 0 and a, b be such that A < a < b < B, then for a sufficiently large λ we have

||S

(e

, t)||

< ∞,

where C[a, b] is the set of all continuous, real-valued functions on [a, b] with the

suppremum norm.

Proof By the definition of S

we have S

(e

, t) = e

Z

W (λ, t, u)e

du

= e

∞

X

m=0

N

m! S

((u − t)

, t).

From Lemma 2.5 we obtain S

(e

, t) = exp

( λ

Z

(

)

t

θ p(θ) dθ

) exp

 β

 g



q(t) + N λ



− t



.

Letting λ −→ ∞ and the application of de L’Hospital’s formula implies that

(7) lim

λ→+∞

S

(e

, t) = e

uniformly in C[a, b].

Hence for a sufficiently large λ

||S

(e

, t)||

< ∞.

Similarly,

||S

(e

, t)||

< ∞.

S

is a positive and linear operator, so we have

||S

(e

, t)||

≤ ||S

(e

, t)||

+ ||S

(e

, t)||

< ∞, because e

< e

+ e

. This completes the proof.

Theorem 3.2 Suppose that m, η are positive numbers, N is a non-negative number and [a, b] ⊂ (A, B). Then

(8) Z

|u−t|≥η

W (λ, t, u)e

du = O(λ

) uniformly on [a, b] as λ → +∞.

Proof By Cauchy-Schwarz’s inequality, (7) and Lemma 2.4 we have Z

|u−t|≥η

W (λ, t, u)e

du

≤ Z

|u−t|≥η

W (λ, t, u) du Z

|u−t|≥η

W (λ, t, u)e

du

!

≤ η

λ

A

(λ, t)

S

(e

, t)

= O(λ

) for λ → +∞,

and (8) is proved.

Theorem 3.3 Let f ∈ C

. Then

(9) lim

λ→∞

S

(f, t) = f (t), uniformly on every [a, b] ⊂ (A, B).

Proof We have

|S

(f, t) − f (t)| ≤ Z

|u−t|<δ

|f (u) − f (t)|W (λ, t, u) du +

Z

|u−t|≥δ

|f (u) − f (t)|W (λ, t, u) du

= I

+ I

for every δ > 0. Let  > 0. By the continuity of f in [a, b] there exists δ > 0 such that |f (u) − f (t)| <  for |u − t| < δ. Hence

I

≤  2 Z

|u−t|<δ

W (λ, t, u) du ≤  2

Z

W (λ, t, u) du =  2 . On the other hand, from (8) for a sufficiently large λ

I

≤ 2||f ||

Z

|u−t|≥δ

W (λ, t, u)e

du ≤ M ||f ||

1 λ < 

2 ,

which proves the theorem.

Theorem 3.4 Let ξ ∈ (A, B). If f ∈ C

and f

(ξ) exists, then

(10) lim

λ→∞

λ(S

(f, ξ) − f (ξ)) = βp(ξ)f

(ξ) + 1

2 p(ξ)f

(ξ).

Proof By Taylor’s formula, (3) and Lemma 2.1

S

(f, ξ) = f (ξ) + f

(ξ)S

(e

, ξ) − ξf

(ξ) + 1

2 f

(ξ)S

(e

, ξ)

−ξf

(ξ)S

(e

, ξ) + 1 2 ξ

f

(ξ) +

Z

W (λ, ξ, u)(u, ξ)(u − ξ)

du.

Lemma 2.1 now implies

S

(f, ξ) = f (ξ) + βp(ξ)f

(ξ)

λ + 1

2

p(ξ)f

(ξ)

λ + 1

2

βp(ξ)p

(ξ)f

(ξ) λ

+ 1

2

β

p

(ξ)f

(ξ) λ

+ I.

Hence

λ (S

(f, ξ) − f (ξ)) = βp(ξ)f

(ξ) + 1

2 p(ξ)f

(ξ) + 1 2

βp(ξ)p

(ξ)f

(ξ) λ + 1

2

β

p

(ξ)f

(ξ)

λ + Iλ.

