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In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis , connected with spectral theory .
This is an example of a non-linear functional. The Riemann integral is a linear functional on the vector space of functions defined on [a, b] that are Riemann-integrable from a to b. In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author).
The Theory of Functional Connections (TFC) is a mathematical framework designed for functional interpolation.It introduces a method to derive a functional— a function that operates on another function—capable of transforming constrained optimization problems into equivalent unconstrained problems.
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T , the aim is to construct an operator, f ( T ), which naturally extends the function f from complex argument to operator argument.
The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, [2] which was the first problem in the field to be formulated and correctly solved, [2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.
The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms.In the next year, Donald H. Hyers [1] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces in the case of additive mappings, that was the first significant breakthrough and a step toward more solutions ...
In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus .