PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 78(99) (2005), 73–77

GENERAL KERNEL CONVOLUTIONS WITH SLOWLY VARYING FUNCTIONS

Slavko Simić

Communicated by Stevan Pilipović

Abstract. We prove a theorem concerning asymptotic behavior of general complex-valued kernel convolutions with slowly varying functions in the sense of Karamata. In applications we showed that the content of some classical theorems can be naturally extended on some parts of complex z-plane.

1. Introduction

A real-valued function L(x) is slowly varying in the sense of Karamata if it is defined for x > 0, positive, measurable and satisfying

(1)  L(λx) ∼ L(x) (x → ∞)

for each λ > 0. Some examples of slowly varying functions L(·) (for x > x_{0}) are

log^{b} x, log^{c}(log x), exp(log^{d} x), exp (\frac{log x}{log log x}); b, c ∈ R, 0 < d < 1.

For 0 < x ⩽ x_{0} we can take L(x) := 1.

There is a plenty of papers investigating convolutions with slowly varying functions and various real-valued kernels. As an example for later analysis we quote Karamata’s Theorem for Laplace Transform

Theorem 1. Assume U(·) ⩾ 0, ρ > −1, Û(s) := s ∫_{0}^{∞} e^{−sx}U(x)dx convergent for s > 0, and ℓ(·) slowly varying. Then

U(x) ∼ x^{ρ}ℓ(x) (x → ∞) implies Û(s) ∼ \frac{Γ(ρ + 1)}{s^{ρ}}ℓ(1/s) (s ↓ 0).

Converse statement also holds under some Tauberian conditions (see [2, p. 43]). We shall also mention here results of Karamata and Bojanić [4] on improper integrals, Aran - delović [1] on Mellin convolutions and a general one of Vuilleumier [3].

2000 Mathematics Subject Classification: Primary 30E15; Secondary 26A12.

For similar assertions in complex z-plane, it is possible to define slowly varying function as a holomorphic one in some angle of z-plane but with additional condition of almost uniform convergence in (1) (see [2, pp. 424–425]).

In this paper we obtain asymptotic behavior of general kernel convolutions with slowly varying functions in some domains D of complex z-plane, without altering its classical definition (1).

2. Results

We shall prove the following theorem.

Theorem 2. Define B_{ρ}(z) := ∫_{0}^{∞} t^{ρ}|K(t, z)| dt for a complex-valued kernel K(t, z). Suppose that for some positive constants a, A, there is a region D in complex z-plane such that B_{ρ}(z) exists for −a ⩽ ρ ⩽ 1 and, for z ∈ D:

(i) | ∫_{0}^{∞} K(t, z) dt| > A;  (ii) B_{0}(z) → 1;
(iii) B_{1}(z) → ∞;  (iv) B_{−a}(z) = O(B_{1}(z))^{−a}.

Then, for L(·) slowly varying and z ∈ D, we have

∫_{0}^{∞} L(t)K(t, z) dt = L(B_{1}(z)) ∫_{0}^{∞} K(t, z) dt (1 + o(1)).

For the proof we need the following two lemmas.

Lemma 1. For any slowly varying L(·), some μ ∈ R^{+} and y → ∞, we have

(i) sup_{0⩽x⩽y}(x^{μ}L(x)) ∼ y^{μ}L(y); (ii) sup_{x⩾y}(x^{−μ}L(x)) ∼ y^{−μ}L(y).

This is well-known assertion from the Theory of Regular Variation (see [2], [5]).

Lemma 2. Under the conditions of Theorem 2, for some ν ∈ (0, 1), and z ∈ D, we have B_{ν}(z) = O(B_{1}(z))^{ν}, where the constant in O(1) does not depend either on z or on ν.

This is a consequence of Holder’s inequality

∫ fg ⩽ ( ∫ f^{p})^{1/p}( ∫ g^{q})^{1/q}, p > 1, \frac{1}{p} + \frac{1}{q} = 1.

Putting there f := (t|K(t, z)|)^{ν}, g := |K(t, z)|^{1−ν}, p := 1/ν, q := 1/(1 − ν), and taking in account the condition (ii), we obtain the assertion of the lemma.

