A Course in Probability and Statistics by Charles J.(Charles J. Stone) Stone PDF

By Charles J.(Charles J. Stone) Stone

ISBN-10: 0534233287

ISBN-13: 9780534233280

This author's glossy process is meant essentially for graduate-level mathematical information or statistical inference classes. the writer takes a finite-dimensional sensible modeling perspective (in distinction to the traditional parametric technique) to bolster the relationship among statistical concept and statistical technique.

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Example text

1. The mappings Land L' . A characterization of frame representation operators is given in the following theorem. 2} is a well-defined topological isomorphism onto a closed subspace of £2 (~). 14 has certain significant consequences worthy of mention. Namely, if L is a frame representation operator then (a) L is injective (one-to-one), (b) L(H) is closed, and (c) L' is surjective (onto). To see the injectivity of the map L suppose {¢n} is a frame for H with bounds A and B and that Lh = Lh. Then which implies that h = h.

Be Computed E:ractl:y? J In this section the question of computing the Fourier transform of an analog signal f on a finite precision digital machine is addressed. In all of its generality this task is an impossible one for arbitrary signals f E L2(JR) for the main reason that a digital machine is inherently constrained to operate on objects represented by a finite number of bits. Because of this, any signal to be digitally processed must first be representable by a finite number of bits. (tn)}:=l' where -'Xl < a <::: tn < tn+l <::: b < 00 and Q is a quantization g function.

Lf f E H}. 2 =L (I, ¢n) (g, L* L = IH, where IH is the identity operator on H. In particular, L is invertible and L -1 = L *. In other words f may be recovered from its orthonormal representation Lf through the application of the operator L*.

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A Course in Probability and Statistics by Charles J.(Charles J. Stone) Stone

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