Mikusinski Operators Research Articles

An Easy Approach to Distributions and Operational Calculus

The Mikusinski space is introduced as the smallest extension of the space of classical continuous functions in which the integration operator is bijective. (This can be viewed as a very small part of the field of Mikusinski operators.) It is shown that the space of Schwartz distributions (of finite order) can be defined as a quotient of the Mikusinski space. A remarkable property of Mikusinski functions is that they admit multiplication by all rational functions. It is demonstrated that this multiplication provides a natural simple basis for Heaviside's operational calculus.

An elementary approach to asymptotic behavior in the Ces\`{a}ro sense and applications to the Laplace and Stieltjes transforms

We present an elementary approach to asymptotic behavior of generalized functions in the Cesaro sense. Our approach is based on Yosida's subspace of Mikusinski operators. Applications to Laplace and Stieltjes transforms are given.

Essay on Fractional Riemann-Liouville Integral Operator versus Mikusinski’s

This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.

Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order

This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.

Flatness of boundary coupled second order partial differential equations

AbstractWe sketch algebraic and trajectory related controllability results for systems described by boundary coupled partial differential equations of second order in one spatial dimension. Those systems may be described by modules over a Bézout domain ℛℚ generated by particular Mikusinski–Operators with compact support. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On matrices with operator entries

In this paper we consider certain matrix equations in the field of Mikusinski operators, and construct a method for obtaining an approximate solution which allows working with numerical constants instead of operators. The theory of diagonally dominant matrices is applied for the analysis, existence and character of the obtained solutions. We introduce a method for determining approximate solutions of a discrete analogue for operational differential equations and give conditions for their existence. The error of the approximation is estimated.

The numerical methods in the field of operators

AbstractThe approximate solution of a class of nonlinear differential equations, in the field of Mikusiński operators is constructed by using the Euler method. The obtained results are applied to a class of partial integro–differential equations. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An algorithm for the approximate solution of operator difference equations

In this paper we consider a second order difference equation with variable coefficients in the field of Mikusinski operators, F, and construct an algorithm for its approximate solution. Also, we estimate the error of approximtion and show that the approximate solutions of this difference equation can be treated as the approximate solution of the corresponding differential equation in the field of Mikusinski, whose difference analogue turns out to be the considered difference equation

The exact solution of the nusselt's model of the cross-flow recuperator

The exact solution of the Nusselt's model of the cross-flow recuperator obtained by the use of the Mikusinski operator calculus is presented in this work. The solution enables formulas for basic recuperator parameters to be derived. Thus it is no longer necessary to perform calculations using formulas for the counter-flow recuperator and the appropriate correction factors.

Analytical and Algebraic Aspects of the Operational Calculus

Schwartz distributions and Mikusinski operators on $R^n $ are brought together in a conglomerate of mappings which also include many well-known linear transformations. Included are the dilatations, exponential shifts, algebraic derivatives, partial differential operators (with constant or variable coefficients) and pseudo-differential operators. The theory of representation of distributions by operators is presented and then used to explain the operational procedures for solving partial differential equations in the Mikusinski field in terms of the analytical procedures for solving the same partial differential equations in the space of distributions. The paper is mainly expository with, however, many proofs given and covers mostly previously published results. No applications are considered.

Linear Transformations in the Operational Calculus

A new development of linear transformations in the one-sided operational calculus is presented. The setting for this is a (noncommutative) ring $\mathcal{R}$ of continuous linear transformations on a familiar test function space from distribution theory. Included in $\mathcal{R}$, both algebraically and topologically, are all the right-sided distributions, all the traditional transformations of the operational calculus, such as the exponential shifts, the dilatations and the algebraic derivative, all translations and multiplications by infinitely differentiable functions, and many other new transformations. The development parallels that given by E. Gesztelyi for linear transformations of the Mikusinski operator field, but is cast in a simpler and more flexible setting. The main tool of the investigation is a representation theorem of the type introduced by V. Dolezal and the results are primarily theorems concerning commutativity properties in $\mathcal{R}$. It is shown that a linear transformation (i) commutes with all translations iff it is a distribution (ii) commutes with differentiation iff it is a distribution (iii) commutes with the algebraic derivative if it is a multiplier and (iv) is a distribution and commutes with every dilatation if it is a number. Because of the latter, it becomes reasonable to define a Laplace transform in $\mathcal{R}$ which encompasses (for right-sided distributions) that givenby D. Price. Some results on inversion in $\mathcal{R}$ are given and a number of unsettled problems, perhaps amenable to solutions in this setting, are mentioned.

