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Partial differential equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
It is known from §10.2 that the set of eigenfunctions {Xnx}n≥1 $ \{ X_{n} \left( x \right)\}_{n \ge 1} $ is complete in the space of square integrable functions. Therefore, the latter is valid only when all coefficients of this sum are zeroes: C˙t+αλnCnt-fnt=0,n=1,2,…. $$ \dot{C}\left( t \right) + \alpha {{\lambda }}_{n} C_{n} \left( t \right) - f_{n} \left( t \right) = 0,\quad n = 1,2, \ldots . $$
Introduction and the Stone-Weierstrass theorem
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
The space of square integrable functions on an interval is an example of a Hilbert space. Hilbert spaces are the infinite dimensional spaces that are closest to finite dimensional spaces, and they are the most tractable. What makes them so tractable are the fact that they have an inner product, and the fact that they are complete metric spaces. In this book, a special emphasis is put on Hilbert spaces, and in this setting integral equations are best understood. We will begin our study of Hilbert spaces in the next chapter.
Fourier Analysis
Published in L. Prasad, S. S. Iyengar, WAVELET ANALYSIS with Applications to IMAGE PROCESSING, 2020
Before we delve into Fourier analysis, we wish to isolate the functions that we are most interested in Fourier analyzing. These functions are mathematical equivalents of those found often associated with phenomena studied in physical and engineering sciences. These are most often finite energy processes and, hence, the associated functions have certain appropriate constraints on them. For instance, a signal with finite energy is represented by a square integrable function. We, therefore, rigorously formulate two important classes of functions below.
Vascular blood flow reconstruction with contrast-enhanced computerized tomography
Published in Inverse Problems in Science and Engineering, 2018
B. Sixou, L. Boussel, M. Sigovan
The aim of the contrast-enhanced computerized tomography is to reconstruct the density of the contrast agent f defined on a 3D spatial bounded domain together with its flow field. The space–time box for the problem is denoted as , where [0, T] is the time domain. We denote the Hilbert space of square integrable functions, , , with values in the Hilbert space of square integrable functions on . The norm on the space is defined as: