Bulletin T.CXXIX de l’Académie serbe des sciences et des arts - 2004 Classe des sciences mathématiques et naturelles sciences mathématiques, N\underline{o} 29

QUASIASYMPTOTIC EXPANSION OF DISTRIBUTIONS WITH APPLICATIONS TO THE STIELTJES TRANSFORM

S. B. GAIKWAD, M. S. CHAUDHARI

(Presented at the 6th Meeting, held on September 26, 2003)

A b s t r a c t. Following the approach of Drozinov, Vladimirov and Zavialov we investigate the quasiasymptotic expansion of distributions and give Abelian type results for the ordinary asymptotic behaviour of the distributional modified Stieltjes transform of a distribution with appropriate quasiasymptotic expansion

AMS Mathematics Subject Classification (2000): 46F12.

Key Words: Modified Stieltjes transform of distributions, quasiasymptotic behaviour of distributions.

1. Introduction

In the papers [9, 10] authors followed the definition of the distributional Stieltjes transform given in [7] which enabled them to use the strong theory of the space of tempered distributions S′. In fact, they generalized slightly the definition of Lavoine and Misra. Using the notion of the quasiasymptotic behaviour of distributions from S′_{+} = {f ∈ S′, suppf ⊂ [0, ∞)}, introduced by Zavialov in [15], they obtained more general results than in [6, 7, 2] for the asymptotic behaviour of the distributional Stieltjes transform at ∞ and 0^{+}.

McClure and Wong [3, 14] studied the asymptotic expansion of the generalized Stieltjes transform of some classes of locally integrable functions characterized by their expansions at ∞ and 0^{+}.

Our approach to the asymptotic expansion of the distributional modified Stieltjes transform which we study in this paper is quite different from the approach given in [3, 14].

In the first part of the paper we give the definition of the quasiasymptotic expansion at ∞ of a distribution from S′_{+} given in [4, p.385]. Also in [4] is given the definition of the qiasi-asymptotic expansion at 0^{+} of an element from S′_{+}. In this paper we give the definition of space M′(r), Stieltjes transformation, Modified Stieltjes transformation T_{r+1}, and Generalized Modified Stieltjes transformation \overline{T_{r+1}}. This enables us to obtain, in the second part of the paper, the asymptotic expansion at ∞ and at 0^{+} of the modified Stieltjes transforms of appropriate distributions from S′_{+}.

Domains in [14] and in this paper on which the modified Stieltjes transform is defined do not contain each other.

Notation: As usually R, C, N are the spaces of real, complex and natural numbers; N_{0} = N ∪ {0}, S′_{+} is the space of tempered distributions with supports in the [0, ∞). The space of rapidly decreasing functions is denoted by S and by S_{m}, m ∈ N_{0} its closure with the norm

γ_{m}(φ) = sup{(1 + |t|^{2})^{m/2}|φ^{(i)}(t)|, t ∈ ℝ^{n}, |i| ≤ m},

S′ denotes the space of all distributions of slow growth.

A positive continuous function L defined on (0, ∞) is called slowly varying at ∞ (0^{+}) if for every k > 0

lim_{t→∞} \frac{L(kt)}{L(t)} = 1, ( lim_{t→0^{+}} \frac{L(kt)}{L(t)} = 1).

We denote by ∑_{∞}(∑_{0^{+}}) the set of all slowly varying functions at ∞(0^{+}). For

the properties of slowly varying functions we refer the reader to [11].

If L is a slowly varying function at ∞(0^{+}), then for every ε > 0 there is A_{ε} > 0 such that x^{−ε} < L(x) < x^{ε} (x^{−ε} > L(x) > x^{ε}) if x > A_{ε} (0 < x < A_{ε}).

This property of L and corresponding properties of S_{m} ([12, p. 93]) imply the following assertion: Let G ∈ L_{loc}^{1}, suppG ⊂ [0, ∞), α > −1, and G(x) ∼ x^{α}L(x) as x → ∞ (x → 0^{+}). Then G(kx)/(k^{α}L(k)) → x_{+}^{α}, k → ∞, in S′_{+} for α + 1 > 0. Recall, for α > −1, x_{+}^{α} = H(x)x^{α}; H is Heaviside’s function. (The symbol ∼ is related to the ordinary asymptotic behaviour).

