Distributional Point Values and Delta Sequences
aa r X i v : . [ m a t h . F A ] F e b DISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES
RICARDO ESTRADA AND KEVIN KELLINSKY-GONZALEZ
Abstract.
Recently Sasane [18] defined a notion of evaluating a distribution at a pointusing delta sequences. In this paper, we explore the relationship between generalizationsof his definition and the standard definition of distributional point values. This allows usto obtain a description of distributional point values via delta sequences and a characteri-zation of when a distribution is actually a regular distribution given by bounded function.We also give a characterization of limits in a continuous variable by the existence of thelimits of certain sequences. Introduction
Distributional point values were first defined in one variable by Lojasiewics [11]. Hisdefinition is given as a distributional limit over a continuous variable. In other words, if f ∈ D ′ ( R ) and x ∈ R then we say that f has a distributional point value, equal to γ, at x if(1.1) lim ε → f ( x + εx ) = γ , in the distributional sense, that is, if for all test functions φ ∈ D ( R ) we have that(1.2) lim ε → h f ( x + εx ) , φ ( x ) i = γ Z ∞−∞ φ ( x ) d x . Similarly [12] point values in several variables are defined as a distributional limit overa continuous variable. Point values have been studied extensively and are the first stepin the study of distributional asymptotic analysis and of the study of local properties ofdistributions [2, 5, 15, 16, 17, 23].It is possible to find in the literature other definitions of distributional point values,based on the use of delta sequences.
For instance, in a recent study, Sasane [18] usesthe alternative definition “ f ( x ) = η ” if for all positive and even test functions φ with R ∞−∞ φ ( x ) d x = 1 one has(1.3) lim n →∞ h f ( x + x ) , φ n ( x ) i = η , where { φ n } ∞ n =1 is the standard delta sequence generated by φ, namely, φ n ( x ) = nφ ( nx ) . Naturally the question arises if the two definitions are equivalent. More generally, if F is afamily of test functions, we would like to consider the relationship between the existenceof the distributional point value and the existence of the limit (1.3) whenever the sequence { φ n } ∞ n =1 belongs to F . Interestingly, the two definitions are not equivalent for many classesof delta sequences, in particular for the family considered in [18]. Nevertheless, we areable to show that for some classes they are actually equivalent.
Mathematics Subject Classification.
Key words and phrases.
Distributional point values, delta sequences.
In order to study this problem, we start by studying a very general question aboutlimits. Indeed, in a metric space X, given a function f : X \ { x } → R , then the limit(1.4) lim x → x f ( x ) = L , exists if and only if(1.5) lim n →∞ f ( x n ) = L , for all sequences { x n } ∞ n =1 in X \ { x } that converge to x . The question we would liketo consider is whether the existence of the limit lim n →∞ f ( x n ) for sequences { x n } ∞ n =1 ofa certain family implies that (1.4) is satisfied. In Section 3, we consider the case where X = (0 , ∞ ] , x = ∞ and f continuous, showing that in such cases the existence of thelimit(1.6) lim n →∞ f ( na ) = F ( a ) , for all a > F is a constant function, F ( a ) = L, for all a > , and that lim x →∞ f ( x ) = L. We give examples of other families of sequences for whichlim x →∞ f ( x ) = L might not hold true. We are also able to present, in Section 4, acorresponding result when the function f in (1.6) is not necessarily continuous but justmeasurable and the limit holds almost everywhere.The plan of the rest of the article is as follows. Basic results on distributional pointvalues are briefly discussed in Section 2, while delta sequences are considered in Section 5.Section 6 gives several useful results on the characterization of distributions and functionsusing normalized positive test functions. The main results are given in Section 7, wherewe give equivalent conditions to the existence of point values obtained from a given familyof delta sequences. In particular, we prove that the existence of the distributional pointvalue is equivalent to the existence of the point value for the family of standard deltasequences generated by a positive normalized test function. Then we show that for thefamily employed in [18] the equivalence is the existence of the symmetric distributionalpoint value. We also consider radial delta sequences in several variables, and finish bystudying the family of all delta sequences of normalized positive test functions.2. Preliminaries
We refer to the texts for the basic ideas about distributions [1, 8, 19, 22]. Ideas onthe local behavior of distributions can be found in [2, 5, 16, 17, 23]. In this article, wewill work mainly in the space D ′ (cid:0) R d (cid:1) of distributions on R d , dual of the space D (cid:0) R d (cid:1) ofstandard test functions, that is, C ∞ functions with compact support, with its inductivelimit topology [20].If f ∈ D ′ ( R ) and x ∈ R then [11] we say that f has a distributional point value, equalto γ, at x if lim ε → f ( x + εx ) = γ, in the strong topology of D ′ ( R ) . Equivalently, sincea sequence of distributions converges strongly if and only if it converges weakly [20], if forall test functions φ ∈ D ( R ) we have that(2.1) lim ε → h f ( x + εx ) , φ ( x ) i = γ Z ∞−∞ φ ( x ) d x . ISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES 3
Interestingly, the existence of the distributional limit lim ε → f ( x + εx ) implies thatthis limit is a constant and that the point value exists. On the other hand, if the limit(2.2) lim ε → + f ( x + εx ) = g ( x ) , exists, then g does not have to be a constant, but it will have the jump behavior [21],that is, g is of the form(2.3) g ( x ) = γ − H ( − x ) + γ + H ( x ) , where H is the Heaviside function and γ ± are some constants. Distributions of the form(2.3) are the most general homogeneous distributions of degree 0 in one variable. Alter-natively, (2.2) and (2.3) hold if and if the lateral limits f ( x ±
0) = lim ε → + f ( x ± εx ) = γ ± , exist in D ′ (0 , ∞ ) and f does not have delta functions at x . In several variables point values are defined similarly [12], namely, if f ∈ D ′ (cid:0) R d (cid:1) , then the distributional point value f ( x ) exists and equals γ if lim ε → f ( x + ε x ) = γ, distributionally. In several variables the limit lim ε → f ( x + ε x ) could exist without beinga constant. In fact, if(2.4) lim ε → + f ( x + ε x ) = g ( x ) , then g is homogeneous of degree 0 . Homogeneous distributions of degree zero are givenby a formula of the type(2.5) h g ( x ) , φ ( x ) i = Z ∞ h α ( w ) , φ ( r w ) i D ′ ( S ) ×D ( S ) r d − d r , for a certain distribution α ∈ D ′ ( S ) [5, Thm. 2.6.2]. The distribution α is the thickdistributional value [3] of f at x , namely, f has no delta functions at x and α is thethick limit(2.6) lim ε → + f ( x + rε w ) = α ( w ) , in the space D ′ ((0 , ∞ ) , D ′ ( S )) , that is, for all ρ ∈ D (0 , ∞ ) , (2.7) (cid:28) lim ε → + f ( x + rε w ) , ρ ( r ) (cid:29) D ′ (0 , ∞ ) ×D (0 , ∞ ) = (cid:18)Z ∞ ρ ( r ) d r (cid:19) α ( w ) , The continuous case
We start with a general known result that will be useful in our analysis.
Proposition 3.1.
Let f : (0 , ∞ ) → R be continuous. Suppose that for each a > thesequence { f ( an ) } ∞ n =1 converges, to F ( a ) . Then F ( a ) does not depend on a, that is, (3.1) F ( a ) = L for all a > , for some L, and actually (3.2) lim x →∞ f ( x ) = L .
Proof.
Clearly the function F is constant in each class of the quotient space R / Q , F ( ra ) = F ( a ) if r ∈ Q . Also, F is continuous or of the first Baire class [6, 13], so that the set D of points of continuity of F is dense in (0 , ∞ ) . Let α ∈ D. Let b > . If ε > RICARDO ESTRADA AND KEVIN KELLINSKY-GONZALEZ exists δ > | a − α | < δ implies | F ( a ) − F ( α ) | < ε and there exist r rationalsuch that | rb − α | < δ. Therefore(3.3) | F ( b ) − F ( α ) | = | F ( rb ) − F ( α ) | < ε , and since ε is arbitrary, F ( b ) = F ( α ) . In order to prove (3.2), observe that if ε > , then for each a ∈ [1 ,
2] there exists n = n ( a ) such that | f ( ka ) − L | < ε for k ≥ n ( a ) . This means that(3.4) [1 ,
2] = ∞ [ n =1 \ k ≥ n { a ∈ [1 ,
2] : | f ( ka ) − L | < ε } . Therefore there exists n such that T k ≥ n { a ∈ [1 ,
2] : | f ( ka ) − L | < ε } contains an inter-val I = [ α, β ] with α = β. Observe now that S ∞ k = n kI contains a ray ( B, ∞ ) , since in fact S ∞ k = n kI is a closed ray if n > / ( β − α ) . Hence if x > B then x = ka for some k ≥ n and some a ∈ I and, consequently, | f ( x ) − L | = | f ( ka ) − L | < ε. (cid:3) It is interesting that there are sequences { ξ n } ∞ n =1 with lim n →∞ ξ n = ∞ such that forsome continuous functions f : (0 , ∞ ) → R the limit(3.5) lim n →∞ f ( aξ n ) = G ( a )exists for all a >
0, but the function G is not constant. Indeed, let f ( x ) = sin (2 π ln x )and ξ n = e ( n +1 /n ) . The limit lim n →∞ f ( aξ n ) = sin (2 π ln a ) , exists but it is not constant,of course. 4. The measurable case
We shall now consider an extension of the results of Section 3 to measurable functions.
Proposition 4.1.
Suppose f : (0 , ∞ ) → R is measurable. For a > , suppose that thefunction F defined by (4.1) F ( a ) = lim n →∞ f ( an ) , is well-defined almost everywhere. Then F is constant almost everywhere.Proof. Let us first suppose that f ∈ L ∞ (0 , ∞ ) . Let φ ∈ D (0 , ∞ ) be a test function. For λ > , let us define(4.2) G ( λ ) = Z ∞ f ( λx ) φ ( x ) d x . The function G is continuous because φ ∈ D (0 , ∞ ). For a fixed λ, let us considerthe sequence { G ( λn ) } ∞ n =1 . Since f is bounded, we can apply the dominated convergencetheorem to see thatlim n →∞ G ( λn ) = Z ∞ lim n →∞ f ( λnx ) φ ( x ) d x = Z ∞ F ( λx ) φ ( x ) d x , exists. The Proposition 3.1 then yields that R ∞ F ( λx ) φ ( x ) d x does not depend on λ, (4.3) Z ∞ F ( λx ) φ ( x ) d x = Z ∞ F ( x ) φ ( x ) d x . Therefore, the regular distribution F is constant since F ( λx ) = F ( x ) , λ > , and onlythe constants are homogeneous of degree 0 in the interval (0 , ∞ ) [5], that is, F ( x ) = C, ISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES 5 as distributions. Notice now that the locally integrable function that gives a regulardistribution is unique almost everywhere, so that F ( x ) = C (a.e.) . Let us now consider the case of a general measurable function f for which the limitlim n →∞ f ( an ) = F ( a ) exists (a.e.). We can then define the bounded function(4.4) h ( x ) = arctan f ( x ) . Then lim n →∞ h ( an ) = arctan F ( a ) exists (a.e.). Consequently, arctan F ( a ) is constant, andtherefore so is F ( a ) . (cid:3) In Proposition 3.1, it is shown that in the continuous case not only is F ( a ) = L forall a > , but actually lim x →∞ f ( x ) = L. This is no longer true in the measurable case;for example, if f = χ B , the characteristic function of a set B of measure zero such that B ∩ ( x, ∞ ) = ∅ for all x > , then lim x →∞ f ( x ) does not exist. Of course, this function f is equal almost everywhere to a function e f , the zero function, for which lim x →∞ e f ( x )exists. An example where (4.1) exists for all a > x →∞ e f ( x ) does not exist for anyfunction f such that f ( x ) = e f ( x ) (a.e.) can be constructed as follows. The strategy willbe to construct an unbounded set, A , with measure 0 such that f ( x ) = 0 for infinitelymany x ∈ A Example 4.2.
