Direct Products and Convolutions of Distributions

  • Ram P. Kanwal


Let R m and R n be Euclidean spaces of dimensions m and n respectively, and let x = (x 1,…,x m ) and y = (y 1,…,y n ) denote the generic points in R m and R n ), respectively. Then a point in the Cartesian product R m +n = R m x R n is (x,y) = (x 1,…, x m , y 1,…, y n ). Furthermore, let us denote by D m , D n , and D m +n the spaces of test functions with compact support in R m ,R n , andR m +n, respectively. When f (x ) and g(y) are locally integrable functions in the spaces R m and R n , then the function f(x)g(y) is also locally integrable function in R m +n. It defines the regular distribution:
$$ \begin{array}{*{20}c} {\left\langle {f\left( x \right)g\left( y \right),\varphi \left( {x,y} \right)} \right\rangle = \int {f\left( x \right)} \int {g\left( y \right)\varphi \left( {x,y} \right)dy dx} } \\ { = \left\langle {f\left( x \right),\left\langle {g\left( y \right),\varphi \left( {x, y} \right)} \right\rangle } \right\rangle } \\ \end{array} $$
$$ \begin{array}{*{20}c} {\left\langle {g(y)f(x),\varphi (x,y)} \right\rangle = \int {g(y)} \int {f(x)\varphi (x,y)dx dy} } \\ { = \left\langle {g(y),\left\langle {f(x),\varphi (x,y)} \right\rangle } \right\rangle } \\ \end{array} $$
for φ (x, y) ∈ Dm+n. Let us denote by s(x) ⊗ t(y) the direct product of the distributions s(x) ∈ D m and t(y) ∈ D n according to (1),
$$ \left\langle {s(x) \otimes t(y),\varphi (x,y)} \right\rangle = \left\langle {s(x),\left\langle {t(y),\varphi (x,y)} \right\rangle } \right\rangle , \varphi (x,y) \in D_{m + n,} $$
and check whether the right side of this equation defines a linear continuous functional over Dm+n . For this purpose , we prove the following lemma: Lemma 1. The function ψ (x) = 〈t(y), φ(x, y)〉, where t ∈ D n and φ(x, y) ∈ Dm+n, is a testfunction in Dm, and
$$ D^k \psi (x) = \left\langle {t(y),D_x^k \varphi (x,y)} \right\rangle $$
for all multiindices k, where D x k implies differentiation with respect to (X1, x2,..., xm) only. Also, if the sequence {φ (x, y)} → φ (x, y) in Dm+n as l → ∞, then the sequence ψl (x) = {〈t( y), φ(x, y)〉} → ψ(x) in Dm as l → ∞.


Integral Equation Direct Product Compact Support Integrable Function Abel Integral Equation 
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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Ram P. Kanwal
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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