The following scale of distributions from S′_{+} has been used in investigations of the quasiasymptotic behaviour of distributions.

f_{α+1} = \{ \begin{matrix}\frac{Ht^{α}}{Γ(α + 1)}, & α > −1 \\ D^{n}f_{α+n+1}, & α ≤ −1, α + n > −1, \end{matrix}  (1.1)

where D is the distributional derivative.

2. Definitions

2.1. Definition of q.a.e.

Definition 1. The quasiasymptotic behaviour of distribution (q.a.b) at infinity. If T be a distribution in S′_{+} such that the distributional limit

lim_{k→∞} \frac{T (kx)}{ρ(k)} = γ(x)

exists in S′ (γ ≠ 0), then T is called to have the quasiasymptotic behaviour at infinity related to the regularly varying function ρ(k) = k^{α}L(k) with the limit γ; we write this as T ∼^{q} γ in S′ as x → ∞.

Here ρ is regularly varying at infinity and the limit γ ∈ S′_{+}, is of the form γ(x) = Cf_{α+1}(x).

We shall repeat in this section some well known facts about the quasiasymptotic behaviour from [13].

Let f ∈ S′_{+}. It is said that f has the q.a.b. at ∞(0^{+}) with the limit g ≠ 0 with respect k^{α}L(k), L ∈ Σ_{∞}((1/k)^{α}L(1/k), L ∈ Σ_{0}), α ∈ R, if

lim_{k→∞} ⟨\frac{f(kt)}{k^{α}L(k)}, ϕ(t)⟩ = ⟨g(t), ϕ(t)⟩, ϕ ∈ S

( lim_{k→∞} ⟨\frac{f(t/k)}{(1/k^{α})L(1/k)}, ϕ(t)⟩ = ⟨g(t), ϕ(t)⟩, ϕ ∈ S).

Let us notice that in [10] is reformulated the definition of the q.a.b. at 0^{+} from [13].

We need in the paper the following Structural theorem (for the q.a.b. at ∞, see [13] and for the q.a.b. at 0^{+}, see [10]).

Structural theorem Let f ∼^{q} g at ∞ (0^{+}) with respect to k^{α}L(k) ((1/k)^{α} L(1/k)). Then there exist F ∈ L_{loc}^{1}, suppF ⊂ [0, ∞), C ≠ 0 and m ∈ N_{0}, m + α > −1, such that

f = D^{m}F, F (k) ∼ Ck^{m+α}L(k), k → ∞
(F (1/k) ∼ C(1/k)^{m+α}L(1/k), k → ∞).  (1.2)

We remark that the q.a.b. at 0^{+} is a local property while the q.a.b. at ∞ is a global property of an f ∈ S′_{+}.

For the quasiasymptotic Expansion of Distributions we extend slightly the definitions of the closed and open quasiasymptotic expansion, in short the q.a.e. at ∞, given in [4] and using the same idea we give the definition of the q.a.e. at 0^{+}.

Definition 2. Let α ∈ R and L ∈ ∑_{∞} (L ∈ ∑_{0}). We put

(f_{L})_{α+1} = \{ \begin{matrix}H(t)L(t) · t^{α}/Γ(α + 1), & α > −1 \\ D^{n}(f_{L})_{α+n+1}, α < −1, & α + n > −1, \end{matrix}  (1.3)

where n is the smallest natural number such that α + n > −1. Obviously, (f_{L})_{α+1} ∼^{q} f_{α+1} at ∞ (0^{+}) with respect to k^{α}L(k) ((1/k)^{α}L(1/k)).

