Carlo Bardaro, Ilaria Mantellini
On pointwise convergence of nets of Mellin-Kantorovich convolution operators
Dedicated to Professor Julian Musielak on his 85th birthday
Abstract. Here we study pointwise approximation and asymptotic formulae for a class of Mellin-Kantorovich type integral operators, both in linear and nonlinear form.
2000 Mathematics Subject Classification: 41A35, 41A25, 47G10.
Key words and phrases: Mellin-Kantorovich convolution operators, Mellin derivati- ves, moments, Lipschitz conditions, Voronovskaja formula.
1. Introduction. Mellin analysis is an actual field of research in the theory of approximation, one of the main topic of Professor Julian Musielak who dedicated to it the last decades of his scientific activity. For many years the theory of Mellin transform and the corresponding Mellin convolution operators, were treated as a part of Fourier analysis, by means of elementary substitution which reduces one to the other. However, for certain kinds of boundary value problems in wedge shaped regions, the direct use of Mellin transform is very suitable (see e.g. [36]). Indeed the solutions of such problems are defined via linear combinations of Mellin convolution operators depending on a parameter, and the study of the approximation of these operators to the boundary data opened the way to a study of approximation theory in Mellin setting by Mellin convolutions (see e.g. [24]). Moreover in certain physical inverse problems involving the light-scattering, the use of Mellin transform revealed its utility, giving origin to the so-called exponential sampling theorem (see [16]).
These studies gave a great impulse to a deep treatment of the Mellin transform and its related approximation theory by Mellin convolution operators, fully independent of Fourier analysis methods. The first approach in this direction was given in [25], in which the author obtained a Mellin version of the classical treatise [17]. However the first general and systematic treatment of Mellin analysis was initiated in [18],
[19], in which the basic theory of Mellin transform, finite Mellin transform, Mellin convolution operators in Lp−spaces was developed, without using Fourier analysis.
The aim was to set in a rigorous frame all the applications described before. In particular in [19] this analysis was applied to a rigorous study of the exponential sampling. This line of research was then developed in various papers (see [8],[9],[10], [11], [12], [13], [14], [26]), in which especially asymptotic formulae were found for various kinds of Mellin convolution operators, and in [1], [2], [5] for the study of approximation properties in various functional spaces.
Here we introduce the following family of Mellin operators,
(Twf )(s) = Z +∞
0
Kw(ts−1)
1
2 log δw
Z tδw
t/δw
f (u)du u
dt t ,
originated from the Kantorovich version of the classical Bernstein polynomials and their generalized versions (see e.g. [21], [20]). Here (Kw)w>0 is a family of kernel functions and δw≥ 1 is a sequence such that δw → 1. We give in particular some quantitative approximation results in the space of uniformly continuous functions, with respect to the invariant measure of the multiplicative group (IR+,·). The main result of the first part is a Voronovskaja type theorem for the above operator, for locally regular functions f.
In the second part we partially extend the above results to the following nonlinear version, namely,
( eTwf )(s) = Z +∞
0
Kew
ts−1,
1
2 log δw
Z tδw
t/δw
f (u)du u
dt t ,
where the kernel functions eKw are now defined in the product space IR+× IR and satisfy certain Lipschitz conditions. In the nonlinear case, it is not possible to obtain an exact asymptotic formula for the pointwise approximation, as in the linear case, so we will express the estimate of the pointwise convergence by means of inequalities (see Section 4).
The use of nonlinear integral operators in approximation theory was introduced, in their classical convolution form, by J. Musielak (see [27], [29], [30], [31]) and later on a lot of research papers were published, concerning approximation by nonlinear operators in general functional spaces, for example, modular spaces (see [28]). An organic treatment of approximation by nonlinear integral operators was given in the monograph [15], in which also a wide literature on the field is given. In recent years, other contributions were given in [3], [4], [6], [7], [8], [22], [23], [32], [33].
The operators studied here may be also considered as special cases of an abstract class of “sampling-type” operators in [33], [34], in which approximation results are obtained in certain modular spaces.
2. Notations and basic definitions. Let IR+be the multiplicative topolo- gical group endowed with the Haar measure
µ(A) = Z
A
dt t ,
for every (Borel) measurable set A ⊂ IR+.We will denote by Lp(µ, IR+) = Lp(µ), 1≤ p≤ +∞, the Lebesgue spaces with respect to the measure µ and we will denote by kfkp the corresponding norm of a function f ∈ Lp(µ).
In what follows we will say that f ∈ Ck locally at the point s ∈ IR+ if there is a neighbourhood Us of the point s such that f is (k − 1)-times continuously differentiable in Usand the derivative of order k exists at the point s.