Now we can prove Theorem 2.

Proof. For a slowly varying L(·) and z ∈ D, we shall estimate the expression

W := |\frac{∫_{0}^{∞} L(t)K(t, z) dt}{L(B_{1}(z)) ∫_{0}^{∞} K(t, z) dt}−1| = \frac{1}{| ∫_{0}^{∞} K(t, z) dt|}| ∫_{0}^{∞}K(t, z)(\frac{L(t)}{L(B_{1}(z))}−1) dt|.

Under the condition (i) and for some σ ∈ (0, 1) we have

W = O(1)(| ∫_{0}^{σB_{1}(z)}(·) dt| + | ∫_{σB_{1}(z)}^{B_{1}(z)/σ}(·) dt| + | ∫_{B_{1}(z)/σ}^{∞}(·) dt|) = W_{1} + W_{2} + W_{3}.

GENERAL KERNEL CONVOLUTIONS WITH SLOWLY VARYING FUNCTIONS 75

Applying part (i) of Lemma 1 and the conditions (iii) and (iv) of Theorem 2, we get

W_{1} = O(1)| ∫_{0}^{σB_{1}(z)} K(t, z)(\frac{L(t)}{L(B_{1}(z))} − 1) dt|

= O(1) ∫_{0}^{σB_{1}(z)} t^{−a}|K(t, z)||\frac{t^{a}L(t)}{L(B_{1}(z))} − t^{a}| dt

= sup_{0⩽t⩽σB_{1}(z)} [\frac{t^{a}L(t)}{L(B_{1}(z))} + t^{a}]O(B_{−a}(z))

= O(σB_{1}(z))^{a} · O(B_{1}(z))^{−a} = O(σ^{a}).

Similarly, using part (ii) of Lemma 1, and Lemma 2, we obtain the estimation of W_{3},

W_{3} = sup_{t⩾B_{1}(z)/σ} [\frac{t^{−ν}L(t)}{L(B_{1}(z))} +t^{−ν}]·O(B_{ν}(z)) = O(\frac{B_{1}(z)}{σ})^{−ν} ·O(B_{1}(z))^{ν}) = O(σ^{ν}).

Finally, applying the condition (ii) of Theorem 2, we have

W_{2} = O(1) ∫_{σB_{1}(z)}^{B_{1}(z)/σ} |K(t, z)||\frac{L(t)}{L(B_{1}(z))} − 1| dt

= sup_{σB_{1}(z)⩽t⩽B_{1}(z)/σ} |\frac{L(t)}{L(B_{1}(z))} − 1| · O(B_{0}(z)) = o(1),

by the Uniform Convergence Theorem (cf. [2, pp. 6–11]). Hence

W = O(σ^{a}) + O(σ^{ν}) + o(1); z ∈ D.

Since a and ν are positive constants and σ can be taken arbitrarily small, we conclude that W = o(1), z ∈ D, z → z_{0}, i.e., Theorem 2 is proved. □

3. Applications

Theorem 2 is can be used to estimate asymptotic behaviours of convolutions with slowly varying functions and various complex-valued kernels. To illustrate this, we give some examples.

First one is an extension of Karamata’s Theorem for Laplace Transform (see Introduction), on some angles in the right complex half-plane.

Proposition 1. For z → 0, | arg z| ⩽ π/2 − ϵ; ϵ, ρ ∈ R^{+}, we have

∫_{0}^{∞} e^{−zt}t^{ρ−1}L(t) dt = \frac{Γ(ρ)}{z^{ρ}}L(\frac{1}{Re z} )(1 + o(1)).

Proof. We shall apply Theorem 2 on the kernel K(t, z) defined as

K(t, z) := \frac{(Re z)^{ρ}}{Γ(ρ)} e^{−zt}t^{ρ−1}, ρ ∈ R^{+}.

