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| Schwartz Distributions And Its Value About A Point
The theory of distributions (generalized functions) was originally
designed to solve differential equations – mostly coming
from mathematical physics – and to
formalize the physicist's intuition about point particles
(which are represented as δ-functions). This
theory has indeed led to many important physical
applications. On the other hand, there are many purely
mathematical developments of this theory that has not yet
led to applications. Moreover, Physicists
strongly believe that many of these new developments
can be interpreted in a physically meaningful way and
hopefully, will lead to new efficient computer programs for
solving important practical problems.
This work contains two important ideas. One is that the
distributions are closer to
physics and engineering than many researchers think. Where
do functions appear in applications? The simplest case is a
“physical function” x(t) that describes how the value of a
certain physical quantity x (e.g., of temperature) depends
on time t. The objective of many physical theories is to
make predictions about such dependence. For example, in
meteorology, we measure the values of temperature, pressure,
etc., at different points in different moments of time, and
then computer programs use these measurement results to
predict the future values of these variables. Every time we
want to measure the value x(t0) of the quantity x at a
certain moment of time t0, then, due to the inevitable
inertia of any measurement device, the measured value is not
x(t0), but rather an “average”
temperature over an interval containing t0, i.e., in precise
terms, an integral ∫f(t)•x(t)dt. Different measuring
instruments correspond to different functions f(t),
some with narrower support, some with wider support. From
this viewpoint, when we say that we have a physical
function, i.e., a quantity depending on time, in reality,
what we measure is a functional f→∫f•x that maps smooth fast
decreasing functions f(t) (corresponding to different
measuring instruments) into real numbers. In other words, a
natural computer representation of a physical function is
such a functional – i.e., a distribution. From this
viewpoint, not only the original definition becomes
physically natural, but also physically meaningful versions
of this notion can be obtained if we take into consideration
which test functions f(t) make physical sense.
Similarly, physical fields – physical functions of several
variables – are naturally represented as corresponding
distributions. Thus, from the computing viewpoint, to study
physical functions, we must study not just real numbers and
traditional mathematical functions x, but also functionals
that map such functions into real numbers. In computer
science, such a transition is very natural: every time we
have two types of objects t1 and t2, we can define a new
type – functions that input objects of type t1 and output
objects of type t2. The corresponding
formalism is called λ -calculus: for example, λ x.x˛ + 1
means a function that takes x as an input and returns x˛+1.
In these terms, distributions can be described as λ x. ∫
f•x.
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The theory of whole computer science & its practical
uses (software) are based on theory of functions y=f(x)
up till now. Moreover, Logicians since 1910 (see,
Hindley & Seldon (1986)) have pointed out that the
function does not provide always output for any
input in a system or sub-system. These Logicians
further pointed out that the operator can provide
always output for any input. Keeping this view in
mind,( Misra (2002),Distribution Theory In Computer
Science,SCI-TEC,2002) has developed a theory of
operators in terms of Schwartz Distribution to carry
on our desired results as stated. In fact, the
importance of the said work can be found above.
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