Schwartz Distribution
Schwartz distribution theory was created by Professor Laurent Schwartz. In order to define the Schwartz distribution, at first, we introduce a set of all C-zero-infinity functions, which is denoted by D.
Schwartz distribution theory was created by Professor Laurent Schwartz. In order to define the Schwartz distribution, at first, we introduce a set of all C-zero-infinity functions, which is denoted by D.
We define a topology of D.
We define a topology of D.
Then we define a Schwartz distribution T, which is a linear and continuous function from D to C. The set of all Schwartz distribution is denoted by D'.
Then we define a Schwartz distribution T, which is a linear and continuous function from D to C. The set of all Schwartz distribution is denoted by D'.
A locally integrable function is naturally understood as an example of Schwartz distribution.
A locally integrable function is naturally understood as an example of Schwartz distribution.
The delta function is an example of Schwartz distribution.
The delta function is an example of Schwartz distribution.
The delta function can be understood as the derivative of a step function.
The delta function can be understood as the derivative of a step function.
The derivative of a Schwartz distribution is defined by a property of partial integration.
The derivative of a Schwartz distribution is defined by a property of partial integration.
The derivative of the delta function is introduced by the definition.
The derivative of the delta function is introduced by the definition.
There are many Schwartz distributions which cannot be represented by any locally integrable function.
There are many Schwartz distributions which cannot be represented by any locally integrable function.
Topology of Schwartz distribution is introduced which is weak* topology of D.
Topology of Schwartz distribution is introduced which is weak* topology of D.
The set of all Schwartz distributions is complete by its topology.
The set of all Schwartz distributions is complete by its topology.
The set of all locally integrable functions is dense in the set of Schwartz distributions.
The set of all locally integrable functions is dense in the set of Schwartz distributions.
A Schwartz distribution is represented by "integration", however, this dw does not mean any integration such as Riemann or Lebesgue.
A Schwartz distribution is represented by "integration", however, this dw does not mean any integration such as Riemann or Lebesgue.
The concept of Schwartz distribution can be extended onto a manifold by using the division of unity.
The concept of Schwartz distribution can be extended onto a manifold by using the division of unity.
The Gelfand zeta function is understood as a Schwartz distribution.
The Gelfand zeta function is understood as a Schwartz distribution.
The Gelfand zeta function can be expanded as a Schwartz distribution.
The Gelfand zeta function can be expanded as a Schwartz distribution.
The Gelfand zeta function can be analytically continued to a meromorphic Schwartz distribution.
The Gelfand zeta function can be analytically continued to a meromorphic Schwartz distribution.
The Gelfand zeta function, the state density function, and the partition function are connected to each other as Schwartz distributions. This connection takes us to statistical learning theory from algebraic geometry.
The Gelfand zeta function, the state density function, and the partition function are connected to each other as Schwartz distributions. This connection takes us to statistical learning theory from algebraic geometry.