Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu ISSN 1083-0464 Vol. No. 01, 1995, pp. 13–16. Submitted: March 16, 1995. Published: June 30, 1995.

DIFFERENTIATED SHIFT-INVARIANT

MULTIVARIATE INTEGRAL OPERATORS

George A. Anastassiou

Abstract. In a recent paper [1] the author, together with H. Gonska, introduced some wavelet-type, very general multivariate integral operators over ℝ^{d}, d ≥ 1, and studied their various properties such as shift-invariance, global smoothness preservation and optimality, convergence to the unit, differentiation, and preservation of continuous multivariate probability distribution functions. These operators are introduced through a convolution-like iteration of another very general multivariate operator with a general multivariate scaling type function. In this paper the author gives sufficient conditions, so that the partial derivatives of the above described multivariate operators possess most of the above listed nice properties of their originals. In particular, a sufficient condition is given so that a corresponding “global smoothness preservation” sharp multivariate inequality is attained. The corresponding convergence of differentiated operators to the differentiated unit operator is given with rates of convergence.

(1991) A.M.S. (MOS) Subject Classification Codes. 60, 62, 28.

Key Words and Phrases. Wavelet integral operators, Probability distribution functions.

Main Results.

Let X := C_{U}(ℝ^{d}), d ≥ 1 be the space of uniformly continuous functions from ℝ^{d} into ℝ.

For f ∈ X, define the first order modulus of continuity of f by

ω_{1}(f, δ) := sup_{\begin{matrix}x,y∈ℝ^{d} \\ ∥x−y∥≤δ \end{matrix}} |f(x) − f(y)|, δ > 0,

where ∥ · ∥ is an arbitrary norm in ℝ^{d}. It is known [1] that ω_{1}(f, δ) < ∞, ∀δ > 0. Here C(ℝ^{d}) denotes the space of continuous functions from ℝ^{d} into ℝ. Let {ℓ_{k}}_{k∈ℤ} be a sequence of linear operators from C(ℝ^{d}) into itself such that

ℓ_{k}(f, x) = ℓ_{0}(f(2^{−k}·), x), all x ∈ ℝ^{d}, f ∈ C(ℝ^{d}).  (1)

Assume that

ℓ_{0} : C^{(ρ)}(ℝ^{d}) → C^{(ρ)}(ℝ^{d}), ρ ∈ ℕ be fixed,  (2)

where C^{(ρ)}(ℝ^{d}) is the space of ρ-times continuously differentiable functions.

In this article we consider only f ∈ C^{(ρ)}(ℝ^{d}) such that \frac{∂^{ρ}f}{∂x_{i_{1}}^{j_{1}} ···∂x_{im}^{jm}} ∈ X, where fixed i_{1}, …, i_{m} ∈ \{1, …, d\}

be such that i_{1} < ⋯ < i_{m}, and fixed j_{1}, …, j_{m} ∈ ℕ be such that ∑_{r=1}^{m} j_{r} = ρ.

From now on we will denote by

f^{∂} := \frac{∂^{ρ}f}{∂x_{i_{1}}^{j_{1}} ⋯ ∂x_{i_{m}}^{j_{m}}} , f ∈ C^{(ρ)}(ℝ^{d}),  (3)

Typeset by AMS-TEX

Department of Mathematics, Memphis State University, Memphis, TN

Email: anastasg@ kermes.msci.memst.edu

Copyright \begin{matrix}c \\ ◯ \end{matrix}1995 by Cameron University

where the mixed partial is defined as above.

Let φ be a bounded real valued function of compact support ⊆ ×_{i=1}^{d}[−a_{i}, a_{i}], a_{i} > 0. We assume that φ ≥ 0, φ is Lebesgue measurable and

\underbrace{∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞}}_{d−fold} φ(x_{1} − u_{1}, x_{2} − u_{2}, …, x_{d} − u_{d}) du_{1} … du_{d} = 1,

all (x_{1}, …, x_{d}) ∈ ℝ^{d}  (4)

((4) same as

∫_{ℝ^{d}} φ(x − u) du = 1, all x ∈ ℝ^{d};

where x = (x_{1}, …, x_{d}), u = (u_{1}, …, u_{d})). One can easily prove that

∫_{ℝ^{d}} φ(u)du = 1.  (5)

Example. For i = 1, …, d take the characteristic function

φ_{i}(x_{i}) := χ_{[− \frac{1}{2} , \frac{1}{2} )}(x_{i}) = \{ \begin{matrix}1, & x_{i} ∈ [−\frac{1}{2}, \frac{1}{2}) \\ 0, & else. \end{matrix}

Define

φ^{∗}(x) := ∏_{i=1}^{d} φ_{i}(x_{i}), all x = (x_{1}, …, x_{d}) ∈ ℝ^{d}.

Note that φ^{∗} is bounded, supp φ^{∗} ⊆ [−\frac{1}{2}, \frac{1}{2}]^{d}, φ^{∗} ≥ 0 and φ^{∗} is Lebesgue measurable. Also

\underbrace{∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞}}_{d−fold} ∏_{i=1}^{d} φ_{i}(x_{i} − u_{i}) du_{1} … du_{d} = 1,

i.e.,

∫_{ℝ^{d}} φ^{∗}(x − u)du = 1, all x ∈ ℝ^{d}.

Let {L_{k}}_{k∈ℤ} be the sequence of linear operators acting on C^{(ρ)}(ℝ^{d}) and defined as follows

L_{0}(f, x) := ∫_{ℝ^{d}} ℓ_{0}(f, u)φ(x − u) du  (6)

and

L_{k}(f, x) := L_{0}(f(2^{−k}·); 2^{k}x),  (7)

all x ∈ ℝ^{d}, k ∈ ℤ. I.e.,

L_{k}(f, x) = ∫_{ℝ^{d}} ℓ_{k}(f, u)φ(2^{k}x − u) du.  (8)

For our convenience we shall use the notation

∫ a⃗ ; −a⃗ • = ∫_{×_{i=1}^{d}[−a_{i},a_{i}]} • .  (9)

Furthermore, we obtain that

L_{k}^{∂}(f, x) = 2^{kρ} · ∫ a⃗ ; −a⃗ ℓ_{k}^{θ}(f, 2^{k}x − u)φ(u) du

= 2^{kρ} · ∫_{ℝ^{d}} ℓ_{k}^{∂}(f, 2^{k}x − u)φ(u) du
= 2^{kρ} · ∫_{ℝ^{d}} ℓ_{k}^{∂}(f, u)φ(2^{k}x − u) du, all x ∈ ℝ^{d}, k ∈ ℤ.  (10)

Assumption. Take f ∈ C^{(ρ)}(ℝ^{d}), ρ ∈ ℕ fixed, such that f^{∂} ∈ X. For fixed α > 0 we assume that

sup_{\begin{matrix}u,y∈ℝ^{d} \\ ∥u−y∥∞≤α \end{matrix}} |ℓ_{0}^{∂}(f, u) − f^{∂}(y)| ≤ ω_{1,∞} (f^{∂}, \frac{mα + n}{2^{r}}) ,  (11)

is true for f as above, m ∈ ℕ, n ∈ ℤ_{+}, r ∈ ℤ, where ω_{1,∞} is the modulus of continuity ω_{1} defined with respect to ∥ · ∥_{∞}.

Here we present our results without proofs due to their length. We intend to show proofs elsewhere. Definition 1. Let f_{α}(·) := f(· + α), α ∈ ℝ^{d} and ϕ be an operator. If ϕ(f_{α}) = (ϕf)_{α}, then ϕ is called a shift invariant operator. Proposition 1. Assume that

ℓ_{0}^{∂} (f(2^{−k} · +α); 2^{k}u) = ℓ_{0}^{∂} (f(2^{−k}·); 2^{k}(u + α)) ,  (12)

all k ∈ ℤ, α ∈ ℝ^{d} fixed, all u ∈ ℝ^{d}; any f ∈ C^{(ρ)}(ℝ^{d}); ρ ∈ ℕ be given. Then L_{k}^{∂}, any k ∈ ℤ is a shift invariant operator.

Next we establish the property of “global smoothness preservation” of operators L_{k}^{∂}, k ∈ ℤ. Theorem 1. For any f ∈ C^{(ρ)}(ℝ^{d}), such that f^{∂} ∈ X, ρ ∈ ℕ fixed, and any u ∈ ℝ^{d} assume that

|ℓ_{0}^{∂}(f, x − u) − ℓ_{0}^{∂}(f, y − u)| ≤ ω_{1}(f^{∂}, ∥x − y∥), all x, y ∈ ℝ^{d},  (13)

where ∥ · ∥ is an arbitrary norm in ℝ^{d}. Then

ω_{1}((L_{k}f)^{∂}, δ) ≤ ω_{1}(f^{∂}, δ),  (14)

any δ > 0.

Optimality of (14) follows. Theorem 2. Let i_{1}, …, i_{m} ∈ {1, …, d} be such that i_{1} < ⋯ < i_{m}, and j_{1}, …, j_{m} ∈ ℕ be such that ∑_{r=1}^{m} j_{r} = ρ. Consider g_{j_{1}}(x) := \frac{x^{j_{1}+1}}{(j_{1}+1)!}, and g_{j_{r}}(x) := \frac{x^{jr}}{j_{r}!} , r = 2, …, m; all x ∈ ℝ. Denote by f^{∂} := \frac{∂^{ρ}f}{∂x_{i_{1}}^{j_{1}} ...∂x_{im}^{jm}}, for any f ∈ C^{(ρ)}(ℝ^{d}). Denote also by pr_{i_{r}} : ℝ^{d} ∋ (x_{1}, …, x_{d}) → x_{i_{r}}, r = 1, …, m the projection onto the i_{r} coordinate. For all x, y, u ∈ ℝ^{d} assume that

(ℓ_{0} ((∑_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}}) (2^{−k}·)))^{∂} (2^{k}x − u)

− (ℓ_{0} ((∏_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}}) (2^{−k}·)))^{∂} (2^{k}y − u) = 2^{−kρ} · (x_{i_{1}} − y_{i_{1}}).  (15)

Then

ω_{1} ((L_{k} (∏_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}}))^{∂} ; δ) = ω_{1} ((∏_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}})^{∂} ; δ) ,  (16)

for any δ > 0; k ∈ ℤ. That is establishing that (14) is sharp!

The convergence of \frac{∂^{ρ}L_{k}}{∂x_{i_{1}}^{j_{1}} ...∂x_{im}^{jm}} (∑_{r=1}^{m} j_{r} = ρ) to \frac{∂^{ρ}}{∂x_{i_{1}}^{j_{1}} ...∂x_{im}^{jm}} operator, as k → +∞, with rates given in the following. Theorem 3. For f ∈ C^{(ρ)}(ℝ^{d}), d ≥ 1, such that f^{∂} ∈ C_{U}(ℝ^{d}), under the assumption (11), it holds

|L_{k}^{∂}(f, x) − f^{∂}(x)| ≤ ω_{1,∞} (f^{∂}, \frac{ma + n}{2^{k+r}}) ,  (17)

where m ∈ ℕ, n ∈ ℤ_{+}, k, r ∈ ℤ, and a := max(a_{i}), i = 1, …, d.

Again, due to their length we intend to present applications of the above results elsewhere.

References

1. G.A. Anastassiou and H.H. Gonska, On some shift-invariant integral operators, multivariate case, Proceedings of International Conference in Approximation Theory and Probability, Santa Barbara, California, May 20-22, 1993 (1994), Plenum, New York, 41–64.

2. H. Bauer, Maβ-und Integrationstheorie (1990), de Gruyter, Berlin.

3. C. Cottin and H.H. Gonska, Simultaneous approximation and global smoothness preservation, Proceedings of the Second International Conference in Functional Analysis and Approximation Theory Acquafredda di Maratea-Potenza-Italy, 1992. Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Numero 33, (1993), 259–279..

4. A.N. Shiryayev, Probability, Springer-Verlag, New York, Berlin, 1984.