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The frame condition was first described by Richard Duffin and Albert Charles Schaeffer in a 1952 article on nonharmonic Fourier series as a way of computing the coefficients in a linear combination of the vectors of a linearly dependent spanning set (in their terminology, a "Hilbert space frame"). [4]
Sahlqvist formulas are built up from implications, where the consequent is positive and the antecedent is of a restricted form.. A boxed atom is a propositional atom preceded by a number (possibly 0) of boxes, i.e. a formula of the form (often abbreviated as for <).
So this Welch bound is met with equality if and only if the set of vectors {} is an equiangular tight frame in . Similarly, the Welch bounds stated in terms of average squared overlap, are saturated for all k ≤ t {\displaystyle k\leq t} if and only if the set of vectors is a t {\displaystyle t} -design in the complex projective space C P n ...
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Let be a Banach space, let ′ be the dual space of , let : ′ be a linear map, and let ′.A vector is a solution of the equation = if and only if for all , () = ().A particular choice of is called a test vector (in general) or a test function (if is a function space).
One can also view the Maurer–Cartan form as being constructed from a Maurer–Cartan frame. Let E i be a basis of sections of TG consisting of left-invariant vector fields, and θ j be the dual basis of sections of T * G such that θ j (E i) = δ i j, the Kronecker delta. Then E i is a Maurer–Cartan frame, and θ i is a Maurer–Cartan coframe.
[1] [2] [3] Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. [4]
In econometrics, the seemingly unrelated regressions (SUR) [1]: 306 [2]: 279 [3]: 332 or seemingly unrelated regression equations (SURE) [4] [5]: 2 model, proposed by Arnold Zellner in (1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially ...