Discrete Function Theory Research Articles

Discrete geometric function theory II

The foundations are laid here for a discrete analytic function theory which can be applied to geometric difference (or q-difference) functions. Using the concept of a q-analytic function. a discrete contour integral is defined and analogues are found for Cauchy integral theorems. Appropriate discrete analytic continuation and multiplication operators are outlined and their properties examined. The development of the theory of q-analytic functions is seen to be closely allied to that monodiffric functions, but significant advantages and distinctive differences are noted

Discrete geometric function theory !

The foundations are laid here for a discrete analytic function theory which can be applied to geometric difference (or q-difference) functions. Using the concept of a q-analytic function. a discrete contour integral is defined and analogues are found for Cauchy integral theorems. Appropriate discrete analytic continuation and multiplication operators are outlined and their properties examined. The development of the theory of q-analytic functions is seen to be closely allied to that monodiffric functions, but significant advantages and distinctive differences are noted

A discrete analogue of the Paley-Wiener theorem for bounded analytic functions in a half plane

In this note we prove a discrete analogue to the following Paley–Weiner theorem: Let f be the restriction to the line of a bounded analytic function in the upper half plane; then the spectrum of f is contained in ([0, ∈). The discrete analogue of complex analysis is the theory of discrete analytic functions invented by Lelong-Ferrand (1944) and developed by Duffin (1956) and others.

A new approach to the theory of discrete analytic functions

A new approach to the theory of discrete analytic functions

A note on a discrete analytic function

An unsolved problem in discrete analytic function theory has been to find a suitable analogue of the function . An analogue z(α), of the function zα, is found here for discrete analytic functions of the first kind (or monodiffric functions). This function resolves a conjecture of Isaacs in the negative, and at the same time it introduces multi-valued functions into the discrete analytic theory.

Convolution products of monodiffric functions

As it has been repeatedly pointed out by mathematicians working in discrete function theory, it is both essential and difficult to find discrete analogs for the ordinary pointwise product operation of analytic functions of a complex variable. R. P. Isaacs [Ill, the initiator of monodiffric function theory, is to be credited with the first results in this direction. Later G. J. Kurowski [13] introduced some new monodiffric analogues of pointwise multiplication while the present author [2] succeeded in sharpening and generalizing Isaacs’ and Kurowski’s results. However, all of these attempts were mainly concerned with trying to preserve as many of the pointwise aspects as possible. Consequently, each of these operations proved to be less than satisfactory either from the algebraic or from the function theoretic viewpoint. In the present paper, a different approach is taken. Influenced by the success of R. J. Duffin and C. S. Duris [5] in a related area, we utilize the line integrals introduced in Ref. [l] and the conjoint line integral of Isaacs [l I] to define several convolution-type product operations. It is shown that if D is a suitably restricted discrete domain then monodiffricity is preserved by each of these operations. Further algebraic and function theoretic properties of these operations are investigated and some examples of their applicability are given. For basic definitions, notational conventions and additional bibliographical references the reader is referred to Ref. [I]. It might also be of interest to the reader that a complete bibliography on monodiffric function theory and related topics was recently compiled by the author and is available upon request.

A discrete Fourier analysis associated with discrete function theory

Discrete function theory has recently received considerable attention from many authors. Among these are Duffin and Duris [l, 21, Ferrand [3], and Washburn [4]. In [I], Duffin and Duris present discrete analogs of the onesided and two-sided Laplace transform. Washburn [4] develops the notion of a D-transform and uses it to find solutions of certain types of equations involving discrete functions, their integrals, and their derivatives. Here, we define a discrete analog of the classical Fourier transform and present some of its properties. Section 2, below, is concerned with the actual definition of the discrete Fourier transform and the inversion formula of Theorem 2.3. We also find that the Fourier transforms of two functions agree only when the functions differ by an additive function of the form (1)” K. In Section 3, a convolution theorem and an analog of Parseval’s indentity are proved. Section 4 lists the basic properties of the discrete Fourier transform, and provides some examples of discrete function-discrete Fourier transform pairs. We indicate a possible method for solving the filtering problem calf : h(t X) Sx = g(t) and solve an illustrative example. In what is to follow, we use the discrete exponential function e(z, t) defined by Ferrand [3] to be

A convolution product for discrete function theory

A convolution product for discrete function theory