Abstract
Let be a topological space equipped with a complete positive -finite measure and a subset of the reals with as an accumulation point. Let be a nonnegative measurable function on which integrates to in each variable. For a function and , define . We assume that converges to in , as in . For example, is a diffusion semigroup (with ). For a finite measure space and , select real-valued , defined everywhere, with . Define the distance by . Our main result is an equivalence between the smoothness of an function (as measured by an -Lipschitz condition involving and the distance ) and the rate of convergence of to .
1. Introduction
One of the questions that arise in harmonic analysis is the connection between the smoothness of a given function and the rate of approximation by members of a specified family of functions. An important example is the relationship between the smoothness of a function and the speed of convergence of its diffused version to itself, in the limit as time goes to zero. As mentioned in the Introduction of [1], for the Euclidean setting and the heat kernel, see for example [2, 3].
In a more general setting, for a diffusion semigroup on a topological space with a positive -finite measure given, for , by an integral kernel operator: , Coifman and Leeb in [1, 4] introduce a family of multiscale diffusion distances and establish quantitative results about the equivalence of a bounded function being Lipschitz and the rate of convergence of to , as . The respective authors of [5–7] consider different aspects of the connection between the smoothness of a function and the rate of convergence of its diffused versions to itself.
As mentioned in, for instance, the Introductions of [5–7], the interest in diffusion semigroups is natural since they play an important role in analysis, both theoretical and applied. Diffusion semigroups include the heat semigroup and, more generally, as discussed in, e.g., [8], arise from considering large classes of elliptic second-order (partial) differential operators on domains in Euclidean space or on manifolds.
For examples of theoretical results involving diffusion semigroups, the interested reader may refer to Chavel [9], Cowling [10], Stein [8], Sturm [11], and Wu [12]. Some applications of diffusion semigroups to dimensionality reduction, embedding, clustering, data representation, manifold parametrization, and multiscale analysis of complex structures can be found in, e.g., [13–22]. Various definitions and procedures for efficient computation of natural diffusion distances can be found in, e.g., [1, 4, 23, 24].
In the present work, we consider a more general family than a diffusion semigroup. For a subset of the reals having as an accumulation point, for , let be a nonnegative measurable function on which integrates to in each variable. For a function and , define . We assume that for every , , as . No assumption is made that the family is symmetric or is a semigroup nor is anything assumed about other than that has as an accumulation point.
For a finite measure space , selecting for every , we define a distance between points by . We next introduce an version of being Lipschitz (relative to ) using this distance . Our main result is that a function is -Lipschitz if and only if we have an estimate of the rate of convergence of to , namely, , where , as .
Our paper is organized as follows. Following a notation and assumptions section (Section 2), we state the main definitions, provide some examples, and establish our results in Section 3. The paper ends with the Conclusions and Acknowledgments sections.
2. Notation and Assumptions
Let be a topological space equipped with a complete positive -finite measure. The measure on will be denoted by and . is a finite measure space, with measure denoted by . We assume all spaces involved are such that Fubini’s theorem holds on any product of these spaces; e.g., the spaces are -finite. All functions are assumed to be real-valued and measurable on the respective spaces; in particular, functions of several variables are assumed to be measurable on the appropriate product spaces.
will denote a subset of the reals, with as an accumulation point. From now on, will mean . For every , let be a nonnegative measurable function on with the property that . For and a function , define by . We assume that for every , , as .
No assumption is made that the family is symmetric or is a semigroup nor is anything assumed about other than that has as an accumulation point.
Note that is indeed bounded on with norms not exceeding one, since
In particular, .
We will define a family of symmetric distances on satisfying the triangle inequality with the following properties for every :(i) for every (ii), as
3. Main Definitions and Results
We start by describing the family of symmetric distances on .
Definition 1. Select a finite measure space. For each, select, defined everywhere, with. Note that somemay be chosen to beidentically. Then, the distanceis given by
Clearly, is symmetric and satisfies the triangle inequality (the latter fact follows from the triangle inequality for ).
Before looking at some examples of such distances, we define our -Lipschitz condition.
Definition 2. For, we say thatis-Lipschitz (relative to) iffor every .
Now let us consider some examples of distances . For the first one, let be a bounded subset of having (some) finite measure . Let , with indicating unit masses assigned at each point of . For , let , for , where is a suitable constant to ensure that . Then, , a multiple of the Euclidean distance on .
For our second example, let be a finite measure space with measure . Let , with , where . For , let . Clearly, . Then,an analog of the distance considered by Coifman and Leeb in [1, 4] for a semigroup. Note that and are normalized in with respect to as well.
To gain some understanding of this distance (although we will use the case of with Lebesgue measure, not a finite measure space), let us calculate the distance for the basic case when with Lebesgue measure and is the heat flow semigroup. (While the derivation right after the statement of Proposition 2.6 in [5] by Coifman and Goldberg has a calculation of this distance, we present a more detailed computation here.)
We easily see that , where is the Euclidean distance between the points and . Thus, .
If , is bounded away from for , so .
If , write
For the first summand, observe that . For the second summand, an easy calculation shows that, for ,
Combining with the estimate for the first summand, and with the case , we obtain that for ,while for ,
Our third example is a variation of our second example above. As in the second example, let be a finite measure space with measure and let , with , where . For , let . Clearly, .
Then, our new distance is given by
(For a related example, see Section 4 of [23] and the very last example in Section 2 of [5].)
Let us specialize to the case of a symmetric diffusion semigroup with the following additional requirement: is constant over (but varies with ). Let denote the value of for every . Under these assumptions, using the semigroup property, we easily obtain
In the very special subcase of equipped with Lebesgue measure and the heat flow semigroup,and we thus obtain the same estimates for as for above.
We now return to the general development. The following simple result is the key tautology to prove our Theorem 6.
Proposition 3. For, .
Proof. Using Fubini’s theorem and the assumption that integrates to in each variable, we see that
For , letting , we obtain the following result.
Proposition 4. .
Proof. Using the definition of , Fubini’s theorem, and Proposition 3, we observe that
Corollary 5. for every and as.
Proof. Since and , the result that , for every , follows from Proposition 4. From one of our initial assumptions that for every , , as , we obtain that , as , for every . Hence, , as , by the dominated convergence theorem.
Recalling Definition 2, we can now prove the following theorem, which is of interest only due to Corollary 5.
Theorem 6. For and, is-Lipschitz if and only if, for every.
Proof. First, suppose that is -Lipschitz. Then, by Proposition 3, we havefor .
Conversely, suppose , for every . Then, by Proposition 3 again,Thus, is -Lipschitz.
It is easy to see that , so Theorem 6 establishes an equivalence between being -Lipschitz and having an estimate of the speed of convergence of to , as .
Note that if is a symmetric semigroup (and ), thenso . Hence, if is a symmetric semigroup, .
4. Conclusions
For a topological space equipped with a complete positive -finite measure, a finite measure space, and selecting everywhere-defined real-valued for every with , we have defined a distance by .
For a subset of the reals having as an accumulation point and for , letting be a nonnegative measurable function on which integrates to in each variable, we have considered bounded operators on given by . Assuming that for every , , as , we have shown that , for every if and only if , where , as .
Data Availability
No data were used to support this study.
Disclosure
Ramapo College had no involvement with the writing or submission for publication of this work.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
As for the last many years, we are grateful to Raphy Coifman for his continued willingness to discuss mathematics with us. The first author was partially supported by sabbatical and Faculty Development Funding from Ramapo College of New Jersey.