Definition 3. An f ∈ S′_{+} has the closed q.a.e. at ∞ (0^{+}) of order (α, L) ∈ R × ∑_{∞} ((α, L) ∈ R × ∑_{0}) and of length ℓ, 0 ≤ ℓ < ∞, with respect

to k^{α−ℓ}L_{0}(k) ((1/k)^{α+ℓ} × L_{0}(1/k)) if f has the q.a.b. it ∞ (0^{+}) with respect

to k^{α}L(k) and if there exist α_{i} ∈ R, L_{i} ∈ ∑_{∞} (L_{i} ∈ ∑_{0^{+}}), c_{i} ∈ C, i =

1, 2, ..., N, α_{1} ≥ α_{2} ≥ α_{N} (α_{1} ≤ α_{2} ≤ ... ≤ α_{N}) and that f is of the form

f(t) = ∑_{i=1}^{N} c_{i}(f_{L_{i}})_{α_{i}+1}(t) + h(t)  (1.4)

such that

\{ \begin{matrix}lim_{x→∞} ⟨\frac{h(kt)}{k^{α−ℓ}L_{0}(k)}, ϕ(t)⟩ = 0, ϕ ∈ S \\ ( lim_{x→∞} ⟨\frac{h(k/t)}{(1/k)^{α+ℓ}L_{0}(1/k)}, ϕ(t)⟩ = 0, ϕ ∈ S). \end{matrix}  (1.5)

Obviously, we can (and we shall) assume that c_{i} ≠ 0, i = 1, ..., N, and that α_{N} ≥ α − ℓ (α_{N} ≤ α + ℓ). Since the sum of the two slowly varying functions is the slowly varying one, we can and we shall always assume that in the representation (1.4) α_{1} > α_{2} > ... > α_{N}(α_{1} < α_{2} < ... < α_{N}). Namely (f_{L_{j}})_{β+1} + (f_{L_{k}})_{β+1} = (f_{L_{j}+L_{k}})_{β+1}. (f_{L_{j}})_{β_{1}+1} and (f_{L_{k}})_{β_{2}+1} have the same q.a.b. at ∞(0^{+}) iff β_{1} = β_{2} and L_{j} ∼ L_{k}. So, we have

Proposition 1. Let f ∈ S′_{+} satisfy conditions of above definition and assume that there are two representation of f

f(t) = ∑_{i=1}^{N} c_{i}(f_{L_{i}})_{α_{i}+1} + h(t)

f(t) = ∑_{i=1}^{M} c̃_{i}(f_{L_{i}})_{α̃_{i}+1} + h̃(t)

for which all the assumptions given above hold. Then M = N, α_{1} = α̃_{1}...α_{N} = α̃_{N}, L_{1} ∼ L̃_{1}...L_{N} ∼ L̃_{N}, α_{1} = α, L_{1} ∼ L.

We shall use the following notation for in f ∈ S′_{+} in Definition 3.

f ∼^{q.e} ∑_{i=1}^{N} c_{i}(f_{L_{i}})_{α_{i}+1} at ∞(0^{+}) of order (α, L) with respect to

k^{α−ℓ}L_{0}(k)((1/k)^{α+ℓ}L_{0}(1/k)).

2.2. Space M′(r)

We extend the definition of the space J′(r) given in [10] and using the same idea we give the definition of space M′(r).

M′(r), r ∈ R \ (−N) denotes the space of all generalized functions f ∈ S′_{+}(R) such that there exist k ∈ N_{0} and a locally integrable function F, suppF ⊂ [0, ∞). so that f is of the form

f = t^{−r}D^{k}F,  (2.1.)

where F is continuous on [0, ∞) and

∫_{R} |F (t)|(t + β)^{r−1−k}dt < ∞, for β > 0,  (2.2)

More precisely M′(r) is the dual space for i.e.

