Harmonic Analysis and the Theory of Probability

CHAPTER 1

APPROXIMATIONS

1.1. Approximation of functions at points

In ordinary Euclidean space ,

for any dimension , the ordinary Lebesgue measure element will usually be denoted by . A function 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

We note that for , (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

and for fixed δ>0 the point set 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.

For a measurable function 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 (1.1.7) this is , 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)-norm, ,

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

the summation extending over all lattice points .

Next, (1.1.12) holds in particular if we have for all ξ 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

for , 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) in , 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 Sk−1 being We 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 ω0 = 2 and , 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 which leaves the origin fixed. Now, for , 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 we take a family 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(x) ; and (iii) it is endowed with a norm such 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 .

But if 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 ’

and it is a basic fact of the Lebesgue theory that their linear combinations are for every 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(u). However, if we take any set of functions each of which is bounded and uniformly continuous, and then form their smallest Banach closure with respect to the norm for a set A of finite Lebesgue measure, then their translation functions are continuous. If we choose for 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, ξ) being first and 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 a function whose values are not ordinary complex numbers but more generally elements of a Banach space the norm of which will be denoted by For we employ the concept of (strong) measurability and (strong) integrability as introduced by this author (the so-called Bochner integral), and is bounded in norm

then for numerical KR(ξj) in L1(Ek) there exist the approximating functions

as functions again with values in . We have

and thus continuity in norm at a point x,

implies convergence in norm

and hence the following conclusion:

THEOREM 1.4.1. Theorems