Since z is in the right complex half-plane, it is obvious that Re z > 0. Note also that Re z/|z| = cos(arg z) ⩾ sin ϵ. Now

| ∫_{0}^{∞} K(t, z) dt| = (\frac{Re z}{|z|} )^{ρ} ⩾ sin^{ρ} ϵ, B_{0}(z) = 1;

and

B_{1}(z) = \frac{ρ}{Re z} → ∞; B_{−ρ/2} = \frac{Γ(ρ/2)}{Γ(ρ)}(Re z)^{ρ/2} = O((B_{1}(z))^{−ρ/2}).

Therefore the conditions of Theorem 2 are satisfied with a = ρ/2; A = (sin ϵ)^{ρ}, and the assertion of the proposition follows. □

If z approaches zero along a half-line arg z = π/4, then Re z = Im z := x > 0, z = x\sqrt{2} exp(iπ/4). Hence, applying Proposition 1 and taking separately real and imaginary parts, we obtain the following consequence.

Proposition 2. For L(·) slowly varying and x, ρ ∈ R^{+}, we have

∫_{0}^{∞} e^{−xt}t^{ρ−1}L(t) cos xt dt ∼ \frac{Γ(ρ)}{2^{ρ/2}} cos(πρ/4)(1/x)^{ρ}L(1/x),
∫_{0}^{∞} e^{−xt}t^{ρ−1}L(t) sin xt dt ∼ −\frac{Γ(ρ)}{2^{ρ/2}} sin(πρ/4)(1/x)^{ρ}L(1/x) (x ↓ 0).

The next example demonstrates the flexibility of definition of the domain D. Namely, complex parameter is fixed here in some strip in z-plane but real parameter tends to infinity.

Lemma 3. [6, p. 115] For −1 < Re u < 1, c > 0, we have

∫_{0}^{∞} \frac{t^{u}}{(t + c)^{2}} dt = \frac{πu}{sin πu}c^{u−1}.

For z = x + iy, −1 < x < 0, c > 0, define K(t, z) := \frac{t^{z}}{(t + c)^{2}} \frac{sin πx}{πx}c^{1−x}. Then,

using Lemma 3, an easy calculation gives:

B_{0}(z) = 1; B_{1}(z) = \frac{1 − |x|}{|x|}c; B_{−(1+x)/2}(z) = \frac{1 + |x|}{|x|} sin (\frac{π|x|}{2})c^{−(1+x)/2},

and

| ∫_{0}^{∞} K(t, z) dt| = \frac{|z|}{| sin πz|} \frac{sin π|x|}{|x|} > \frac{sin π|x|}{cosh πy} .

Since z, defined as above is fixed, we see that the conditions of Theorem 2 are satisfied when c → ∞. Therefore

Proposition 3. For −1 < Re z < 0, c > 0, we have

I(z) := ∫_{0}^{∞} \frac{t^{z}}{(t + c)^{2}} L(t) dt = \frac{πz}{sin πz} L(c) c^{z−1}(1 + o(1)) (c → ∞).

In particular

\frac{∂^{m}}{∂z^{m}} I(z) = ∫_{0}^{∞} \frac{t^{z}}{(t + c)^{2}} log^{m} t dt = \frac{πz log^{m} c}{sin πz}c^{z−1}(1 + o(1)) (c → ∞).

References

[1] D. Aran - delović, Sur un theoreme mercerien asymptotique general pour des fonctions a comportement regulier, Publ. Inst. Math., Nouv. Sér. 20(34) (1976), 29–36.

[2] N. H. Bingham et al. Regular Variation, Cambridge University Press, 1989.

[3] M. Vuilleumier, Sur le comportement asymptotique des transformations lineaires des suites, Math. Zeitsch. 98 (1967), 126–139.

[4] R. Bojanić, J. Karamata, On slowly varying functions and asymptotic relations, Tech. Report 432, Math. Res. Center, Madison, Wis. 1963.

[5] E. Seneta, Functions of Regular Variation, Springer-Verlag, New York, 1976.

[6] D. S. Mitrinović, Calculus of Residues, P. Noordhoof, Groningen, 1966.

Matematički institut SANU (Received 24 12 2001) Kneza Mihaila 35 (Revised 11 05 2004) Beograd, p.p. 367 Serbia and Montenegro

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