Harmonic Analysis and the Theory of Probability
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About this ebook
Mathematician Salomon Bochner wrote a pair of landmark books on the subject in the 1930s and 40s. In this volume, originally published in 1955, he adopts a more probabilistic view and emphasizes stochastic processes and the interchange of stimuli between probability and analysis. Non-probabilistic topics include Fourier series and integrals in many variables; the Bochner integral; the transforms of Plancherel, Laplace, Poisson, and Mellin; applications to boundary value problems, to Dirichlet series, and to Bessel functions; and the theory of completely monotone functions.
The primary significance of this text lies in the last two chapters, which offer a systematic presentation of an original concept developed by the author and partly by LeCam: Bochner's characteristic functional, a Fourier transform on a Euclidean-like space of infinitely many dimensions. The characteristic functional plays a role in stochastic processes similar to its relationship with numerical random variables, and thus constitutes an important part of progress in the theory of stochastic processes.
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Harmonic Analysis and the Theory of Probability - Salomon Bochner
Indexes
CHAPTER 1
APPROXIMATIONS
1.1. Approximation of functions at points
,
will also be written briefly as f(ξ) or f(ξj), and we will also put
We take a family of functions
also called ‘kernels’, subject to the following assumptions. The index R ranges over 0 < R< ∞ and has continuous and occasionally only integer values. For each R, KR(ξj) is defined and Lebesgue integrable over Ek, so that the integrals
with K0 independent of R; and, what is decisive, for each δ>0, no matter how small, we have
, (1.1.2) implies (1.1.3) with K0= 1.
Starting from an integrable function K(ξ1, …ξk) with
then this is a family as just described, since by the change of variables Rξj → ξj, j = 1, …, k, we obtain
converges to the empty set as R→∞. Sometimes a statement will be intended only for such a special family of kernels, as will be indicated by the context.
in Ek we introduce, if definable, the approximating functions
and since (1.1.2) implies
and our first statement is as follows:
THEOREM 1.1.1. IF f(x) is bounded in Ek
at every point x at which f(x) is continuous Also, if f(x) is continuous in an open set A, then the convergence is uniform in every compact subset .
Proof. We have
and by continuity of f(x) at x this is small for δ sufficiently small. However, for δ fixed (small) we have
and by , which is small for large
R by explicit assumption (1.1.4), q.e.d.
The global requirement (1.1.7) was only needed for obtaining
and it can be relaxed if we correspondingly tighten the assumptions on KR(ξ). For any measurable set A in Ek we can introduce the Lp(A,
and for A = Ek this simply is
Also, if A is the set
and if f(ξ1, …, ξk) is (multi)-periodic with period 1 in each variable, then the Lp(Tk)-norm is the Lp-norm of f(x) over a fundamental domain of periodicity. Now, it follows from the Holder inequality
that if, for a given KR(ξ), (1.1.8) holds for every function for which
then it also holds for every function with finite Lp(Ek) or Lp(Tk)-norm. Now, (1.1.11) means that |f(x) | becomes bounded after having been averaged over a Tk-neighbourhood of each point, and for such an f(x), the integral
is definable, whenever we have
.
Next, Ek
for some C > 0, no matter how large, and some ρ >0, no matter how small. Also, if we form the special family (1.1.5), then the estimate
, and this secures relation (1.1.8) under the assumption (1.1.13). We do not claim that the mere condition (1.1.12) would secure (1.1.8), but it could be shown that the condition
which falls between (1.1.12) and (1.1.13) would already suffice, and hence the following theorem:
THEOREM 1.1.2. For a family of the form (1.1.5), if the kernel K(ξ) satisfies (1.1.13), or only (1.1.14), then theorem 1.1.1 also applies if, globally, f(x) has a finite norm Lp(Ek) or only Lp(Tk).
and in particular for the simple exponential
The kernel (1.1.15) is a product kernel, in the sense that we have
where K⁰(ξ) is a kernel in E1.
Another product kernel is the (nonperiodic) Fejer kernel
which, however, although it satisfies (1.1.12), does not satisfy (1.1.14) and thus could not be used in theorem 2. With a kernel K⁰(ξ) we may also form the multi-index kernel
and most statements would be valid if R1, …,Rk tend to ∞ independently of each other, but we will not pursue this possibility.
Of paramount importance is the Gaussian kernel
which in addition to being a product kernel is also, antithetically, a radial function, meaning that there is a function H(u, such that
We will take as known the formula
and this time we obtain for the function (1.1.17) the approximation
For any radial kernel (1.1.21) it is profitable to introduce in
polar coordinates
in which case the volume element dυξ is the product of tk−1 dt with the volume element dωη on
the total volume of SkWe then obtain
is the spherical average of our function at distance t from the given point x. By Fubini’s theorem, fx(t) exists for almost all t, and we are always permitted to put fx(0)= f(Xj), and a glance at the term I1(R ; x) in the proof to theorem 1.1.1 leads to the following conclusion:
THEOREM 1.1.3. In theorems 1.1.1 and 1.1.2, if K(ξj) is a radial function, then locally it suffices to assume that the spherical average fx(t) → fx(0) as t → 0, which is a weaker assumption than continuity proper.
For k = 2 we have w1 = 2π and
However, for k = l we have ω, and radiality means evenness, K(−ξ) = K(ξ). For k = 1, a function is even if it is invariant with respect to the (then only nontrivial) orthogonal transformation ξ′ = −ξ , radiality means invariance with respect to the entire group of such orthogonal transformations and fx(t) was an average over this group. However, if K(ξj) is invariant with respect to a subgroup only, then the function f(x+ξ) may be averaged correspondingly. Thus if K(ξ1, …, ξk) is even with respect to each ξj separately, then it suffices to assume in theorems 1.1.1 and 1.1.2 that the averaged function
shall be continuous at ξ = 0.
Turning for a moment to the smoothing operation (1.1.16) we note that by iterating it (or by some such procedure) we obtain the following result:
LEMMA 1.1.1. A continuous function f(Xj) in Ek which is 0 outside a compact set is a limit, uniformly in Ek, of suchlike functions each of which is of class C(r) (that is, has continuous partial derivatives of order ) for any fixed r.
The conclusion also holds more precisely in the class C∞, but of this we will not make use in primary contexts.
1.2. Translation functions
In Ek of functions {f(x)} with the following properties: (i) it is a group of addition, meaning that if f, g then f − g ; (ii) it is invariant with regard to translations, that is, if f(x, and if for any u = (u1, …, uk,) in Ek, we define
then fu(xsuch that
and, what is important, this norm is invariant, that is,
With any f we associate a certain non-negative function in Ek: (u1, …, uk), namely, the function
and we call it the translation function of f. It has the following properties. First,
by (1.2.1). Next, due to
Next, we have
the last by (1.2.3), and on putting υ = − u we obtain
and finally for f, g we have
but we also have
and if herein we replace − u by u + v we obtain
Combining (1.2.10) and (1.2.11) and also using (1.2.5) we obtain
and hence the following conclusion:
LEMMA 1.2.1. If a translation function is continuous at the origin it is uniformly continuous throughout.
Next, by the use of (1.2.6) and (1.2.8) we now obtain by a familiar reasoning the following conclusion:
LEMMA 1.2.2. If 0 is a dense (in norm) subset of and if τf(u) is continuous in u for f in 0 it is continuous for f in .
is a normed vector space, more can be stated.
LEMMA 1.2.3. If is a normed vector space and if τf(u) is continuous in u for a set 0 whose linear combinations are dense in , then it is continuous for all of .
This follows from
Now, for all finite multi-intervals
we introduce the ‘characteristic functions ’
dense in the Lp(Ek)-space with the norm
On the other hand, we have
and it is easy to verify that this tends to 0 as | u | → ∞. Similarly, if we introduce for the periodic functions
the Lp(Tk)-norm
then linear combinations of periodic functions of the form (1.2.13) are again dense in norm. Hence the following conclusion:
THEOREM 1.2.1. For functions in Lp(Ek) and (periodic) functions f in Lp(Tk, the translation functions are (bounded) and continuous.
We note that the general norm as defined by formula (1.1.9) is invariant with respect to translations, but we do not at all claim that every function with a finite norm of this kind has a continuous τf(ueach of which is bounded and uniformly continuous, and then form their smallest Banach closure with respect to the norm for a set A the simple exponentials
and for A the set Tk, then the smallest closure is composed of the almost periodic functions of the Stepanoiff class Lp, to which we will sometimes refer incidentally.
1.3. Approximation in norm
We will now state a certain proposition first in a general version heuristically and then in a specific version precisely.
THEOREM 1.3.1 (heuristic). If τf(u) is continuous then the approximating function
converges to f(x) in norm as R → ∞.
Reasoning. For a finite discrete sum we have
and this suggests for (1.3.1) the estimate
But the last term is the value of
for x = 0, and for bounded continuous τf(ξ) this tends to 0 as R → ∞ as in theorem 1.1.1.
Assume now specifically that f(x) is in L1(Ek). The function
is measurable in (x, ξ), and we have
and since the last term is finite, it follows by Fubini’s theorem that the integral (1.3.1) exists for almost all x and is an integrable function in x. This being so, we now obtain
and this time rigorously. This argument also works for (periodic) f L1(Tk), if we replace one of the two symbols Ek by Tk, and thus we obtain the following theorem, at first for p = 1 :
THEOREM 1.3.2. If f(x) belongs to Lp(Ek) or to periodic or Stepanoff almost periodic Lp(Tk), then the integral (1.3.1) exists for almost all x, is a function of the same class with
For p > 1 it is necessary to apply the Holder-Minkowski inequality
for H(x, ξand then
and B = Ek and A = Ek or Tk.
1.4. Vector-valued functions
Theorem 1.1 on convergence at a point and theorem 1.3.2 on convergence in strong average can be brought together by a third theorem embracing them both.
We define in Ek is bounded in norm
then for numerical KR(ξj) in L1(Ek) there exist the approximating functions
. We have
and thus continuity in norm at a point x,
implies convergence in norm
and hence the following conclusion:
THEOREM 1.4.1.