Then \ mathcal { H } contains all bounded functions that are measurable with respect to \ sigma ( \ mathcal { A } ), the sigma-algebra generated by \ mathcal { A}
22.
The operators " H " ?, " R " are convolutions by bounded functions of compact support, so their operator norms are given by the uniform norm of their Fourier transforms.
23.
(3 ) If f _ n \ in \ mathcal { H } is a sequence of non-negative functions that increase to a bounded function f, then f \ in \ mathcal { H}
24.
When the Banach space is separable, the unit ball of the dual, equipped with the weak *-topology, is a metrizable compact space, and every element in the bidual defines a bounded function on:
25.
Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated.
26.
To avoid any confusion, any bounded function on a " compact set " in "'R "'should be Riemann integratable ( note that this applies to any finite set for instance ).
27.
Similarly one can form the space of essentially bounded functions, with the norm given by the essential supremum, and the positive elements of the dual of this space are given by bounded contents that vanish on sets of measure 0.
28.
A proposition in Royden says " A bounded function f on [ a, b ] is Riemann integrable if and only if the set of points at which f is discontinuous has measure zero . " But, doesn't f have to be bounded to be Riemann integrable?
29.
There are even worse examples . " I " "'Q "'is equivalent ( that is, equal almost everywhere ) to a Riemann integrable function, but there are non-Riemann integrable bounded functions which are not equivalent to any Riemann integrable function.
30.
A common example would be restricting \ mathcal { H } to linear functions : this can be seen as a reduction to the standard problem of linear regression . \ mathcal { H } could also be restricted to polynomial of degree p, exponentials, or bounded functions on L1.
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