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Lauricella's Theorem
In the theory of orthogonal functions, Lauricella's theorem provides a condition for checking the closure of a set of orthogonal functions, namely: ''Theorem.'' A necessary and sufficient condition that a normal orthogonal set \ be closed is that the formal series for each function of a known closed normal orthogonal set \ in terms of \ Convergence_of_random_variables#Convergence_in_mean, converge in the mean to that function. The theorem was proved by Giuseppe Lauricella in 1912. References *G. Lauricella: ''Sulla chiusura dei sistemi di funzioni ortogonali'', Rendiconti dei Lincei, Series 5, Vol. 21 (1912), pp. 675–85. {{Functional analysis Theorems in functional analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Functions
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval (mathematics), interval as the domain of a function, domain, the bilinear form may be the integral of the product of functions over the interval: : \langle f,g\rangle = \int \overlineg(x)\,dx . The functions f and g are bilinear form#Reflexivity and orthogonality, orthogonal when this integral is zero, i.e. \langle f, \, g \rangle = 0 whenever f \neq g. As with a basis (linear algebra), basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot-product; two vectors are mutually independent (orthogonal) if their dot-product is zero. Suppose \ is a sequence of orthogonal functions of nonzero L2-norm, ''L''2-norms \left\, f_n \right\, _2 = \sqrt = \left(\int f_n ^2 \ dx \right) ^\frac . It follows th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure Of A Set
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For S as a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r ( is allowed). Another way to express this is to sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Function
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: : \langle f,g\rangle = \int \overlineg(x)\,dx . The functions f and g are orthogonal when this integral is zero, i.e. \langle f, \, g \rangle = 0 whenever f \neq g. As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot-product; two vectors are mutually independent (orthogonal) if their dot-product is zero. Suppose \ is a sequence of orthogonal functions of nonzero ''L''2-norms \left\, f_n \right\, _2 = \sqrt = \left(\int f_n ^2 \ dx \right) ^\frac . It follows that the sequence \left\ is of functions of ''L''2-norm one, forming an orthonormal sequence. To have a defined ''L'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Of Random Variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. Background "Stochastic convergence" formalizes the idea that a sequence of essentially rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giuseppe Lauricella
Giuseppe Lauricella (15 December 1867 – 9 January 1913) was an Italian mathematician who contributed to analysis and theory of elasticity.Lauricella, Giuseppi — Treccani, Dizionario-Biografico Born in (Sicily), Lauricella studied at the , where his professors included , |