M′(r) = {ψ ∈ C^{∞} \ (0, ∞); ψ = t^{r}φ; ψ ∈ S([0, ∞))}

with the multinorm ∥ψ∥_{α} = sup{t^{r}φ^{(i)}(t)(1 + t)^{k}; i ≤ k, t ∈ [0, ∞), k ∈ N_{0}}. The Stieltjes transformation S_{r}(f)(s), r ∈ R \ (−N) is complex valued

function, defined by

S_{r}(f(t))(s) = ∫_{0}^{∞} \frac{f(t)}{(s + t)^{r+1}} dt.  (2.3)

s ∈ C \ (−∞, 0], 0 < t < ∞, r ∈ R \ (−N)

Modified Stieltjes transformation introduced by Marichev is defined as

(T_{α}f)(x) = \frac{1}{Γ(α)} ∫_{0}^{∞} (1 + \frac{x}{y})^{−α} \frac{1}{y} f(y)dy, x ∈ C \ (−∞, 0],  (2.4)

0 < y < ∞, α ∈ R \ (−N). It can be written as

T_{α}(f)(x) = \frac{1}{Γ(α)} ∫_{0}^{∞} \frac{y^{α−1}f(y)}{(x + y)^{α}} dy, x ∈ C \ (−∞, 0].  (2.5)

Now, we shall find relation between (2.3) and (2.5). Putting r = α − 1, f(t) = y^{α−1}f(y) in (2.3) we get,

S_{α−1}(y^{α−1}f(y))(x) = ∫_{0}^{∞} \frac{y^{α−1}f(y)}{(x + y)^{α}} dy, x ∈ C \ (−∞, 0].  (2.6)

By (2.5) and (2.6), we have

T_{α}(f)(x) = \frac{1}{Γ(α)} S_{α−1}(y^{α−1}f(y))(x), x ∈ C \ (−∞, 0].

Interchanging x by z and α by r + 1, it follows that

Γ(r + 1)T_{r+1}(f)(z) = S_{r}(y^{r}f)(z), z ∈ C \ (−∞, 0].  (2.7)

2.3. Modified Stieltjes transformation T_{r+1}

Let us define the T_{r+1}-transformation r ∈ R \ (−N). Assume that f ∈

M′(r) that is f = t^{−r}D^{k}F and ∫_{R} |F (t)|(t + β)^{−r−1−k}dt < ∞, for β > 0.

Modified Stieltjes transformation is defined.

T_{r+1}(f), r ∈ R \ (−N) is complex valued function defined by

Γ(r + 1)T_{r+1}(f)(s) = (r + 1)_{k} ∫_{0}^{∞} \frac{F (t)}{(s + t)^{r+1+k}} dt,

r ∈ R \ (−N), s ∈ C \ (−∞, 0], 0 < t < ∞.

2.4. Generalized Modified Stieltjes Transformation T̃_{r+1}

The T̃_{r+1}-transformation of a distribution f ∈ S′_{+}(R) is complex valued function T̃_{r+1}(f) defined by

Γ(r + 1)T̃_{r+1}(f)(s) = lim_{w→∞}⟨f(t), η(t)(s + t)^{−r−1} exp(−wt)⟩.

w ∈ R, s ∈ Λ ⊂ (C \ (−∞,]), η ∈ A(s).

Here Λ is the set of complex numbers for which this limit exists; A(s) is the family of all smooth functions, defined on R for which there exists ε = ε_{η,s} > 0 such that 0 ≤ η(t) ≤ 1, t ∈ R, η(t) = 1 if t belongs to the ε-neighbourhood of \overline{R}_{+}, η(t) = 0 if it belongs to the complement of the 2ε-neighbourhood of \overline{R}_{+}, where ε > 0 is arbitrary if Im s ≠ 0 and 0 < 2ε < max Res, if Im s = 0. Moreover, assume |D^{p}η(t)| ≤ Cp, t ∈ R. If η(t) ∈ A(s), s ∈ (C \ \overline{R}_{−}), then η(t)(s + t)^{−r−1} exp(−wt) ∈ S(R) for w ∈ R_{+}, r ∈ R.

3. Main results

For the main results of this section we need the following assertion from [10].

Theorem 1. Let f ∈ S′_{+}, and f = D^{m}F. Then if f has the q.a.b at ∞ with respect to k^{α}L(k), it follows that

\frac{F (kt)}{k^{α+m}L(k)} → Cf_{α+m+1} in S′_{p} for p > α + m + 1 as k → ∞. (3.1)

In the case of q.a.b. at 0^{+}

\frac{F (t/k)}{(1/k)^{α+m}L(1/k)} → Cf_{α+m+1} in S′_{p} for p > α + m + 1 as k → ∞.

