Zemanian Space Research Articles

Convolution for a pair of quadratic-phase Hankel transforms

In this work, a pair of quadratic-phase Hankel (QPH) transforms is introduced and defined their inversion. Moreover, some differential operators are given, and two variants of Bessel differential operators are defined. Further two different types of Zemanian spaces are defined and discussed the continuity of QPH-transform and given differential operators on these spaces. Next, we extend the QPHT to generalized functions. QPH-translation and convolution are also defined and studied their properties. Furthermore, an application of QPH-transform to a generalized non-linear parabolic equation is given.

Generalized functions and Laguerre expansions

In this paper we study and characterize expansions of distributions in the Zemanian spaces, \({\mathcal {H}}_{\mu }\) and its dual \({\mathcal {H}}_{\mu }^{\prime }\) (\(\mu \ge -\frac{1}{2}\)) with respect to Laguerre functions. We obtain as applications of this result, the kernel Theorem and a structure Theorem for \({\mathcal {H}}_{\mu }^{\prime }\). We also introduce a new algebra of generalized functions in the sense of J. F. Colombeau such that it satisfies interesting properties involving the Hankel transformation and Hankel convolution.

The distributional continuous fractional generalized Hankel-Clifford transformation

The main objective of this paper is to study the fractional generalized Hankel-Clifford transformation and some of their basic properties. The continuous fractional generalized Hankel-Clifford transformation, its inversion formula are also studied. The finite continuous fractional generalized Hankel-Clifford transformation is also studied in Zemanian space. Applications of the finite fractional generalized Hankel-Clifford transformation in solving generalized nth order linear non-homogeneous partial differential equation is given.

A perspective on fractional Laplace transforms and fractional generalized Hankel-Clifford transformation

In this study a relation between the Laplace transform and the generalized Hankel-Clifford transform is established. The relation between distributional generalized Hankel-Clifford transform and distributional one sided Laplace transform is developed. The results are verified by giving illustrations. The relation between fractional Laplace and fractional generalized Hankel-Clifford transformation is also established. Further inversion theorem considering fractional Laplace and fractional generalized Hankel-Clifford transformation is proved in Zemanian space.

Interpolation by Hankel translates of a basis function: inversion formulas and polynomial bounds.

For μ ≥ −1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis function Φ in certain spaces of continuous functions Y n (n ∈ ℕ) depending on a weight w. The functions Φ and w are connected through the distributional identity t 4n(h μ′Φ)(t) = 1/w(t), where h μ′ denotes the generalized Hankel transform of order μ. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian space ℋ μ in order to derive explicit representations of the derivatives S μ mΦ and their Hankel transforms, the former ones being valid when m ∈ ℤ + is restricted to a suitable interval for which S μ mΦ is continuous. Here, S μ m denotes the mth iterate of the Bessel differential operator S μ if m ∈ ℕ, while S μ 0 is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (h μ′Φ)(t) = 1/t 4n w(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation space Y n.

An n-dimensional pseudo-differential operator involving the Hankel transformation

An n-dimensional pseudo-differential operator (p.d.o.) involving the n-dimensional Hankel transformation is defined. The symbol class H m is introduced. It is shown that p.d.o.’s associated with symbols belonging to this class are continuous linear mappings of the n-dimensional Zemanian space \(H_\mu(I^n)\) into itself. An integral representation for the p.d.o. is obtained. Using the Hankel convolution, it is shown that the p.d.o. satisfies a certain L 1-norm inequality.

The hankel transform of gevrey ultradistributions

The space generalizing the Zemanian space is defined. Some properties of this space are studied. It is shown that the conventional Hankel transform is an automorphism of . The generalized Hankel transform of Gevrey ultradistributions is defined and it is found that the generalized Hankel transform is an automorphism of . Multiplication on and convolution on are investigated.

On hankel convolutors on zemanian spaces

In this paper we introduce the spaces of Hankel convolutors on the Zemanian spaces and characterize the dual spaces as spaces of Hankel convolutors. Also we establish some new properties of in terms of the convolutive dual space of . Here the Hankel transformation and the Hankel convolution play an important role.

Convolution Hankel Transforms on the Zemanian Spaces

In this paper we investigate the convolution Hankel transforms on the Zemanian spaces of Hankel transformable functions and distributions. The convolution Hankel transform is defined on generalized functions by using the adjoint method. Our new definition includes as special cases other known definitions of the convolution Hankel transform of distributions. Finally we establish a distributional inversion formula for the transformation under consideration involving Bessel differential operators.

A new characterization of the bounded operators commuting with Hankel translation

In this note we prove that a bounded operator in the Zemanian space \( {\cal H}_\mu \) commutes with the Hankel translation if, and only if, it commutes with the Bessel operator \(S_\mu = x^ {-\mu -1/2}Dx^ {2\mu +1}Dx^ {-\mu -1/2}\).

McKennon-Type Spaces for the Hankel Transformation

Forμ>−\\frac12;, the Zemanian space Hμof Hankel-transformable functions is expressed as the projective limit of a chain {Spμ}p∈Nof reflexive Banach spaces where the Hankel transformation Sμis an isometric isomorphism. Moreover, Hμis dense in Spμ(p∈N) and Spμis dense in Sqμ(p, q, ∈N, p⩾q). The images of these results in the dual spaces then follow. A fundamental rôle in the construction is played by the space S0μ, consisting of all thoseφ=φ(t) (t∈I=]0, ∞[) such that bothφand Sμφlie in the spaceLμof functions Lebesgue integrable onIwith respect to the weighttμ+1/2. We show that, under Hankel convolution, S0μis a Banach algebra and a dense ideal ofLμ, namely that formed by the functions inLμwhich are Hankel–Stieltjes transforms of Radon measuresγonIwith the property that ∫∞0tμ+1/2d|γ|(t)<∞. New results on the inversion of the Hankel transformation and the positivity of certain Hankel transforms are also established.