On the other hand,

I ≤ Z

A

W (λ, t, u)

(u, ξ) du

!

Z

A

W (λ, t, u)(u − ξ)

du

!

. Applying Theorem 3.3 we see that

lim

λ→∞

Z

W (λ, t, u)

(u, ξ) du = 0.

Therefore

Z

W (λ, t, u)(u − ξ)

du

!

=  A

(λ, ξ) λ



= O(λ

)

by Lemma 2.4. Thus I = o(λ

), and (10) is proved.

Before the next theorem we recall some notations. Let < a, b > denote the interval [a, b] or (a, b) or (a, b] etc. If f :< a, b >−→ R, then

ω(f ; h, a, b) = sup {|f (t) − f (t + δ)|; t, t + δ ∈< a, b >, |δ| ≤ h}

and

ω

(f ; h, a, b)

= sup {|f (t) − 2f (t + δ) + f (t + 2δ)|; t, t + 2δ ∈< a, b >, |δ| ≤ h}

for h > 0.

Theorem 3.5 Suppose that f ∈ C

and A < a < a

< b

< b < B. Then for every m > 0 there exists a constant K

such that

||S

(f, ·) − f ||

≤ K

h

λ

ω(f ; λ

, a, b) + ω

(f ; λ

, a, b) + λ

||f ||

i . Proof Let δ > 0 and δ ≤

min{a

− a, b − b

}. Define

g

(x) = 1 2δ

Z

−^δ₂

Z

−^δ₂

[f (x + u + v) + f (x − u − v)] du dv.

Since S

is linear we get

kS

(f, ·) − f k

≤ kS

(f − g

, ·)k

+ kS

(g

, ·) − g

k

+ kg

− f k

= I

+ I

+ I

.

Let η =

min{a

− a, b − b

}, then for x ∈ [a

− η, b

+ η] we get

|f (x) − g

(x)| =     

f (x) − 1 2δ

Z

−^δ₂

Z

−^δ₂

[f (x + u + v) + f (x − u − v)] du dv     

=     

1 2δ

Z

−^δ₂

Z

−^δ₂

[f (x + u + v) − 2f (x) + f (x − u − v)] du dv

.

As |u + v| ≤ δ, the above equality shows that

(11) |f (x) − g

(x)| ≤ 1

2 ω

(f ; δ, a, b).

Thus

(12) I

≤ 1

2 ω

(f ; δ, a, b).

The estimation of I

follows from (11) and (8). Indeed, for t ∈ [a

, b

] we have

|S

(f − g

, t)| ≤ Z

|u−t|<η

W (λ, t, u)|f (u) − g

(u)| du +

Z

|u−t|≥η

W (λ, t, u)|f (u) − g

(u)| du

= L

+ L

.

Since |u − t| < η and η =

min{a

− a, b − b

}, (11) yields

|f (u) − g

(u)| ≤ 1

2 ω

(f ; δ, a, b).

Hence

L

≤      1

2 ω

(f ; δ, a, b) Z

|u−t|<η

W (λ, t, u) du     

≤ 1

2 ω

(f ; δ, a, b).

Moreover,

L

=      Z

|u−t|≥η

W (λ, t, u)e

|(f − g

)(u)|e

du      .

As f − g

∈ C

we have |(f − g

)(u)|e

≤ 4kf k

. We conclude from (8) that L

≤ 4M λ

kf k

,

and finally that

(13) I

≤ M

(ω

(f ; δ, a, b) + λ

kf k

).

To estimate I

we first compute g

(x) = 1

2δ

Z

−^δ₂

[f (x − u − δ

2 ) − f (x − u + δ

2 )] − [f (x + u − δ

2 ) − f (x + u + δ 2 )] du.

Hence for x ∈ [a

− η, b

+ η] we have

(14) kg

k

≤ δ

ω(f ; δ, a, b).

Moreover,

g

(x) = δ

[f (x − δ) − 2f (x) + f (x + δ)]

and

(15) kg

k

≤ δ

ω

(f ; δ, a, b).

Condition (3) now shows that

|S

(g

, t) − g

(t)| =     

Z

W (λ, t, u)(g

(u) − g

(t)) du     

, t ∈ [a, b].

By the above and by Taylor’s formula we have

|S

(g

, t) − g

(t)| ≤     

g

(t) Z

A

W (λ, t, u)(u − t) du      +

     Z

|u−t|<η

W (λ, t, u)g

(ξ)(u − t)

du      +

     Z

|u−t|≥η

W (λ, t, u)g

(ξ)(u − t)

du     

= J

+ J

+ J

.

From Lemma 2.4, (5) and (14) for t ∈ [a

− η, b

+ η] we obtain J

= |g

(t)λ

A

(λ, t)| ≤ M λ

δ

ω(f ; δ, a, b).

On the other hand, (5), Lemma 2.4 and (15) show that

J

≤ δ

ω

(f ; δ, a, b)|λ

A

(λ, t)| ≤ Kλ

δ

ω

(f ; δ, a, b), t ∈ [a

, b

].

It is clear that g

∈ C

and kg

k

≤ 4kf k

. From this, Cauchy-Schwarz’s inequality and Lemma 2.4 we deduce that

J

≤ 4kf k

Z

|u−t|≥η

W (λ, t, u)e

du

!

λ

A

(λ, t)

≤ Lλ

kf k

, and finally that

(16) I

≤ M

[δ

λ

ω(f ; δ, a, b) + δ

λ

ω

(f ; δ, a, b) + λ

kf k

].

Put δ = λ

into (12), (13) and (16). This ends the proof.

Let 0 < α ≤ 1 and A ≤ a < b ≤ B. The Lipschitz class Lip(α, a, b) is defined by

Lip(α; a, b) = {f : ω(f ; h, a, b) ≤ M h

}.

From Theorem 3.5 we have

Corollary 3.6 If A < a < a

< b

< b < B and f ∈ Lip(α; a, b), then kS

(f, ·) − f k

≤ M λ

.

Theorem 3.7 Let f ∈ C

, 0 < α ≤ 1. If kS

(f, ·) − f k

≤ M λ

, then

f ∈ Lip(α, A, B).

We divide the proof into a sequence of the following lemmas.

Lemma 3.8 Let f ∈ C

, 0 < α ≤ 1. If kS

(f, ·) − f k

≤ M λ

, then there exists M

> 0 such that

ω(f ; h, A, B) ≤ M

[λ

+ hλω(f ; λ

, A, B)]

for 0 < h ≤ 1 and λ > 1.

Proof Let x, y ∈ (A, B) be such that |y − x| ≤ h. Then

|f (y) − f (x)| ≤ |f (y) − S

(f, y)| + |S

(f, y) − S

(f, x)| + |S

(f, x) − f (x)|

≤ 2M λ

+ P.

It remains to estimate the term P . Let us first examine |S

(f, t)|

|S

(f, t)| =     

Z

∂

∂t W (λ, t, u)f (u) du     

=     

λ p(t)

Z

W (λ, t, u)(u − t)f (u) du − β Z

A

W (λ, t, u)f (u) du      . Since

λ p(t)

Z

W (λ, t, u)(u − t)f (t) du = f (t)

p(t) A

(λ, t) = βf (t), we have

|S

(f, t)| ≤     

λ p(t)

Z

W (λ, t, u)(u − t)[f (u) − f (t)] du     

+ |β[f (t) − S

(f, t)]|

≤ λ

p(t) Z

A

W (λ, t, u)|u − t|ω(f ; λ|u − t|λ

, A, B) du + βM λ

≤ λM

Z

A

W (λ, t, u)|u − t|[λ|u − t| + 1]ω(f ; λ

, A, B) du + βM λ

≤ λM

ω(f ; λ

, A, B)

"

A

(λ, t)

λ +

Z

W (λ, t, u)|u − t| du

#

+ βM λ

.

