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We prove the existence of series , whose coefficients are in and whose terms are translates by rational vectors in of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener’s algebra , , , , for every , and the space of measurable functions. Applying this theory to particular situations, we establish approximations by such series to solutions of the heat and Laplace equations as well as to probability density functions.
► The existence of universal series for Wiener’s algebra is proved. ► The existence of universal series for further function spaces is proved. ► The theory of universal series is applied to the heat and Laplace equations.
Starting from any random variable and taking appropriate averages, we end up in the limit with a normal distribution. Conversely, starting from the normal distribution is it possible to reach, by some limiting process, any random variable? The answer is in the affirmative, because any probability density can be approximated in the norm by a convex combination of normal densities , . It turns out that this approximation result can be put in the framework of universal series, strengthened significantly, and generalized. More precisely, in the present paper we prove the following results (see Theorem 3.1).
Let be an approximation to the identity on , and let be an enumeration of . If is an enumeration of all , then there exists a sequence in such that the partial sums of the series are dense in , the so-called Wiener algebra.
Moreover, thanks to the density properties of to several classical spaces, we also show the following (see Theorem 3.2).
The partial sums of the series are also dense, with respect to their natural topologies, in for every , the space of continuous functions vanishing at infinity , the space of continuous and bounded functions , and the space of measurable functions .
Similar results can be obtained if we replace by , or the space of all real sequences. We show, however, that the space cannot be replaced by . The proof is based on the fact that, if is an approximation to the identity and is a function in on , then approximates , as in various ways, and that each convolution , being an integral, can be approximated by a Riemann sum. Thus some linear combinations of translates of approximate . This fact, according to a recently developed abstract theory of universal series , , implies the generic existence of such series, and in particular implies the above-mentioned approximation results.
The same argument extends naturally to obtaining universal series for solutions to the heat or Laplace equations. Here, what we deal with is that every such solution fits together with initial or boundary data through a convolution with an appropriate approximation of the identity: the Gauss kernel for the parabolic case and the Poisson kernel for the elliptic one. The new element here is that we do not approximate the initial or boundary state but rather the solution itself. The series obtained are valuable in the context of partial differential equations (PDEs) since the derivatives of all orders of the approximating partial sums converge to the corresponding derivatives of the solution.
The above approximation results and in particular those related to probability and partial differential equations are new contributions to the theory of universal series and lead outside the initial range of applications of Baire’s category theorem, as described in the excellent surveys on universal series , .
This paper is organized as follows. Section 2 contains mainly background material on universal series and Wiener’s algebra . In Section 3, we prove the main approximation results of the paper. In Section 4, we state the results of Section 3 in terms of probability densities. In Section 5, extending the approximation result of Section 3, we prove the existence of series whose partial sums converge uniformly on compact sets, along appropriate subsequences, to bounded solutions of the heat equation, in . We also show that the derivatives of all orders of the partial sums converge to the derivatives of the corresponding solution. Finally, in Section 6, we obtain analogous approximation results to the ones obtained in Section 5 for solutions of the Dirichlet problem for Laplace’s equation in the upper half space of as well as in the unit ball of . In all series, the coefficients belong to.
Let be a real vector space, and consider the metric space , where is compatible with and is invariant under translations. If is a fixed sequence of elements of , by we denote the class of sequences in for which the partial sums form a dense set in . Observe that if and then there exists a sequence of natural numbers , which can be taken to satisfy for all , such that
Thus each , whenever , generates an unrestricted universal series.
Of special interest is the case where the coefficients of the universal series are required to belong to a particular subspace of the space of all real sequences, , where by we denote the set of positive integers. These are the restricted universal series for which the following result is established in , .
Let be a vector subspace of the set of all sequences in and let be a metric on compatible with that is invariant under translations. Setting , we moreover assume that the following hold.
Then we have the following.
In this paper, we deal with series whose coefficients are sequences belonging to the intersection of all spaces for .
Let be fixed and let . We define the metric space by
with and being the usual norm.