Let { N k } ∞ k =1 be a sequence of positive integers such that(4.5) kN k < N k +1 . For each k let us choose a non empty open interval B k ⊂ ( N k − , N k ) . Then if j ∈ N ,j (cid:0) k , (cid:1) ∩ B k = ∅ only if N k ≤ j < N k +1 . Therefore, if x ∈ (cid:0) k , (cid:1) , then(4.6) χ B k ( jx ) = 0 for all j ∈ N , x / ∈ A k , where A k = S N k +1 − j = N k j B k . Let now { η k } ∞ k =1 be a sequence of strictly positive numberssuch that the series P ∞ k =1 η k converges and let us further restrict the sets B k by requiringthat µ ( A k ) < η k , for all k. Let A = lim sup k →∞ A k = T ∞ k =1 S ∞ q = k A q . Then µ ( A ) = 0 . Let us now define the function f : (0 , ∞ ) → R by(4.7) f ( x ) = ∞ X k =1 χ B k ( x ) . If e f ( x ) = f ( x ) almost everywhere, then the limit lim x →∞ e f ( x ) does not exist. Onthe other hand, lim n →∞ f ( nx ) exists almost everywhere. Indeed, it is enough to showthe existence almost everywhere in (0 , , and the limit of f ( nx ) exists and equals 0 if x ∈ (0 , \ A since if x / ∈ A then there exists k such that x / ∈ A k for k ≥ k andconsequently, f ( nx ) = 0 whenever n ≥ N k . An example involving continuous functions can be obtained by a slight modification.
Example 4.3.
Let g be a continuous function in (0 , ∞ ) such that 0 ≤ g ( x ) ≤ f ( x ) andsuch that there exist points ξ k ∈ B k , for all k, such that g ( ξ k ) = 1 . Then lim n →∞ g ( nx ) = 0almost everywhere, but not everywhere , since lim x →∞ g ( x ) does not exist. RICARDO ESTRADA AND KEVIN KELLINSKY-GONZALEZ
The examples show that it is possible for lim n →∞ f ( an ) to be equal to a constant L almosteverywhere but without lim x →∞ f ( x ) existing. We do have a convergence in measure typeresult. Proposition 4.4.
Suppose f : (0 , ∞ ) → R is measurable. Suppose that (4.8) lim n →∞ f ( an ) = L ( a.e. ) . Then for all ε > and all C > , (4.9) lim x →∞ µ ( { t ∈ [ x, Cx ] : | f ( t ) − L | > ε } ) µ ([ x, Cx ]) = 0 , where µ denotes the Lebesgue measure of a set.Proof. Let us denote by G ( x ) the quotient µ ( { t ∈ [ x, cx ] : | f ( t ) − L | > ε } ) / (1 − C ) x. Notice that G is a continuous function in (0 , ∞ ) . Let a > f n ( x ) = f ( nx ) in the interval [ a, Ca ] . Since f n converges to L almost everywhere in this finite interval, it converges to L in measure. This means thatfor all ε > { s ∈ [ a, Ca ] : | f n ( s ) − L | > ε } tends to zero. But thetransformation t = ns gives µ ( { s ∈ [ a, Ca ] : | f n ( s ) − L | > ε } ) µ ([ a, Ca ]) = µ ( { s ∈ [ a, Ca ] : | f ( ns ) − L | > ε } ) µ ([ a, Ca ])= µ ( { t ∈ [ na, Cna ] : | f ( t ) − L | > ε } ) µ ([ na, Cna ])= G ( na ) , so that lim n →∞ G ( na ) = 0 . Proposition 3.1 then yields (4.9). (cid:3) Delta sequences
A sequence { f n } ∞ n =1 of distributions is called a delta sequence if f n ( x ) → δ ( x ) in eitherthe strong or the weak topology of D ′ (cid:0) R d (cid:1) , since the two notions are equivalent [20]. Inother words, { f n } ∞ n =1 is a delta sequence if(5.1) lim n →∞ h f, φ i = φ ( ) , for all φ ∈ D (cid:0) R d (cid:1) . In this article we will be interested mostly in the case when the distri-butions f n are actually smooth functions, but general delta sequences are also of interest,of course. They have been employed in several problems [1], such as the definitions ofpoint values [18] or the definition of products of distributions [9, 10, 14].There are many ways to construct delta sequences. A simple one is the following. Let f be a fixed distribution of rapid decay at infinity, that is, f ∈ K ′ (cid:0) R d (cid:1) . Then all themoments µ k = (cid:10) f ( x ) , x k (cid:11) , exist for k ∈ N d since all polynomials belong to K (cid:0) R d (cid:1) , andthe moment asymptotic expansion (5.2) f ( λ x ) ∼ ∞ X q =0 X | k | = q µ k D k δ ( x ) k ! 1 λ q + d , as λ → ∞ , ISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES 7 holds in K ′ (cid:0) R d (cid:1) [5]. Therefore, when µ = 0 if { ξ n } ∞ n =1 is any sequence of positive numberswith lim n →∞ ξ n = ∞ then(5.3) g n ( x ) = ξ dn µ f ( ξ n x ) , is a delta sequence, generated by f and { ξ n } ∞ n =1 . When ξ n = n for all n, we call thissequence the standard delta sequence generated by f. Another useful construction of delta sequences is provided by the ensuing well knownresult.
Lemma 5.1.
Suppose { ψ n } ∞ n =1 is a sequence of normalized positive test functions in D ′ (cid:0) R d (cid:1) such that supp ψ n ⊂ { x : | x | < r n } , where lim n →∞ r n = 0 . Then { ψ n } ∞ n =1 is adelta sequence.Proof. Let φ be any test function. Then by the first mean value theorem for integrals,(5.4) h ψ n , φ i = Z supp ψ n ψ n ( x ) φ ( x ) d x = φ ( x n ) , for some x n ∈ supp ψ n . Since | x n | ≤ r n → , we obtain that x n → , and consequently, φ ( x n ) → φ ( ) . Thus ψ n ( x ) → δ ( x ) . (cid:3) We now give a notion of point value of a distribution based on delta sequences. Ourdefinition applies to several spaces of distributions, but the cases A = D , E , or S seemthe most relevant. Definition 5.2.
Let A (cid:0) R d (cid:1) be a space of test functions. Let F be a family of deltasequences whose elements belong to A (cid:0) R d (cid:1) . If f ∈ A ′ (cid:0) R d (cid:1) and x ∈ R d we say that thevalue f ( x ) exists and equals γ with respect to F if(5.5) lim n →∞ h f ( x + x ) , φ n i = γ , for all { φ n } ∞ n =1 ∈ F . When this holds we write(5.6) f ( x ) = γ ( F ) . The definition of point value employed by Sasane [18] corresponds to the case when d = 1 , f ∈ D ′ ( R ) , and F is the family of all delta sequences whose elements are thestandard sequences generated from a positive, normalized, and symmetric test functionof D ( R ) . Several lemmas
In this section, we present several results on how positive test functions allow us tostudy many properties of distributions. In particular, we see how positive test functionstell us if a distribution is a regular distribution given by a bounded measurable functionand give us the essential supremum and infimum of such a function. In this section, andonly in this section, we will make a notational difference between a regular distribution f ∈ D ′ (cid:0) R d (cid:1) and the locally integrable function f that generates it as(6.1) h f ( x ) , φ ( x ) i = Z R d f ( x ) φ ( x ) d x , φ ∈ D (cid:0) R d (cid:1) . RICARDO ESTRADA AND KEVIN KELLINSKY-GONZALEZ
In the rest of the article, we will use the same notation, f, for the distribution and thefunction.Let us start with following simple result. Lemma 6.1.
The set of functions of the form (6.2) φ = c ψ − c ψ , where c and c are constants and where ψ and ψ are normalized positive test functionsis the whole space D (cid:0) R d (cid:1) . When d = 1 , the corresponding space with ψ and ψ normalized positive symmetrictest functions is the space of all even test functions.Proof. It is enough to show that the real valued elements of D (cid:0) R d (cid:1) have the form (6.2) forsome positive constants c and c . Let ζ ∈ D (cid:0) R d (cid:1) be such that ζ ( x ) ≥ max { φ ( x ) , } and let ζ = ζ − φ. Then we write ζ j = c j ψ j where the ψ j are normalized positive testfunctions and c j = R R d ζ j ( x ) d x . In the symmetric case we just also ask ζ to be even. (cid:3) Our first characterization using positive normalized test functions is the following.
Lemma 6.2.
Let f ∈ D ′ (cid:0) R d (cid:1) . Then f is a regular distribution in an open set U ⊂ R d , given by a bounded function f ∈ L ∞ ( U ) if and only if there exists a constant M > suchthat for all positive, normalized test functions φ ∈ D ( U ) we have (6.3) |h f ( x ) , φ ( x ) i| ≤ M .
