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For n > 1, there are additional conditions, i.e. Sp(2n, F) is then a proper subgroup of SL(2n, F). Typically, the field F is the field of real numbers R or complex numbers C . In these cases Sp(2 n , F ) is a real or complex Lie group of real or complex dimension n (2 n + 1) , respectively.
Property (c) says the operator M is bounded on L p (R n); it is clearly true when p = ∞, since we cannot take an average of a bounded function and obtain a value larger than the largest value of the function. Property (c) for all other values of p can then be deduced from these two facts by an interpolation argument.
Because the remainder R m,n in the Euler–Maclaurin formula satisfies , =, + (), where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form: (!
This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from L p (R d) to itself for p > 1. That is, if f ∈ L p (R d) then the maximal function Mf is weak L 1-bounded and Mf ∈ L p (R d). Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x | f(x) > t}. Now we have:
Arithmetic groups over function fields have very different finiteness properties: if is an arithmetic group in a simple algebraic group of rank over a global function field (such as ()) then it is of type F r but not of type F r+1.
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where R × is the multiplicative group of R (that is, R excluding 0 when R is a field). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).
The Fourier transform is a linear isomorphism F:𝒮(R n) → 𝒮(R n). If f ∈ 𝒮(R n) then f is Lipschitz continuous and hence uniformly continuous on R n. 𝒮(R n) is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers. Both 𝒮(R n) and its strong dual space are also: complete Hausdorff locally convex spaces ...