We note that the definition above makes sense since , whenever ; hence . The space is a Fréchet space, and one can check with no difficulty that the postulates (A1)–(A4) are satisfied.
Let be a real vector space with translation-invariant metric compatible with , and let be a fixed sequence in . Let be the set of scalar sequences defining universal series. Let be fixed. The following statements are equivalent.
The direction is obvious. We prove . Appealing to Theorem 2.1(2), it suffices to show that, given and , there exist scalars so that the following approximations are accomplished:
Choose, first, so that . Applying, next, (1) with , where, as before, , and since , there exist so that
Setting ,and recalling that , whenever , we compute, by the choice of ,
Thus , and the proof is complete. □
As a kind of universal function space, we introduce the following.
The space of all for which , endowed with the norm
is a Banach space and is called Wiener’s algebra .
In what follows, whenever we write we will always mean the norm. Simplifying notation, we set for .
One immediately sees that and that , whenever . Moreover, is continuously and densely embedded in for (see [5, Chapter 6.1]).
The proofs in this paper make use of several convenient properties of this space; in particular, the fact that it is densely and continuously embedded in several common function spaces, see for example  or , plays an important role.
In the proof of the main outcome, we will need the following straightforward result.
Let . Then, for each , the translation is in , and
Moreover, for a fixed , the map
Since, for each , , assuming first that , we conclude that . Otherwise, each cube , of unit edge, is contained in a cube with integral vertices with double-size edge. From this observation, (2.2) follows.
Let and be given. Then
where is arbitrary. The first sum may become arbitrarily small, less than say, for the right choice of as the tail of a convergent series. Indeed, let be such that . Choose such that , for , where , say. Then
The second sum may become less than , for sufficiently close to , by the uniform continuity of . □
This lemma shows that Wiener’s algebra is a Segal algebra with the translation-invariant and equivalent norm
In fact, it is the first example of a Segal algebra. See also [10, Chapter 6.2].
A family of integrable functions on satisfying
where and are positive constants independent of , is called an approximation to the identity, or approximate identity. The reason is explained by the fact that, if is an integrable function and , that is
then , as , in various ways. In the context of Wiener’s algebra, we have the following.
Assume that the family satisfies conditions (2.3). Let and . Then and , in the norm, as .
This result is shared by all Segal algebras. In , Proposition 6.2.4 shows that Segal algebras are ideals in , and the proof of Proposition 6.2.8 in the same book shows the approximation property. Moreover, Lemma 2.3 is valid for all homogeneous Banach spaces, if the functions are generated by one function (see ), as is done below.
The typical way of constructing approximate identities is that, given a function satisfying and , one defines
Then each satisfies conditions (i) and (ii) of (2.3).
We consider a family of continuous functions satisfying (2.3), where and are constants independent of . Note that, due to the assumed continuity and property (2.3)(ii), it is easy to show that for all .
We set to be an arbitrary but fixed enumeration of the countable collection of the functions ; that is,
Let be as in(3.1). Then there exists a sequence in with the property that, given a function , there exists an increasing sequence in such that
Moreover, the set of such universal sequences is a dense set in , and it contains a dense vector subspace of except the zero sequence.
We prove the theorem by showing that condition (2) of Theorem 2.1 is fulfilled. For a given approximation error , we choose a so that for all (see Lemma 2.3). We will show that, due to the properties of Wiener’s algebra , and in fact this would work for every Segal algebra, it is possible to approximate the convolution by a Riemann sum, and furthermore to use only translates by rational vectors. First, we approximate the convolution by an integral over a rational cube .
via (2.2). For large enough, the right-hand side gets smaller than . Since we can interpret the convolution as a Riemann integral, we can approximate it in the sense by Riemann sums, which are, in this situation, weighted sums of translates of . It is enough to take the translations on a uniform, rational grid in with ‘gridsize’ , i.e. , where, for with , the intersections are empty sets. Reshuffling sums and integrals gives
from which, using the continuity result of Lemma 2.2, and the fact that is uniformly continuous, by taking small enough, leads us to
So we have found that
for an and the right choice of the coefficients . Notice that there are fewer than coefficients in this sum, all smaller than or equal to . So the -norm of the coefficients is smaller than or equal to , which could be made arbitrary small by again choosing sufficiently small. Lemma 2.1 finishes the proof. □
We note at this point that there exists no universal series with coefficients. Indeed, supposing that such a series exists, we are led via (2.4) to
Thus the series converges almost everywhere on to an function. This is, however, in contradiction with the fact that its partial sums are dense in .