Proof. If f ∈ L ∞ ( U ) . Then when φ ∈ D ( U ) , |h f ( x ) , φ ( x ) i| = (cid:12)(cid:12)(cid:12)(cid:12)Z U f ( x ) φ ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k L ∞ ( U ) k φ k L ( U ) , so that if φ is normalized, |h f ( x ) , φ ( x ) i| ≤ k f k L ∞ ( U ) . Therefore (6.3) holds with M = k f k L ∞ ( U ) . Conversely, if (6.3) is satisfied for some
M > U then |h f ( x ) , ψ ( x ) i| ≤ M k ψ k L ( U ) for all real test functions ψ ∈ D ( U ) , becauseof Lemma 6.1 (or 4 M if complex). This means that f is continuous in D ( U ) , a densesubspace of L ( U ) with the topology induced by L ( U ) in its subspace. Hence, f admitsan extension f ∈ ( L ( U )) ′ ≃ L ∞ ( U ) , and this means that(6.4) h f ( x ) , ψ ( x ) i = Z U f ( x ) ψ ( x ) d x , for all ψ ∈ D ( U ) . Therefore, f is a regular distribution given by the bounded function f in the open set U. (cid:3) In the proof we can see that inf { M : (6.3) holds } ≤ k f k L ∞ ( U ) . In fact, we have more.
Lemma 6.3. If f ∈ D ′ (cid:0) R d (cid:1) is a regular distribution in U, given by a bounded function f ∈ L ∞ ( U ) then (6.5) k f k L ∞ ( U ) = inf { M : (6.3) holds for all positive, normalized test functions } , and (6.6) k f k L ∞ ( U ) = sup {|h f , φ i| : φ ∈ D ( U ) positive, normalized test function } . ISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES 9
Proof.
Clearly inf { M : (6.3) holds for all positive, normalized test functions } is equal tosup {|h f , φ i| : φ ∈ D ( U ) positive, normalized test function } ; let us call this K. We knowthat K ≤ k f k L ∞ ( U ) . To prove the converse inequality, let s < k f k L ∞ ( U ) . Then there exists x ∈ U such that the distributional point value f ( x ) exists and s < | f ( x ) | . If φ is apositive normalized test function, then so are the test functions ϕ λ ( x ) = λ d φ ( x + λ x )for all λ > , and if λ is big enough, ϕ λ ∈ D ( U ) . Since lim λ →∞ h f , ϕ λ i = f ( x ) , wecan find λ such that |h f , ϕ λ i| > s. Consequently,
K > s, and because s < k f k L ∞ ( U ) isarbitrary, K ≥ k f k L ∞ ( U ) . (cid:3) In fact, the same argument in the proof of Lemma 6.3 allows us to obtain the ensuing.
Lemma 6.4. If f ∈ D ′ (cid:0) R d (cid:1) is a real regular distribution in U, given by a function f ∈ L ( U ) then the essential supremum and infimum of f are also given as (6.7) esssup x ∈ U f ( x ) = sup φ ∈D ( U ) ,φ ≥ , R φ =1 h f ( x ) , φ ( x ) i , and (6.8) essinf x ∈ U f ( x ) = inf φ ∈D ( U ) ,φ ≥ , R φ =1 h f ( x ) , φ ( x ) i . We notice that when f ∈ L ( U ) then (6.7) could be + ∞ and (6.8) could be −∞ . Comparison of definitions
We will now study whether the existence of the distributional point value f ( x ) isequivalent to the existence of f ( x ) ( F ) for several families of delta sequences F . Standard delta sequences generated by a positive normalized test function.
In this section we consider the family F of standard delta sequences generated by a positivenormalized test function of D (cid:0) R d (cid:1) . Proposition 7.1.
