Empirical Analysis of Duality Spaces Connecting Distinct Optical form Transforms

This paper presents a novel technique of construction a precise functional frame in presence of the new proposed constraints during the planning straightforward extension of excessive considerable dimensional generalizations using a empirical relationship of two absolutely distinct transforms having diverse kernels transform for the Laplace Stieltjes spaces consisting of analytical signals from two dimensions at any point heavily affecting the successful development for the view of the Gelfand Shilov techniques a subspace of a Schwartz space simple objective function along with their duals implies continuity having functional analyst approach under many classical conventional transforms arise naturally as Laplace Stieltjes transform of certain distributions extensively used in many applications like magnetic field theory follows from the belongings of strong continuity at origin lean heavily in constructing multidimensional S type spaces based on the testing function spaces upto some desired order for infinitely differentiable functions t, x with Gelfand Shilov concept under one umbrella.

Further results on the neutrix composition of distributions involving the delta function and the function cosh+-1(x1/r+1)$\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)$

The neutrix composition F(f (x)) of a distribution F(x) and a locally summable function f (x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {Fn(f (x))} is equal to h(x), where Fn(x) = F(x) * δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function (x). The function $\cosh _ + ^{ - 1}\left( {x + 1} \right)$ is defined by$$\cosh _ + ^{ - 1}\left( {x + 1} \right) = H\left( x \right){\cosh ^{ - 1}}\left( {\left| x \right| + 1} \right),$$where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition ${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right]$] exists and$${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right] = \sum\limits_{k = 0}^{s - 1} {\sum\limits_{j = 0}^{kr + r - 1} {\sum\limits_{i = 0}^j {{{{{( - 1)}^{kr + r + s - j - 1}}r} \over {{2^{j + 2}}}}\left( {\matrix{{kr + r - 1} \cr j \cr } } \right)} } } \left( {\matrix{j \cr i \cr } } \right)\left[ {{{\left( {j - 2i + 1} \right)}^s} - {{\left( {i - 2i - 1} \right)}^s}} \right]{\delta ^{(k)}}(x),$$for r, s = 1, 2, . . . . Further results are also proved.Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh−1+(x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629–7640].

Generalization of Kravchenko—Kotelnikov theorem by spectra of compactly supported infinitely differentiable functions

New generalization of Kravchenko–Kotelnikov theorem by spectra of compactly supported infinitely differentiable functions is discussed. These functions are solutions of linear integral equations of special form. The spectrum of is a multiple infinite product of the spectra of atomic functions. dilated by the argument. Constructed generalized series has fast convergence. This property is confirmed by the presented truncation error bound formula and the results of a numerical experiment.

On the neutrix composition of x−−slnmx− and ln(1 + x+)

The neutrix composition [Formula: see text], [Formula: see text] is a distribution and [Formula: see text] is a locally summable function, is defined as the neutrix limit of the sequence [Formula: see text], where [Formula: see text] and [Formula: see text] is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function [Formula: see text]. The neutrix composition of the distributions [Formula: see text] and [Formula: see text] is evaluated for [Formula: see text] Further related results are also deduced.