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1) On the extension of the methods of Harmonic Analysis with the
application of the technique of Daniel Integration.
(1992).**

In this paper we show that by applying the Daniel scheme of integration , we get new results in harmonic analysis, and in particular Fourier Transformation in the non-separable Hilbert space ,which is obtained by extension of the almost periodic functions.

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2) Differential and Infinitesimal Calculus on
finite multi-resolution number systems. (2002)**

In this paper we introduce resolution-sensitive
differential and infinitesimal calculus on finite lattices or grids. The finite
lattices or grids in the rational numbers , are considered as algebras with
addition , multiplication , order and rounding equivalence relation as equality.
Such algebras are called finite resolution-fields A differential calculus, in
order to be defined requires at least two resolution fields, a basic for the
definition of the values of the function, and a secondary for the definition of
the first derivative. Any higher derivative requires a resolution of not coarser
equal pixels (usually finer) . The size of the
pixels are the **keys** of the resolutions specific, differential calculus.
Continuity, differentiability and solvability of differential equations by
functions ,is always only up-the specific (variable or unspecified numerically
in same cases but fixed!) resolution keys! There is a resemblance
with E. Nelson's (in Princeton) internal and external real numbers except that
the present approach is strait, elementary, transparent and does not require
concepts like models of set theory or even axioms of logical character. It also
reminds something of the analysis on surreal numbers, except the present is the
final state and interpretation of similar as the previous creative
formulations. Any ordinary or partial differential equations is understood as,
an equation of the rounding equivalence relation of the basic resolution. Even
with fixed boundary or initial conditions the determination of a solution
function is only up-to-the rounding equivalence relation. Simple and obvious
algorithms produce and propagate on the resolutions such functions practically
for any ODE, or PDE , of implicit or explicit forms, such that they satisfy the
defining differential equations. (In a plotted function on a resolution the
roots are found simply with "find" command of the list in the
computer). There is not at all any need to consider disctetization in to
difference equations, then piece-wise linear approximations, then convergence of
solutions, rates of convergence etc, as such questions are of a different
conceptual approach. They are substituted, by the requirement that at each
step the algorithm that produces the solution, must verify that the
(up-to-rounding) initial equation does hold. Any function is always defined only
up to the key-resolutions. This approach does not make the distinction between
symbolic calculus or analysis and numerical analysis. It is simply a
symbolic calculus which is exactly a numerical calculus too! This restores the
original intention of the technique of differential calculus, which was a
complexity reduction tool, in computations, and not just the reverse, that has
resulted in the present research, which is mainly carrier-seeking
through publications, and redundant complexity multiplier. What is
remarkable, is in addition that functional analysis calculus is also handled in
the same way, which puts theorems, and applications at the same area, and
proves unfortunately much of the contemporary analysis either meaningless or of overwhelming
redundant symbolic complexity not interpretable for computations and
applications. This is even more dramatic at the case of stochastic differential
calculi. We
discus applications in physics, micro and macro economics, especially in cases,
that the usual differential calculus through limits , gives contradictory
differential equations for the same phenomenon!. *A phenomenon may have system of
equations on high resolutions, and some of its equations may turn in to null
as the resolution gets coarser. This is a concept of layered or hierarchical, in
respect to resolution, system of equations or laws of social or physical
phenomenon. The hierarchy, depends on the hierarchy of resolutions or accepted
quantitative accuracy or significance level.*

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