We will say that a function f : IR+ → IR is log-uniformly continuous in IR+ if for every ε > 0 there exists η > 0 such that |f(s1)− f(s2)| < ε whenever | log s1− log s2| < η. Note that there are functions uniformly continuous in IR+ in the usual sense but not in the log-sense and conversely. It is clear that the two notions are equivalent on every compact interval in IR+.
We denote by BC0 the space containing the log-uniformly continuous and bo- unded functions in IR+.
We recall that the Mellin derivative of f is defined by Θf (s) = sf0(s), s∈ IR+,
provided the usual derivative f0(s)exists. The Mellin differential operator of order r∈ IN is defined inductively by putting Θ1= Θ, Θr= Θ◦ Θr−1, Θ0= I, I being the identity operator. From [18] we have the following representation result:
Θrf (s) = Xr k=0
S(r, k)f(k)(s)sk
where S(r, k), r ∈ IN, 0 ≤ k ≤ r, denote the Stirling numbers of the second kind.
For a bounded and log-uniformly continuous function f we define the modulus of continuity of f by, for η > 0,
ω(η) = sup{|f(s1)− f(s2)| : | log s1− log s2| ≤ η}.
It is easy to see that the function ω satisfies all the classical properties of a modulus of continuity (see for example [20]). In particular ω(η) is finite for every η > 0 and limη→0ω(η) = 0if and only if f is log-uniformly continuous.
The following Taylor formula with the remainder in the form of Peano will be useful in the following (see [25] and [10])
Proposition 2.1 Let f ∈ Cn locally at the point s ∈ IR+. Then there exists δ > 1 such that for t ∈]1/δ, δ[
f (st) = f (s) + Θf (s) log t +Θ2f (s)
2! log2t +· · · + Θnf (s)
n! lognt + h(t) lognt where h(t) → 0 as t → 1. Moreover, if f ∈ L∞(µ), the above formula holds for every t ∈ IR+,and the function h is bounded on IR+.
3. Linear Mellin-Kantorovich operators. Let Kw be a kernel function Kw: IR+ → IR satisfying the following conditions :
1) Kw ∈ L1(µ), and kKwk1 ≤ D for every w > 0 and an absolute constant D > 0,
2) putting Uδ = [1/δ, δ],for δ > 1 we have
w→+∞lim Z
IR+\Uδ
|Kw(t)|dt t = 0.
Let (δw)w>0 be a net of numbers such that δw> 1and δw→ 1. For any w > 0, let us consider the linear Mellin-Kantorovich convolution operator
(Twf )(s) = Z +∞
0
Kw(ts−1)
1
2 log δw
Z tδw
t/δw
f (u)du u
dt t
=
Z +∞ 0
Kw(t)
1
2 log δw
Z stδw
st/δw
f (u)du u
dt t where f ∈ DomT = T
w>0DomTw, and DomTw is the space of all measurable functions f : IR+ → IR for which the integral is well defined. Note that for every 1≤ p ≤ +∞ we have Lp(µ)⊂ DomTwfor every w > 0.
For every j ∈ IN we define the logarithmic moment of order j of the functions Kw by
mj(Kw) :=
Z +∞ 0
Kw(t)(log t)jdt t .
Moreover, we define the absolute logarithmic moment of order j ∈ IN by Mj(Kw) :=
Z +∞
0 |Kw(t)|| log t|jdt t . Theorem 3.1 Let f ∈ BC0,then
w→+∞lim kTwf− fk∞= 0.
Proof Since f is log-uniformly continuous, given ε > 0 let δ > 1 be such that
| log x − log y| < 6 log δ implies |f(x) − f(y)| < ε. There exists w such that for every w≥ w we have 1 < δw< δ.For such w and u ∈ [δstw, stδw]we have
| log u − log s| ≤ | log u − log st
δw| + | log st
δw − log stδw| + | log stδw− log s|
≤ 4 log δw+| log stδw− log s| = 4 log δw+| log tδw| ≤ 4 log δ + | log tδw|.
If t ∈ Uδ we have log1δ < logδδw < log tδw < 2 log δ hence for u ∈ [stδ, stδ] we get
|f(u) − f(s)| ≤ ε. Thus
|(Twf )(s)− f(s)| ≤ (Z
Uδ
+ Z
IR+\Uδ
)
|Kw(t)|
1
2 log δw
Z stδw st/δw
|f(u) − f(s)|du u
dt t
< εD + 2kfk∞
Z
IR+\Uδ
|Kw(t)|dt t
and so the assertion follows from assumption 2).