On the Topology of the Space of Hankel Convolution Operators

Let Hμbe the Zemanian space of Hankel transformable functions, let O′μ,#be its space of convolution operators, and let Oμ,#be the predual of O′μ,#. We prove that the topology of uniform convergence on bounded subsets of Hμand the strong dual toplogy coincide on O′μ,#. Our technique, involving Mackey topologies, differs from, and is simpler than, those usually employed with the same purpose for other spaces of convolution operators, to which it is also applicable. As a consequence, the properties of Oμ,#being reflexive, complete, nuclear, and Montel are established.

A Class of Pseudo-Differential Operators Associated with Bessel Operators

A class of pseudo-differential operators (p.d.o.), generalizing Bessel differential operator d2/dx2 + (1 − 4μ2)/(4x2), is defined. Symbol classes Hm and Hm0 are introduced. It is shown that p.d.o.′s associated with symbols belonging to these classes are continuous linear mappings of the Zemanian space Hμ into itself. An integral representation of p.d.o.′s is obtained. Using Haimo′s theory of the Hankel convolution it is shown that p.d.o.′s satisfy a certain L1 - norm inequality.

On the convolution theorem of the finite Legendre transform in a Zemanian space

On the convolution theorem of the finite Legendre transform in a Zemanian space

Some Properties of Hankel Convolution Operators

AbstractLet be the Zemanian space of Hankel transformable generalized functions and let be the space of Hankel convolution operators for . This is the dual of a subspace of for which is also the space of Hankel convolutors. In this paper the elements of are characterized as those in and in that commute with Hankel translations. Moreover, necessary and sufficient conditions on the generalized Hankel transform are established in order that every such that in .

The Hankel Convolution and the Zemanian Spaces βμ and β'μ

In this paper we give structure theorems for the elements of the Zemanian spaces β'μ and β'μ,a' Also, bounded sets and convergent sequences in β'μ,a' are characterized through representations as derivatives of measurable functions. Finally, we analyze the Hankel convolution on the above spaces.

Fractional powers of Hankel transforms in the Zemanian spaces

In a recent paper [3], we developed a theory of fractional powers for the Hankel transform x in the Hilbert space L*(R + ). For v > 1, we constructed a C, group (&‘r, ‘AER} of unitary operators on L2(Rf) and showed that when n E Z, 2’:” is the n th iterate of 2”. It is now of interest to consider what happens if v 6 1 (i.e., when the classical Hankel transform is not defined on the whole of L*(R+)). Zemanian [S, 63 has constructed a family {H,, v E R} of Frechet spaces in such a way that for each VER, the Hankel transform 2” is a homeomorphism on H,. When vb $, H, is a subspace of L’(R + ) n L*(R + ) and x coincides with the classical transform. When v 1 (cf. [ 16, p. 1651). In our first two sections, we shall introduce our Frtchet spaces and look at the behaviour of some linear operators in them. Section 3 will be concerned with the case v > i although concepts which are valid for v E R will be included wherever possible. The operators { %F, CI E R, v 2 $} will be the restrictions of those in [3] to an appropriate subspace of L2(R+). We shall be interested mainly in the homeomorphic properties of { %‘z} and the property of strong continuity with respect to LX Finally, in Section 4, the fractional operators (2:) will be defined recursively for v < 4 and our earlier results will be shown to hold for the more general case v E R.

Generalization of Zemanian Spaces of Generalized Functions which Have Orthonormal Series Expansions

We define the space of generalized functions $\exp _p \mathcal{A}'$, $p \in N$. If $f \in \exp _p \mathcal{A}'$, then f can be uniquely expanded into a series of the form $( * )\qquad f = \sum_{n = 0}^\infty {a_n \psi _n } ,$ where $(\psi _n )_{n = 0}^\infty $ is an orthonormal base of the corresponding space $L^2 (I)$, I is an interval in R, $\{ {b_n } \}_{n = 0}^\infty $ is a sequence of complex numbers, such that for some $k \in N$, $( * * )\qquad b_n = O\left( {\left( {\underbrace {\exp \exp \cdots \exp }_p\tilde \lambda _n } \right)^k } \right),\quad n = 0,1, \cdots ;$$\{ {\lambda _n } \}_{n = 0}^\infty $ is a sequence of eigenvalues of a corresponding operator $\mathcal{R}(\mathcal{R}\psi _n = \lambda _n \psi _n )$, and $\tilde \lambda _n = | {\lambda _n } |$, if $\lambda _n \ne 0$ and $\tilde \lambda _n = 1$ if $\lambda _n = 0$. The series in $( * )$ converges in the sense of weak topology in $\exp _p \mathcal{A}'$. Conversely, if a sequence $\{ {b_n } \}_{n = 0}^\infty $ satisfies $( * * )$ for some $k \in N$, a unique element in $\exp _p \mathcal{A}'$ is defined by the series in $( * )$. We give representation theorems for elements from $\exp _p \mathcal{A}'$, $p = 1,2, \cdots $, which show that spaces $\exp A;\exp _2 \mathcal{A}', \cdots $ are natural generalizations of the space $A'$ from [8].