Theory of Statistics - George Mason University

734 0 Statistical Mathematics
**************domain *** real line versus complex plane ******************
refer to proof of Theorem 1.17) on page 49 as an example of analytic continuation.
The property of being analytic is quite different for real and complex
functions. In the case of a complex function of a complex variable f(z), if the
first derivative of f exists at all points within a region D, then the derivatives
of all orders exist. Furthermore, the Taylor series converges to the function
value within the region over which the function is analytic. (These facts can
be shown using the Cauchy integral formula; see Churchill, 1960, Chapters 5
and 6, for example.) The definition of an analytic complex function is usually
different from that of a real function. An analytic complex function is defined
as one whose (first) derivative exists over a region.
0.1.9 L p Real Function Spaces
Definition 0.1.42 (L p space)
Given the measure space (Ω, F, ν) and the real number p ≥ 1. The space of
all measurable functions f on Ω for which |f| p dν < ∞ is called the L p (ν)
space, or just the L p space.
Although the measure ν is needed to define the integral, we often drop the
ν in L p (ν). If the integral is taken only over some D ∈ F, we may denote the
space as L p (D), and a more complete notation may be L p (ν, D).
An L p space is a linear space (exercise).
An important fact about the L p spaces is that they are Banach spaces
(that is, among other things, they are complete). This fact is called the Riesz-
Fischer theorem and is proved in most texts on real analysis.
There are several types of useful operations on functions in a given function
space F. Most binary operations require that the domains of the two functions
be the same. Function composition, equation (0.1.68), is a very common
operation that only requires that the range of one function be in the domain of
the other. Many interesting binary operations on functions involve integration
of the functions, and so require that the functions be in some L p space.
We now describe some of these operations. Each operation is a mapping.
Some binary operations map L p × L p to L q or map L p × L p to IR, often for
p = 2. Some useful unary operations map L p to IR, to [−1, 1], or to ĪR+. The
transforms described in Section 0.1.12 map L 1 to L 1 .
0.1.9.1 Convolutions and Covariances and Correlations
The convolution of the functions f and g is
(f ⋆ g)(t) = f(x)g(t − x)dx. (0.1.69)
Theory of Statistics c○2000–2013 James E. Gentle
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