The assertions in (1), (2), (3), and (4) are easily shown by the fact that is densely and continuously embedded in and with respect to the natural topologies, respectively.
(5) A combination of Lusin’s theorem and Urysohn’s lemma renders a sequence of continuous functions with compact support converging to almost everywhere. (2) implies that for each there exists such that . Then, the sequence of partial sums converges to almost everywhere.
(6) The proof is similar to the one in (5). □
In this section, we consider approximations of the identity generated, via (2.5), by probability densities. To simplify the presentation, we consider the one-dimensional case. Possible such densities are the Laplace density: , the standard normal density: , the Cauchy density: , or any other continuous density. Hence, we prove the existence of universal series whose terms are modified rational translates of these densities. Translating Theorem 3.2 in this setting, we have the following.
Let be a continuous probability density and let be an arbitrary but fixed enumeration of the set . For , we set , and we consider the family . Then, there exists a sequence in such that the series has the properties of the series of Theorem 3.1, Theorem 3.2. Moreover, the set of such universal sequences is a dense set in , and it contains a dense vector subspace of except the zero sequence.
Notice that, if is the standard normal density, then
the normal density with mean and variance . In analogy to statement (1) of Theorem 3.2 for , we recover the known result that any probability distribution is approximated in the norm by a finite linear combination of normal distributions. By Proposition 4.1 it follows, however, that this approximation is accomplished by other probability distributions as well and in every norm, .
We consider the initial value problem for the heat equation
where is the Laplacian and is a continuous function. Let
and set . Then the family is an approximation to the identity. This is the Gauss or heat kernel. We recall that the function is a solution of (5.1a), for each , and, if is a continuous and bounded function, then the unique bounded solution of (5.1a) and (5.1b) is given by
and the initial condition is attained as . Moreover, the map
is well defined and continuous if we regard the topologies of uniform convergence on compact sets on and the uniform convergence on compact sets of arbitrary derivatives on . The latter topology is given by the semi-norms
so the continuity inequalities
are valid for the right choice of . Furthermore, is a complete metric space with metric
Let be an enumeration of all rational vectors in and let us set . Then there exist series with , having the property that, given any bounded solution of the heat equation with continuous and bounded initial data , there exists an increasing sequence in such that
uniformly on compact sets of , for every multi-index with and for every nonnegative integer . Moreover, the set of such universal sequences is a dense set in , and it contains a dense vector subspace of except the zero sequence.
We again show assertion (2) of Theorem 2.1 by inferring Lemma 2.1. Let be the bounded solution of the heat equation with boundary data , or , and . For a compact set there is an such that on the one hand, and on the other . Therefore, holds for all . Since and its derivatives are uniformly continuous on , we can further choose such that
for all . By the reasoning in the proof of Theorem 3.1, we can also take in a way that for there are coefficients and points such that
for a fixed and corresponding to the appropriate constant of inequality (5.3). Due to the continuity inequality in (5.3) mentioned above, and using the well-known property of the Gaussian kernel , we derive
follows, since . By the triangle inequality, we now establish
Again, by continuity and density, this result is also valid for . Lemma 2.1 finishes the proof. □
If a sequence of partial sums of the series approximates the solution , then the same sequence approximates the initial condition for small, uniformly on compact sets of . Indeed, assuming that as , then, for small, via Lemma 2.3, we have . Hence, for , we finally obtain
as long as .