Remark 3.2 For what concerns the pointwise convergence we can prove, using the same method, that if f ∈ L∞(µ)and it is continuous at the point s then
w→+∞lim (Twf )(s) = f (s).
This last result can be extended also to functions f ∈ Lp(µ)with p ≥ 1, with the
condition Z
IR+\Uδ
|Kw(t)|dt
t = o(log1/pδw), w→ +∞.
Indeed it is enough to estimate the term
J :=
Z
IR+\Uδ
|Kw(t)|
1
2 log δw
Z stδw
st/δw
|f(u) − f(s)|du u
dt t . We have
J ≤ 1
2 log δw
Z
IR+\Uδ
|Kw(t)|
Z stδw
st/δw
(|f(u)| + |f(s)|)du u
dt t
≤ 1
2 log δw
Z
IR+\Uδ
|Kw(t)|
Z stδw
st/δw
|f(u)|du u
dt t
+ 1
2 log δw
Z
IR+\Uδ
|Kw(t)|
Z stδw
st/δw
|f(s)|du u
dt
t = J1+ J2. Using Hölder inequality, we have
J1≤ kfkp (2 log δw)1/p
Z
IR+\Uδ
|Kw(t)|dt t = o(1) and
J2≤ |f(s)|
Z
IR+\Uδ
|Kw(t)|dt t . Thus the assertion follows. 2
Theorem 3.3 Assume that M2(Kw) =O(log2δw), w→ +∞. If the function f is log-uniformly continuous, then
kTwf− fk∞≤ Cω(log δw).
Proof We have
|(Twf )(s)− f(s)| ≤ Z ∞
0 |Kw(t)|
1
2 log δw
Z stδw
st/δw
|f(u) − f(s)|du u
dt t . Since
|f(u) − f(s)| ≤ ω(| logu s|) ≤ ω
log δw| logus| log δw
≤
1 +| logus| log δw
ω(log δw),
then
|(Twf )(s)− f(s)| ≤ Z ∞
0 |Kw(t)|
1
2 log δw
Z stδw st/δw
1 +| logus| log δw
ω(log δw)du u
dt t
= ω(log δw) 2 log δw
Z ∞
0 |Kw(t)|
Z tδw t/δw
1 + |log v| log δw
dv v
dt t
≤ Dω(log δw) +ω(log δw) 2 log2δw
Z∞
0 |Kw(t)|
Z tδw t/δw
| log v|dv v
dt t. For the inner integral the following estimate holds
Z tδw
t/δw
| log v|dv
v ≤ log2t + log2δw
and so
|(Twf )(s)− f(s)| ≤ (D +1
2)ω(log δw) +ω(log δw) 2 log2δw
M2(Kw).
So from the assumption the assertion follows.
Now we give a Voronovskaja-type asymptotic expansion for the operator Tw.
Theorem 3.4 Let (Kw)w be fixed and f ∈ L∞(µ).Let us assume that there exists α > 0 such that
w→+∞lim wαmj(Kw) = `j, j = 1, 2,· · · , n, (1)
and
wα
n+1X
k=0
n + 1 k
(log δw)n−k(1 + (−1)n+1−k)Mk(Kw)≤ M
uniformly with respect to w ≥ w0. If f ∈ Cn locally at the point s ∈ IR+ then
w→+∞lim wα[(Twf )(s)− f(s)] = Xn j=1
Θjf (s) j! `j.
Proof Since f ∈ L∞(µ)and f ∈ Cn locally at the point s. By Proposition 2.1 we have
f (u) = f (s)+Θf (s) log(u
s)+Θ2f (s) 2! log2(u
s)+· · ·+Θnf (s) n! logn(u
s)+h(u
s) logn(u s)
where h(t) → 0 as t → 1 and h is a bounded function. So we can write (Twf )(s)− f(s) =
Z ∞ 0
Kw(t)
1
2 log δw
Z stδw
st/δw
[f (u)− f(s)]du u
dt t
= Z ∞
0
Kw(t)
1
2 log δw
Z stδw
st/δw
Xn
j=1
Θjf (s) j! logj(u
s) + h(u
s) logn(u s)
du u
dt t
= Xn j=1
Z ∞ 0
Kw(t)
1
2 log δw
Z stδw
st/δw
Θjf (s) j! logj(u
s)du u
dt t +
Z ∞ 0
Kw(t)
1
2 log δw
Z stδw
st/δw
h(u
s) logn(u s)du
u
dt
t = I1+ I2. Now we consider I1.We get
I1= Xn j=1
Θjf (s) j!