If f ∈ M′(r), and f = t^{−r}D^{m}F then F ∈ S′_{r+m+1}. For given z ∈ C \ (−∞, 0], we denote by A(z) the space of all η ∈ C^{∞}, such that η ∈ A(z), if 0 ≤ η ≤ 1, η(t) = 1 for t > −ϵ, η(t) = 0 for t < −2ϵ, where ε > 0 is arbitrary if z ∉ (0, ∞) and z ∈ (0, ∞) then we choose ε such that 0 < 2ε < z. Clearly, for a given z ∈ C \ (−∞, 0] and every η ∈ A(z)

R ∋ t → η(t)(t + z)^{−r−m−1} ∈ S_{p} for p < r + m + 1.  (3.2)

Let f ∈ M′(r). we have (x > 0, t > 0),

Γ(r + 1)(T_{r+1}f)(tx) = x(r + t) ∫_{0}^{∞} Γ(r + 2)(T_{r+2}f)(xu)du

and if

Γ(r + 2)(T_{r+2})(x) ∼ x^{−(r−α)−1}L(x)  (3.3)

as x → ∞(x → 0^{+}), then

Γ(r + 1)(T_{r+1}f)(x) ∼ \frac{(r + 1)}{(1 − α)}x^{−(r−α)}L(x) as x → ∞(x → 0^{+}).

Now we are ready to prove.

Theorem 2. Let f have the closed q.a.e at ∞ of order (α, L) and of length ℓ related to k^{α−ℓ}L_{0}(k) (see the notation in Definition 3). Let r > α, r ∈ R \ (−N). Then

(i) f ∈ M′(r), (f_{L})_{α+1} ∈ M′(r);

(ii) If we put Γ(r+1)T_{r+1}(f_{L})_{α+1}(x) = S_{α,L}(x), then for L ∼ L̃, S_{α,L}(x) ∼

S_{α,L̃}(x) ∼ \frac{Γ(r − α)}{Γ(r + 1)}x^{a−r}L(x), x → ∞;

(iii) Γ(r + 1)(T_{r+1}f)(x) − C \frac{Γ(r − α)}{Γ(r + 1)}x^{α−r}L(x) = 0(x^{α−ℓ−r}L_{0}(x)), (3.4)

x → ∞.

P r o o f. We shall prove the theorem by using the similar idea in the proof of the main theorem in [9]. Obviously, (i) follows from (1.2).

(ii) Let β < r − 1, x ∈ R, L ∈ ∑_{∞} . Let m be the smallest element from

N_{0} such that β + m > −1. Then

Γ(r + 1)(T_{r+1}(f_{L})_{β+1})(x) = (r + 1)_{m} ∫_{0}^{∞} \frac{f_{β+m+1}(t)L(t)}{(x + t)^{r+m+1}}dt

= (r + 1)_{m} < f_{β+m+1}(t)L(t), \frac{η(t)}{(x + t)^{r+m+1}} >, η ∈ A(x),

where < f_{β+m+1}(t)L(t), \frac{η(t)}{(x+t)^{r+m+1}} > is observed as a pair from (S′_{r+m}, S_{r+m}). Obviously this pair does not depend on η ∈ A(x). Since r + m > β + m + 1, we have

\frac{Γ(r + 1)T_{r+1}(f_{L})_{β+1}(kx)}{k^{β−r}L(k)} =

= (r + 1)_{m}⟨f_{m+β+1}L(t), \frac{η(t)}{k^{β+m+1}L(k)(x + (t/k))^{r+m+1}} ⟩

= (r + 1)_{m}⟨\frac{f_{β+m+1}(kt)L(kt),}{k^{β+m}L(k)}, \frac{η(kt)}{(x + t)^{r+m+1}} ⟩.