Let be the upper half space in , that is, , and consider the boundary value problem
where is a bounded, continuous, real-valued function on . The corresponding solution is given by
and the boundary value is realized as the limit of the integral as . The function in the integrand is the Poisson kernel for the upper half space, and it is defined by
where is the surface area of the unit sphere in . is a harmonic function, and, defining , the family is also an approximate identity, and hence a result that is analogous to the Theorem 5.1 result can be proved.
If is an enumeration of the rational vectors in and , then there exist series with coefficients in having the property that, given any function harmonic on , and with continuous and bounded boundary data, there exists an increasing sequence in such that
uniformly on compact sets of , for every multi-index , . Moreover, the set of such universal sequences is a dense set in , and it contains a dense vector subspace of except the zero sequence.
Let be a function harmonic on and continuous on the closure of , and let be compact. Since , for all , the integral may become arbitrarily small outside a compact set containing . Hence the integral in (6.2) is approximated uniformly on by Riemann sums
where are cubes covering . Hence, by Theorem 3.2(4), there exists a series with and , so that a sequence of partial sums converges uniformly on compact subsets of to . The proof of the statement for the derivatives follows from the fact that the sequence of harmonic functions , being convergent to uniformly on compact subsets of , also satisfies that, for any multi-index , in the same sense. The proof is complete. □
Let us replace in (6.1) by , the unit ball of , and by , the unit sphere in , where now is a continuous function on . We recall that the function
where is the area measure on , is harmonic on and continuous on . It is therefore the solution of the boundary value problem. The expression
in the integrand is the Poisson kernel for . For , the function is harmonic on and, moreover, . We can then write the solution of (6.1) as
If, on the other hand, is a continuous function on and harmonic on , then, by the uniqueness of the solution of (6.1), , where is the restriction of on . Although the integral representation in (6.3) is not a convolution of the boundary data with the Poisson kernel for the ball, the integral can be broken into a finite number of integrals over pieces of and then approximated uniformly over compact subsets of by Riemann sums, where the belong to a countable dense subset of . Hence a theorem like Theorem 6.1 can be proved. Alternatively we may work as follows.
Writing each point in spherical coordinates with , where for and , we have the following.
Let be an enumeration of all rational vectors in and let . Let . Then there exist series with , having the property that, given a function harmonic on and continuous on , there exists an increasing sequence in such that
uniformly on compact sets of for every multi-index , . Moreover, the set of such universal sequences is a dense set in , and it contains a dense vector subspace of except the zero sequence.
The proof is similar to that of Theorem 6.1, and is omitted.
We conclude this section with a few comments.
Although Theorem 6.2 is stated and proved in the context of boundary value problems for the steady-state equation, the result holds true in a more general setting. In particular, the series obtained is universal in the space of functions harmonic in the unit ball . Indeed, every such function is approximated in the topology of uniform convergence on compact sets of by functions harmonic on and continuous on . The same result also holds for harmonic functions on . The analogous statement is also true for solutions of the heat equation in .
We note, as in Remark 3.1, that no universal series exists with coefficients. The supposition that it does leads to
for the upper half space , and
for the ball, where , via the absolute convergence of the series . But this contradicts that the partial sums are dense in the relevant spaces of harmonic functions.
An earlier version of this work was completed while the first and the last author were visiting the Department of Mathematics and Statistics of the University of Cyprus. It appeared as technical report: V. Nestoridis and V. Stefanopoulos, Universal Series and Approximate Identities, TR-28-2007, Department of Mathematics and Statistics, University of Cyprus. When some authors refer to “V. Nestoridis and V. Stefanopoulos, Universal Series and Approximate Identities”, they mean the above-mentioned technical report. In particular, “A.G. Bacharoglou, Approximation of Probability Distributions by Convex Mixtures of Gaussian Measures, Proc. AMS, 138, (2010), no. 7, 2619–2628” uses a lemma which does not exist in the present version.
V.N. and V.S. are grateful for hospitality at the University of Cyprus. We also thank K. Fokianos and A.M. Olevskii for several fruitful discussions we had while preparing this paper.
V.N. and V.S. were partially supported by Caratheodory program#C164. S.S. was supported by the FWF GrantSISE S10602.
Communicated by Hans G Feichtinger