The Mikusinski Operators as a Topological Space

The Mikusinski Operators as a Topological Space

An extension of Dedekind’s linear independence theorem

Dedekind's theorem on the linear independence of isomorphisms of a field is extended to the case of linear independence of compositions of isomorphisms and powers of a derivation, D,, for fields of characteristic zero which contain an element s such that D(s)=1. In this paper we present an extension of the following classical theorem on the linear independence of isomorphisms of a field [1, p. 25]. THEOREM 1 [DEDEKIND]. Let E and P be fields. Then the collection {FJF is an isomorphism of E into P} is left linearly independent over P; i.e., 1pip F=O, pi in P, n> 1 an integer, and Fi 0 Fj for i #j implies pi=O for i= I , 2, ***,n. For fields of characteristic zero, the above theorem may be extended to the case of compositions of isomorphisms and powers of a derivation on E as in the following theorem. This result is useful in the study of linear transformations on the field of Mikusinski operators [2]. THEOREM 2. Let E and P befields of characteristic zero, D a derivation of E into E, and s an element in E such that D(s)= 1. Then, the collection {FD'IF is an isomorphism of E into P and n>O is an integer} is left linearly independent over P. In the following we consider elements of a field E as endomorphisms of E defined for b in E by b(x)=bx for each x in E. (A field E can be embedded in the ring of group endomorphisms on the additive group E.) For the proof of Theorem 2, we will need a lemma. LEMMA. Let E and P be fields of characteristic zero, D a derivation of E into E, and s an element in E such that D(s)= 1. If F and G are homomorphisms of E into P such that FDv=GD7 for some integers p, q>O, then p=q and F=G. Received by the editors August 28, 1972. AMS (MOS) subject classqifcations (1970). Primary 13B10.

A simple approximate method for simulating ESR spectra of randomly oriented samples

The use of Mikusinski operators allows for developing a fast, approximate method for the simulation of ESR spectra of randomly oriented samples. We present an example of using this procedure for the simulation of the ESR spectrum of a borone ceramic.

On continuous endomorphisms of Mikusinski's operator field

On continuous endomorphisms of Mikusinski's operator field

Mathematical analysis of distortion effects of spectrometers

By use of Mikusinski's operators, in case of spectrometers with linear response, a simple invertable relationship can be set up between true and recorded spectrum, respectively. The explicit form of this relation valid for ESR spectrometers is presented together with its inverse form.

Two singular integral equations involving confluent hypergeometric functions

Widder(1) obtained an inversion of the convolution transformby the method of the Laplace transform, Ln(x) being the Laguerre polynomial. Buschman (2) inverted a similar transform with a generalized Laguerre polynomial as kernel and also solved (3) the singular integral equationusing Mikusinski operators. Srivastava(4, 4a) solved singular integral equations with kernels involving and Whittaker functions Mk,μ(x).

A generalization of the Mikusinski operational calculus

0. Introduction. In his version of the operational calculus Mikusinski uses as a starting point the familiar theorem from algebra tlhat every integral domain can be embedded isomorphically in a field. He shows that the class of complex-valued continuous functions defined on [0, cO) forms an integral domain when addition and multiplication are taken to be pointwise addition and convolution of functions, respectively. The resulting field is called the field of Mikusinski operators. In the present paper we generalize Mikusinski's basic results by developing an operational calculus for strongly continuous functions defined on [0, cc) and having values in any Banach algebra with unit element. The Mikusinski operational calculus will then be a special case of our theory, obtained by taking the Banach algebra to be the complex numbers. The paper is divided into four parts. The first gives an algebraic construction; the second shows how the algebraic construction may be used to develop the operational calculus mentioned above; the third indicates some basic results which carry over from the (Mikusinski) operational calculus to this more general setting. Finally, in the last section, we generalize within the framework of the operational calculus a recent theorem of T. K. Boehme. We assume the reader is familiar with the basic notations and results in the books by Mikusinski [3] and Erdelyi [2].

A Phenomenological Theory of Dielectric Response

The phenomenological theory of the electret effect, as developed by Wiseman and Feaster and extended by Perlman and Meunier, is cast into a form which allows a unified treatment of a variety of experiments. Introducing Mikusinski operators, a superposition principle expresses the electric displacement operator as εE+Pb with Pb(t)= ∫ −∞0χ(t−T)E(T)dT,and ε=ε0+χi+χ, where the operator χ characterizes the slow polarization response and the constant χi gives the fast response. Then, the field E within a dielectric of conductivity c is related to the electrode current density J by the operator equation E=Z(J—J̄), where Z=1/(c+sε) determines the impedance operator and the operator J̄={dPb/dt}−εiE(−0) gives the effects of fields applied before t=0. Applications include a derivation of interesting relations discovered by Gross, an evaluation of the accuracy of the quasiequilibrium approximation used by Perlman and Meunier, and the construction of an equivalent circuit.