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Distributions (or generalized functions) are mathematical objects that allow the extension of the concept of the derivative to a much larger class of (not necessarily continuous) functions. Many classical calculus tools, such as convolution or Fourier transform, can be defined for distributions and, more importantly, the theory of differential equations can be developed,sac longchamp. This provides the theoretical background for many important non-continuous problems in physics and engineering. In fact, engineers and physicists performed computations using distributions even before the complete mathematical theory was formulated,gianmarco lorenzi. Using a few properties of otherwise mysterious objects they were able to find the right answers to some practical problems,gianmarco lorenzi. Now,gianmarco lorenzi boots, the spaces of distributions,Distribution mathematics, called Sobolev spaces, are a fundamental concept both in theoretical and applied sciences,gianmarco lorenzi prix.
A well known example of a distribution is the Dirac delta. In physics,Alviero Martini outlet, it may represent a mass distribution (a useful idealization),sac a main longchamp pas cher. In signal theory and engineering it is used as an approximate (idealised) model for an “impulse” or “flash” signal: a very short and strong excitation of an electrical circuit,Alviero Martini Bags. In mathematics,gianmarco lorenzi pas cher, it can be thought of as of the “derivative” of the Heaviside function, a function that is constant everywhere,longchamp pas cher, but makes the unit jump at x = 0,Alviero Martini outlet. Obviously,longchamp soldes, if the “derivative” is understood in the classical way as the rate of change,Distribution mathematics, it is infinite at this point (formally,Alviero Martini outlet, the Heaviside function is not differentiable).
A widely used intuitive description of the Dirac delta reads “this is a function that is equal to zero everywhere except for x = 0,gianmarco lorenzi; the integral of this function is equal to 1″,gianmarco lorenzi pas cher.
Strictly speaking,gianmarco lorenzi pas cher, these two intuitive properties are contradictory. It does not matter what value is put at x = 0, the integral of such a function will always be equal to zero. Moreover, if one puts “”,sac longchamp pas cher, as it is often stipulated, then a serious problem arises in applications,sac longchamp. For instance,Alviero Martini outlet, when one deals with signals represented as and 2 say,sac longchamp solde, how can one make a practical difference between and ? Definitely, Dirac’s delta is not a usual function (strictly speaking,sac longchamp pas cher, it is not defined on the real line) and a rigorous mathematical background for such an object is therefore essential.
Dirac’s intuitive ideas were placed on firm mathematical footing by S. L. Sobolev in 1936, who studied the uniqueness of solutions of the Cauchy problem for linear hyperbolic equations,chaussure pas cher. In 1950 Laurent Schwartz published his Théorie des Distributions. In this book he systematizes the theory of generalized functions unifying all earlier approaches and extending the results.
Nowadays,Alviero Martini Bags, the physicist’s definition of the Dirac delta function
is recognized by the mathematician as a linear functional acting on a set of “well-behaved” test functions
In order to understand and generalize this, the concept of “test functions” is needed,longchamp pas cher. Let the set K consist of all real functions with continuous derivatives of all orders and bounded support,sac longchamp. This means that the function vanishes outside some bounded region,Alviero Martini Bags, which may differ for different functions. The set K is the space of test functions. It can be shown that K is a linear space,longchamp pas cher.
Secondly,sac longchamp pas cher, the concept of linear functional is needed. We call f a continuous linear functional on K, if f maps all elements of K onto a real number (f,Alviero Martini Bags, such that
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