Z ∞ 0
Kw(t)
1
2 log δw
Z tδw
t/δw
logjvdv v
dt t
= Xn j=1
Θjf (s) (j + 1)!
1 2 log δw
Z ∞ 0
Kw(t)
logj+1tδw− logj+1 t δw
dt t . Thus, using the binomial theorem, we get
I1 = Xn j=1
Θjf (s) (j + 1)!
1 2 log δw
Z ∞ 0
Kw(t)
j+1X
k=0
j + 1 k
logkt×
logj+1−kδw(1− (−1)j+1−k)dt t
= Xn j=1
Θjf (s) (j + 1)!
1 2
Z ∞ 0
Kw(t) Xj k=0
j + 1 k
logkt× logj−kδw(1− (−1)j+1−k)dt t
= 1
2 Xn j=1
Θjf (s) (j + 1)!
Xj k=0
j + 1 k
logj−kδw(1− (−1)j+1−k) Z ∞
0
Kw(t) logktdt t
= 1
2 Xn j=1
Θjf (s) (j + 1)!
Xj k=0
j + 1 k
logj−kδw(1− (−1)j+1−k)mk(Kw).
Now we consider I2=
Z ∞ 0
Kw(t)
1
2 log δw
Z tδw
t/δw
h(v) logn(v)dv v
dt t .
Since h(t) → 0 for t → 1, for a fixed ε > 0 there exists η > 1 such that |h(t)| ≤ ε for t ∈ Uη= [1/η, η].Let η > 1 and w so large that ηδw≤ η, for every w ≥ w. Thus for t ∈ Uη = [1/η, η] and v ∈ [t/δw, tδw]one has 1/η < 1/ηδw< t/δw < v < tδw<
ηδw< η.Then I2≤
Z
Uη
+ Z
IR+\Uη
|Kw(t)|
1
2 log δw
Z tδw
t/δw
|h(v)|| logn(v)|dv v
dt
t = I21+ I22.
For I21,as in Theorem 2, we have Z tδw
t/δw
| logn(v)|dv v ≤
Z 1 t/δw
(−1)nlogn(v)dv v +
Z tδw
1
logn(v)dv v
= 1
n + 1
n+1X
k=0
n + 1 k
logkt(log δw)n+1−k(1 + (−1)n+1−k).
Thus we have
I21≤ ε 2(n + 1)
n+1X
k=0
n + 1 k
(log δw)n−k(1 + (−1)n+1−k)Mk(Kw) and so by assumptions,
wαI21≤ M ε 2(n + 1).
For the second term I22, taking into account that for every k ≤ n + 1 one has Z
IR+\Uη
|Kw(t)|| logkt|dt
t ≤ 1
(log η)n+1−k Z
IR+\Uη
|Kw(t)|| logn+1t|dt
t ≤ Mn+1(Kw) (log η)n+1−k we have analogously
I22≤ khk∞ n+1X
k=0
n + 1 k
(log δw)n−k Z
IR+\Uη
|Kw(t)|| logkt|dt t
≤ khk∞
(log η)n+1−kMn+1(Kw)
n+1X
k=0
n + 1 k
(log δw)n−k
and using the assumptions we get
wαI22→ 0.
So we have the assertion, on account of (1) and the fact that logj−kδw→ 0 for any
0≤ k < j.
4. An extension to the nonlinear case. We denote by IL the class of all nets of Borel measurable functions (Lw)w>0, Lw: IR+→ IR+0 such that Lw∈ L1(µ).
Let ( eKw)w>0be family of functions, eKw: IR+×IR → IR, such that the following assumptions hold:
1. for every u ∈ IR, eKw(·, u) is Borel measurable and eKw(t, 0) = 0 for every t∈ IR+ and w > 0,
2. there exists a family (Lw)w>0∈ IL such that
| eKw(t, u)− eKw(t, v)| ≤ Lw(t)|u − v|
for every t ∈ IR+, u, v∈ IR and w > 0.
We will say that the family ( eKw)w>0is “singular ”, if the following assumptions are satisfied:
(i) There is a constant D > 0 such that, for every w > 0, t ∈ IR+, Z +∞
0
Lw(t)dt t ≤ D, (ii) for every δ > 1, we have, with Uδ= [1δ, δ],
w→+∞lim Z
IR+\Uδ
Lw(t)dt t = 0, (iii) For every u ∈ IR we have
w→+∞lim
Z +∞
0
Kew(t, u)dt t − u
= 0.
