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Hilbert function spaces
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
Let H be a Hilbert function space on a set X. As point evaluation is a bounded functional, the Riesz representation theorem implies that there is, for every x ∈ X, an element kx ∈ H such that f(x) = 〈f, kx〉 for all f ∈ H. The function kx is called the kernel function at x. We define a function on X × X by k(x,y)=ky(x).
Fourier Series and Orthogonal Functions
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
In the preceding sections of this chapter the trigonometric sequence (2) was used for the formation of Fourier series. During the nineteenth and early twentieth centuries many mathematicians and physicists became aware that one can form series similar to Fourier series by using any orthogonal sequence of functions. These generalized Fourier series turned out to be indispensable tools in many branches of mathematical physics, especially in quantum mechanics. They are also of central importance in several major areas of twentieth century mathematics, in connection with such topics as function spaces and theories of integration.13
Function Spaces
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
A function space is the general name for essentially any set of functions satisfying some particular criteria, which may differ between different function spaces. In physics, we will often be dealing with very particular function spaces as classes of functions to which we expect that the solutions to our physical problems belong. The main rationale for referring to these sets of functions as spaces rather than just calling them function sets is the fact that they will often turn out to be abstract vector spaces, which are generally of infinite dimension, and we will therefore start this chapter by examining some properties of these.
Uniqueness for an inverse coefficient problem for a one-dimensional time-fractional diffusion equation with non-zero boundary conditions
Published in Applicable Analysis, 2023
William Rundell, Masahiro Yamamoto
For the mathematical formulations, we need to introduce function spaces and relevant operators; all functions considered are assumed to be real-valued. Let be a usual Lebesgue space and let and denote the scalar product and the norm respectively in , and let be the scalar product in other Hilbert spaces X when we so specify.