From Cauchy-Schwarz’s inequality and Lemma 2.4 we conclude that

|S

(f, t)| ≤ M

λ

+ λω(f ; λ

, A, B) . We are now in a position to estimate P . We have

P ≤

Z

|S

(f, t)|dt

≤ M

λ

+ hλω(f ; λ

, A, B) ,

and the lemma follows.

Lemma 3.9 Let f ∈ C

and 0 < α ≤ 1. If

ω(f ; h, A, B) ≤ M

[λ

+ hλω(f ; λ

, A, B)]

for 0 < h ≤ 1 and λ > 1, then f ∈ Lip(α, A, B).

Proof It is sufficient to show that ω(f ; h, A, B) ≤ M

h

for 0 < h < 1, where M

is a positive constant. Let K > 1 be such that 2M

< K

. Choose M

= max{ω(f, 1, A, B); 2M

K

}. Define h

= K

, n = 1, 2, . . ..

By induction it is easy to check that for every positive integer n (17) ω(f ; h

, A, B) ≤ M

h

.

On the other hand, for every 0 < h ≤ 1 exists an integer n > 0 such that h

< h ≤ h

. From this

ω(f ; h, A, B) ≤ ω(f ; h

, A, B) ≤ M

h

= M

K

h

≤ M

h

,

and the lemma is proved.

Theorem 3.10 A kernel W can be obtained by the partial differential equation (4) and the condition (3).

Proof Let W satisfy (4) and (3). We define ξ as follows ξ(λ, t, u) = exp



−λ Z

c

u − θ p(θ) dθ + βt



W (λ, t, u).

Then ∂ξ(λ, t, u)

∂t = 0,

hence ξ(λ, t, u) is depending only on λ and u. On the other hand,

(18) W (λ, t, u) = exp

 λ

Z

u − θ p(θ) dθ − βt

 C(λ, u).

From (3) we have exp

 λ

Z

θ

p(θ) dθ + βt



= Z

A

exp {λuq(t)} C(λ, u)du and

(19) exp

( λ

Z

θ

p(θ) dθ + βg(t) )

= Z

A

exp {λut} C(λ, u)du.

This and (18) give W .

4. Examples.

Example 4.1 Let p(t) ≡ 1, c = 0, A = −∞, B = +∞. Then q(t) = g(t) = t. From (19) we have

e

= Z

−∞

e

C(λ, u)du.

Hence

e

(

)

= e

Z

−∞

e

C(λ, u)du.

Put z = t +

, then we obtain e

= e

Z

−∞

e

e

C(λ, u)du.

From this and [1] we conclude that

e

C(λ, u) = r λ

2π e

. Therefore

C(λ, u) = r λ

2π e

. Applying (18) we get

W (λ, t, u) = r λ

2π e

. Combining this with (1) we obtain

S

(f, t) = r λ

2π Z

−∞

e

f (u)du

= r λ

2π Z

−∞

e

λ

(

)

2

f (u)du.

Now consider the function

(20) U (x, t) :=

Z

−∞

K(u, t, x)f (u)du, where

K(u, t, x) = 1 2 √

πt e

.

Theorem 4.2 If f ∈ C

, then the function U given by (20) belongs to C

[Ω], where Ω = {(x, t) : x ∈ R, t > 0}. Moreover, U is a solution of the equation

∂

U

∂x

+ 2β ∂U

∂x = ∂U

∂t

in Ω and lim

U (x, t) = f (x).

Proof From Theorem 3.3 we conclude that lim

U (x, t) = f (x).

Let t

, T

, x

, X

∈ R be such that 0 < t

< T

and x

< X

. Consider the set Ω(t

, T

, x

, X

) = {(x, t) : x

≤ x ≤ X

, t

≤ t ≤ T

}.

By induction we deduce that the integral (21)

Z

−∞

∂

∂t

∂x

K(u, t, x)f (u) du is a linear combination of integrals of the form:

(22)

Z

−∞

β

(u − x)

t

e

f (u) du, where q, r, s ∈ N.

Now we prove that the integral (22) is uniformly convergent in Ω(t

, T

, x

, X

).

Note that for any p, k, l ∈ N from the definition of the norm in C

we have Z

−∞

β

(u − x)

t

e

f (u)    

du

≤ β

kf k

Z

−∞

(u − x)

t

e

du.

On the other hand, for (x, t) ∈ Ω(t

, T

, x

, X

) we have β

kf k

Z

−∞

(u − x)

t

e

du

≤ M β

kf k

Z

−∞

(u − x)

e

du.

Put z = u − x, then Z

−∞

|u − x|

e

du

= e

Z

−∞

|z|

e

dz

= e

Z

−∞

|z|

e

dz

= e

Z

−∞

|z|

e

dz.

Hence for v = z − 2t(β + N ) we get Z

−∞

|u − x|

e

du

= e

Z

−∞

|v + 2t(β + N )|

e

dv

≤ M

Z

−∞

|v + α|

e

dv,

where M

, α, γ are positive constants depending only on the set Ω(x

, X

, t

, T

), β and N . This implies that

Z

−∞

β

(u − x)

t

e

f (u) du    

≤ M

Z

−∞

|v + α|

e

dv, where M

is a positive constant. Observe that the integral

Z

−∞

|v + α|

e

dv

is convergent, hence the integral (22) is uniformly convergent on Ω(t

, T

, x

, X

).

Thus

(23) ∂

∂t

∂x

Z

−∞

K(u, t, x)f (u) du = Z

−∞

∂

∂t

∂x

K(u, t, x)f (u) du.

Consequently, the function U is of the class C

in Ω. It is easy to check that

∂

K(u, t, x)

∂x

+ 2β ∂K(u, t, x)

∂x = ∂K(u, t, x)

∂t .

This completes the proof.

Example 4.3 For p(t) = t, c = 1, A = −∞, B = +∞ we have q(t) = ln t, g(t) = e

. From (19) we get

e

= Z

−∞

e

C(λ, u) du.

Hence

e

= e

Z

−∞

e

C(λ, u) du.

From this and [1] we obtain e

C(λ, u) =

∞

X

k=0

(λ + β)

k! δ(k − λu), where δ is Dirac distribution. Therefore

C(λ, u) = e

∞

X

k=0

(λ + β)

k! δ(k − λu).

Applying (18) we have

W (λ, t, u) = t

e

∞

X

k=0

(λ + β)

k! δ(k − λu).

Combining this with (1) we get S

(f, t) =

Z

−∞

t

e

∞

X

k=0

(λ + β)

k! δ(k − λu)f (u) du

= e

∞

X

k=0

(λ + β)

t

k! f  k

λ



.

Acknowledgement. The authors is thankful to the referee for giving useful comments.

References

[1] J. Antoniewicz, Tables of functions for engineers (in Polish), PWN Warszawa 1980.

[2] M. E. Ismail i C. R. May, On a Family of Approximation Operators, J. Math. Anal. 63 (1978), 446-462.

[3] C. R. May, Saturation and Inverse theorems for combinations of a class of exponential type operators, Canad. J. Math. 28 (1976), 1224-1250.

Agnieszka Tyliba, Eugeniusz Wachnicki

Institute of Mathematics, Pedagogical University ul. Podchora¸ ˙zych 2, 30-084 Krak´ow, Poland E-mail: tyliba@ap.krakow.pl, euwachni@ap.krakow.pl

(Received: 22.12.2003; revised: 17.10.2004)