Let f ∈ D ′ (cid:0) R d (cid:1) . Then f has a thick distributional point value at x ifand only if for all standard delta sequences generated by a positive normalized test functionof D (cid:0) R d (cid:1) , { φ n } ∞ n =1 , the limit (7.1) lim n →∞ h f ( x + x ) , φ n ( x ) i = γ { φ n } , exists.Proof. A standard delta sequences generated by a normalized positive test function φ isof the form φ n ( x ) = n d φ ( n x ) . If the distributional thick point value f x ( w ) = γ ( w )exists, γ ∈ D ′ ( S ) , thenlim n →∞ h f ( x + x ) , φ n ( x ) i = lim n →∞ (cid:10) f ( x + x ) , n d φ ( n x ) (cid:11) = lim n →∞ h f ( x + (1 /n ) x ) , φ ( x ) i = Z ∞ h γ ( w ) , φ ( r w ) i D ′ ( S ) ×D ( S ) r d − d r , exists. Conversely, let φ be a normalized positive test function. If the limit (7.1) existsfor all standard delta sequences generated by a positive normalized test function, it will exist for φ { a } n ( x ) = n d a d φ ( na x ) for all a > . Consequently, if the function Φ is definedas(7.2) Φ ( a ) = (cid:10) f ( x + x ) , a d φ ( a x ) (cid:11) , a > , then(7.3) lim n →∞ Φ ( na ) = lim n →∞ (cid:10) f ( x + x ) , φ { a } n ( x ) (cid:11) = γ n φ { a } n o , exists for all a. Since Φ is continuous, Proposition 3.1 yields that γ n φ { a } n o = γ ( φ ) isindependent of a and actually lim λ →∞ Φ ( λ ) = γ ( φ ) . Hence,(7.4) lim ε → + h f ( x + ε x ) , φ ( x ) i = γ ( φ ) , for all normalized positive test functions. Therefore, Lemma 6.1 yields that the limitlim ε → + h f ( x + εx ) , φ ( x ) i = γ ( φ ) exists whenever φ ∈ D (cid:0) R d (cid:1) . The formula h γ , φ i = γ ( φ ) , defines a distribution γ ∈ D ′ (cid:0) R d (cid:1) , and γ is homogeneous of degree 0 , that is, γ ( t x ) = γ ( x ) , t > . As explained in Section 2, using [5, Thm. 2.6.2] we conclude that γ is obtained from a distribution γ ∈ D ′ ( S ) by the formula(7.5) h γ , φ i = Z ∞ h α ( w ) , φ ( r w ) i D ′ ( S ) ×D ( S ) r d − d r , and that α is the thick distributional value of f at x . (cid:3) Let { φ n } ∞ n =1 be a sequence of test functions. If T is an orthogonal transformation of R d , that is, with | det T | = 1 , then the sequence (cid:8) φ Tn (cid:9) ∞ n =1 , where φ T ( x ) = φ ( T x ) , is alsoa delta sequence. We have then the following result. Proposition 7.2.
Let f ∈ D ′ (cid:0) R d (cid:1) . Then the distributional point value f ( x ) existsif and only if for all standard delta sequences generated by a positive normalized testfunction of D (cid:0) R d (cid:1) , { φ n } ∞ n =1 , the limit lim n →∞ h f ( x + x ) , φ n ( x ) i = γ { φ n } exists and forall orthogonal transformations T of R d , γ { φ Tn } = γ { φ n } . Proof.
This follows immediately from Proposition 7.1 if we observe that a homogeneousfunction or distribution of degree 0 is a constant if and only if it is invariant with respectto orthogonal transformations. (cid:3)
Notice that in one variable, Proposition 7.1 says that lim n →∞ h f ( x + x ) , φ n ( x ) i = γ { φ n } exists for all standard delta sequences generated by a positive normalized test func-tion if and only if there are constants γ + and γ − such that(7.6) lim ε → + h f ( x + εx ) , ψ ( x ) i = γ − Z −∞ ψ ( x ) d x + γ + Z ∞ ψ ( x ) d x , for all ψ ∈ D ( R ) . On the other hand, since the only orthogonal transformations in di-mension one are the identity and x − x, Proposition 7.2 says that the distributionalpoint value f ( x ) exists if and only if for all standard delta sequences generated by a pos-itive normalized test function of D ( R ) , { φ n } ∞ n =1 , the limit lim n →∞ h f ( x + x ) , φ n ( x ) i = γ { φ n } exists and γ { φ n ( − x ) } = γ { φ n ( x ) } . Our results also give the ensuing equivalence.
ISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES 11
Proposition 7.3.
Let f ∈ D ′ (cid:0) R d (cid:1) . Then the distributional point value f ( x ) exists andequals γ if and only if for F the family of standard delta sequences generated by a positivenormalized test function (7.7) f ( x ) = γ ( F ) . Standard delta sequences generated by an even positive normalized testfunction.