If k → ∞ from equation (3.1) it follows

\frac{Γ(r + 1)T_{r+1}(f_{L})_{β+1}(kx)}{k^{β−r}L(k)} → ⟨f_{β+m+1}(t), \frac{η(t)}{(x + t)^{r+m+1}} ⟩ =

= \frac{(r + 1)_{m}}{Γ(β + m + 1)} ∫_{0}^{∞} \frac{t^{β+m}}{(x + t)^{r+m+1}} · dt = \frac{Γ(r − β)}{Γ(r + 1)}x^{β−r}.

On putting x = 1 we obtain that (ii) holds for all α < r − 1. Let us suppose that r − 1 ≤ β < r. Then by the same arguments given above, we have

Γ(r + 2)T_{r+2}(f_{L})_{β+1}(x) ∼ \frac{Γ(r + 1 − β)}{Γ(r + 2)}x^{β−r−1}L(x), x → ∞.

Now we completed the proof of result (ii).

(iii) We can assume that α < r − 1 because if r − 1 ≤ α < r we have as in (ii), to observe Γ(r + 2)T_{r+2}(f_{L})_{β+1} firstly and after that to use (3.3). Since f − C(f_{L})_{α+1} ∈ S′_{r+m}, equation (3.1) implies that

\{\frac{Γ(r + 1)(T_{r+1}f)(xk) − C(Γ(r + 1)T_{r+1}(f_{L})_{α+1})(kx)}{x^{α−ℓ−r}L_{0}(x)}\}
=< {f(kt) − C(f_{L})(kt)}/(k^{α−ℓ}L_{0}(k)), η(t)(x + t)^{−r−m−1} >→ 0

as k → ∞. On putting x = 1 the assertion (iii) follows. The similar assertion holds for closed q.a.e. at 0^{+} but with more restrictive assumptions.

Theorem 3. Let f have the closed q.a.e. at 0^{+} of order (α, L) and of length ℓ with respect to (1/k)^{α+ℓ}L_{0}(1/k). If (α + ℓ) < r and f ∈ M′(r), then

Γ(r + 1)(T_{r+1}f)(x) − C \frac{Γ(r − α)}{Γ(r + 1)}x^{α−r}L(x) = 0(x^{α+ℓ−r}L_{0}(x)), x → 0, (3.5)

P r o o f. The proof of this theorem is very similar to the proof of Theorem 2. We only notice that we must observe firstly S_{r+1}f and after that to use (3.3). From f ∈ M′(r) we have F ∈ S′_{r+m+1} and this implies that we have to observe the dual pair (S′_{r+m+1}S_{r+m+1}) (η(t)(x + t)^{−r−m−2} ∈ S_{r+m+1} as a function of t).

4. The uniform Behaviour of T_{r+1}(f)

Let F be a continuous function with supp F ⊂ [0, ∞), r > α > −1 and F (x) ∼ x^{α} as x → ∞. Denote by λ_{a,ϵ}, a > 0, ϵ > 0, a subset of C defined by Λ_{a,ϵ} = {a + Re^{iϕ}; R ≥ 0, −π + ε ≤ ϕ ≤ π − ε}. If z = a + Re^{iϕ} ∈ Λ_{a,ε} and t ∈ [0, ∞), then we have

|z + t|^{r+1} > (\frac{1 − cos ϕ}{2})^{(r+1)/2}(R + a + t)^{r+1}.  (4.1)

This follows from the elementary inequalities:

(a + t)^{2} + 2(a + t)R cos ϕ + R^{2} ≥ (a + t)^{2} − 2(a + t)R cos ε + R^{2}

= (a + t)^{2} + R^{2} + ((a + t)^{2} + R^{2}) cos ε − (a + t + R)^{2} cos ε
≥ ((a + t)^{2} + R^{2})(1 + cos ε) − 2((a + t)^{2} + R^{2}) cos ε
= ((a + t)^{2} + R^{2})(1 − cos ε).