The family ( eKw)w>0,is said to be “strongly singular ”, if condition (iii) is replaced by
(iv) Putting
rw:= sup
u6=0
Z +∞ 0
Kew(t, u)dt t − u
, we have rw= o(1), w→ +∞.
For every singular family ( eKw)w>0 we define the net of the nonlinear Mellin type operators on putting
( eTwf )(s) :=
Z +∞
0
Kew
ts−1,
1
2 log δw
Z tδw
t/δw
f (u)du u
dt t for every f ∈ DomeT =T
w>0Dom eTw.
For the above operators we can give in a similar manner the nonlinear versions of Theorems 3.1 and 3.3, under the assumption of strong singularity on the family ( eKw)w>0. In particular Theorem 2 reads now as
Theorem 4.1 Assume that ( eKw)w>0is strongly singular, and M2(Lw) =O(log2δw), w→ +∞. If the function f is log-uniformly continuous, then
k eTwf− fk∞≤ Cω(log δw) + rw. Proof Using the properties of strong singularity, we have
|( eTwf )(s)− f(s)| ≤ Z ∞
0
Lw(t)
1
2 log δw
Z stδw
st/δw
|f(u) − f(s)|du u
dt t + rw. Thus we can proceed as in the proof of Theorem 3.3, in order to have the assertion.
Remark 4.2 If in Theorem 4.1 we assume rw=O(w−α),for α > 0, we clearly get
|( eTwf )(s)− f(s)| ≤ C(ω(log δw) + w−α) (w→ +∞).
The family of functions ( eKw)w>0 will be called "α-singular"if (i) holds and, for a fixed α > 0, assumptions (ii) and (iii) are replaced by:
(v) For every δ > 1, we have, with Uδ= [1δ, δ],
w→+∞lim wα Z
IR+\Uδ
Lw(t)dt t = 0.
(vi) For every u ∈ IR we have
w→+∞lim wα
Z +∞ 0
Kew(t, u)dt t − u
= 0.
Now, we give an extension of the Voronovskaja formula for the family of operators ( eTw)w>0.
Theorem 4.3 Let ( eKw)be an α− singular family, with α > 0 be such that
wα
log δw+M2(Lw) log δw
≤ M,
for an absolute constant M > 0, uniformly for w ≥ w0 > 0. Then for f ∈ L∞(µ) and f ∈ C1 locally at a point s ∈ IR+, we have
lim sup
w→+∞wα|( eTwf )(s)− f(s)| ≤ B|Θf(s)|, for an absolute constant B > 0.
Proof Let s ∈ IR+ be fixed and f ∈ C1 locally at the point s. As before, using the Lipschitz condition of the function eKw,we have
|( eTwf )(s)− f(s)| ≤ Z ∞
0
Lw(t)
1
2 log δw
Z stδw
st/δw
|f(u) − f(s)|du u
dt t +
Z +∞
0
Kew(t, f (s))dt t − f(s)
= J1+ J2.
For J1,using the Taylor formula of the first order, f (u) = f (s) + Θf (s) log(u
s) + h(u s) log(u
s),
where h is a bounded function, such that h(v) → 0 for v → 1, we get
J1≤ Z ∞
0
Lw(t)
"
1 2 log δw
Z stδw
st/δw
|Θf(s)|| log(u s)|du
u
#dt t +
Z ∞ 0
Lw(t)
"
1 2 log δw
Z stδw
st/δw
|h(u
s)|| log(u s)|du
u
#dt
t = J11+ J12.
Since Z stδw
st/δw
| log(u s)|du
u ≤ log2t + log2δw,
J11≤ |Θf(s)|
2 log δw
Z +∞
0
Lw(t)(log2t + log2δw)dt t
= |Θf(s)|M2(Lw) 2 log δw
+D|Θf(s)| log δw
2 .
For J12,arguing as in the proof of Theorem 3 for the term I2, with n = 1, we have wαJ12→ 0. Finally, the term J2 is estimated immediately using the α−singularity, obtaining wαJ2→ 0. Summing up, we obtain
lim sup
w→+∞wα|( eTwf )(s)− f(s)| ≤ B|Θf(s)|,
for a suitable constant B > 0.
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Carlo Bardaro
Department of Mathematics and Informatics, University of Perugia Via Vanvitelli 1, 06123 Perugia, Italy
E-mail: bardaro@unipg.it Ilaria Mantellini
Department of Mathematics and Informatics, University of Perugia Via Vanvitelli 1, 06123 Perugia, Italy
E-mail: mantell@dmi.unipg.it
(Received: 19.08.2013)