Assumptions on F imply F (x) < C(1 + x^{α}), x ≥ 0 For z ∈ Λ_{a,ϵ} and suitable C_{1}

| ∫_{0}^{∞} \frac{F (t)}{(z + t)^{r+1}} dt| ≤ C(\frac{2}{1 − cos ε})^{(r+1)/2} ∫_{0}^{∞} \frac{(1 + t^{α})}{(R + a + t)^{r+1}} dt

≤ C_{1}[\frac{1}{r} + \frac{Γ(r − α)Γ(α + 1)}{Γ(r + 1)}][\frac{1}{a + R}]^{r−α}.  (4.2)

So, we have proved the following

Lemma 1. Let F satisfy the conditions given above. The function z^{r−α}Γ(r + 1)(T_{r+1}F)(z) is bounded in Λ_{a,ϵ}, a > 0, ϵ > 0.

We use this lemma for the proof the following theorem.

Theorem 4. Let f satisfy the conditions of Theorem 2 and let all the slowly varying functions in Theorem 2 be equal to 1. Then

i) A_{f,r}(z) = Γ(r + 1)(T_{r+1}f)(x) − C \frac{Γ(r − α)}{Γ(r + 1)} \frac{z^{α−r}}{z^{α−ℓ−r}}

is bounded analytic function in any Λ_{a,ε}, a > 0, ε > 0; (ii) A_{f,r}(z) converges uniformly to zero in Λ_{a,ε} when |z| → ∞.

P r o o f: (i) It follows from the Structural theorem (cf. 2.1) and Lemma 1.

(ii) It follows from (i) and Theorem 2 which enable us to use the Montel theorem [1, p.5].

Theorem 5. Let f satisfy the conditions of Theorem 3 and let all the solwly varying functions in Theorem 3 be equal to 1. Let

A_{f,r}(z) = \{Γ(r + 1)(T_{r+1}f)(z) − \frac{Γ(r − α)}{Γ(r + 1)}z^{α−r}\}/z^{α+ℓ−r}.

Then

(i) A_{f,r}(z) is a bounded function in Λ_{0,ε} ∩ B(0, R_{0}), ε > 0, R_{0} > 0, where B(0, R_{0}) = {z; |z| < R_{0}};

(ii) A_{f,r}(z) converges uniformly to zero in Λ_{0,ε} when |z| → 0.

For the proof of this theorem, we need

Lemma 2. Let F ∈ L_{loc}^{1}, suppF ⊂ [0, ∞), r > α > −1,

F (x) ∼ x^{α}, x → 0^{+}, and ∫_{0}^{∞} |F (t)(z + t)^{−r−1}|dt < ∞,

z ∈ Λ_{0,ε} ∩ B(0, R_{0}). Then z^{r−α}Γ(r + 1)(T_{r+1}F)(z) is bounded in Λ_{0,ε} ∩ B(0, R_{0}), R_{0} > 0.

P r o o f of Theorem 4:

(i) From the Structural theorem and Lemma 2 it follows that A_{f,r}(z) is bounded in Λ_{0,ε} ∩ B(0, R_{0}).

(ii) Let Ã_{f,r}(z) = A_{f,r}(1/z), z ∈ C \ (−∞, 0]. The function A_{f,r}(w), w ∈ Λ_{0,ε} ∩ {w : |w| > 1/R_{0}}, ε > 0, R_{0} > 0 is analytic and bounded. As well, we have Ã_{f,r}(x) → 0 as x → ∞. This implies that the same assertions hold for A_{f,r} in the domain Λ_{a,ϵ} ∩ {w; |w| > 1/R_{0}}, a > 0, R_{0} > 0. So by the Montel theorem it follows that Ã_{f,r}(z) converges uniformly to 0, in Λ_{a,ε} as |z| → ∞. Further on this implies that A_{f,r}(z) converges uniformly to 0 in Λ_{0,ε} as |z| → ∞ and so that the assertion (ii) holds.

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S. B. Gaikwad New Arts, Com. & Sci. College Parner Dist. Ahmednagar 414302.(M.S.) India M. S. Chaudhari Department of Mathematics, Shivaji University, Kolhapur 416004.(M.S.) India