We now consider the case of symmetric standard delta sequences, the familyconsidered by Sasane [18].We first need to explain the idea of symmetric point values. Let f ∈ D ′ ( R ) and x ∈ R . The symmetric distributional point value of f exists at x and equals γ if(7.8) lim ε → f ( x + εx ) + f ( x − εx )2 = γ , in D ′ ( R ) . Each distribution can be written as the sum of an even one and an odd one,(7.9) g = g e + g o , where(7.10) g e ( x ) = g ( x ) + g ( − x )2 , g o ( x ) = g ( x ) − g ( − x )2 . Applying this to g ( x ) = f ( x + x ) , we see that the distributional symmetric value f ( x )exists and equals γ if and only if the distributional value g e (0) exists and equals γ. Notice also that if φ is a test function and we write φ = φ e + φ o , then(7.11) h g, φ i = h g e , φ e i + h g o , φ o i . Therefore we have the following result.
Lemma 7.4.
A distribution f ∈ D ′ ( R ) has a symmetric distributional value γ at x ifand only if (7.12) lim ε → h f ( x + εx ) , φ e ( x ) i = γ Z ∞−∞ φ e ( x ) d x , for all even test functions φ e . Proof.
Indeed, if (7.8) is satisfied, thenlim ε → h f ( x + εx ) , φ e ( x ) i = lim ε → h f ( x + εx ) − g o ( εx ) , φ e ( x ) i = lim ε → (cid:28) f ( x + εx ) + f ( x − εx )2 , φ e ( x ) (cid:29) = γ Z ∞−∞ φ e ( x ) d x . Conversely, if (7.12) holds, then for any test function φ = φ e + φ o , lim ε → h g e ( εx ) , φ ( x ) i = lim ε → h g e ( εx ) , φ e ( x ) i = lim ε → h f ( x + εx ) , φ e ( x ) i = γ Z ∞−∞ φ e ( x ) d x = γ Z ∞−∞ φ ( x ) d x . Hence g e (0) = γ, so that the symmetric distributional value of f at x equals γ. (cid:3) We can now give an equivalence to the existence of the point value f ( x ) = γ ( F sy ) , where F sy is the family of standard delta sequences generated by a positive normalizedeven test function of D ( R ) . Proposition 7.5.
Let f ∈ D ′ ( R ) . Then the following are equivalent:1. If F sy is the family of standard delta sequences generated by a positive normalizedeven test function then (7.13) f ( x ) = γ ( F sy ) .
2. The symmetric distributional point value of f exists at x and equals γ. Proof.
Indeed, if (7.13) holds then(7.14) lim n →∞ h f ( x + x ) , φ n ( x ) i = γ , for all standard delta sequences { φ n } ∞ n =1 generated by a positive normalized even testfunction φ e , and use of Proposition 3.1 yields that(7.15) lim ε → h f ( x + εx ) , φ e ( x ) i = γ , for such normalized even test functions φ e . This last statement is equivalent to the factthat (7.12) holds for all even test functions because of Lemma 6.1, and Lemma 7.4 yieldsthat in turn this is equivalent to the symmetric distributional point value being equal to γ. (cid:3) Actually, using the same ideas as in the proof of this Proposition we see that thelimit lim n →∞ h f ( x + x ) , φ n ( x ) i = γ { φ n } exists for all standard delta sequences { φ n } ∞ n =1 generated by a positive normalized even test function φ e if and only if this limit is aconstant γ and (7.13) is satisfied.7.3. The family of standard delta sequences generated by a radial positivenormalized test function.
We now consider the family F rad of standard sequencesgenerated by a radial positive normalized test function.Let us start with some notation. We denote r = | x | the radial variable in R d . A testfunction φ ∈ D (cid:0) R d (cid:1) is called radial if it is a function of r, φ ( x ) = ϕ ( r ) , for someeven function ϕ ∈ D ( R ) ; the space of all radial test functions of D (cid:0) R d (cid:1) is denotedas D rad (cid:0) R d (cid:1) . Similarly, we denote as D ′ rad (cid:0) R d (cid:1) the space of all radial distributions; adistribution f ∈ D ′ (cid:0) R d (cid:1) is radial if f ( T x ) = f ( x ) for any orthogonal transformation of R d , and this actually means [4, 7] that f ( x ) = f ( r ) for some distribution of one variable ISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES 13 f . Notice, however, that while ϕ is uniquely determined by φ, for a given f there areseveral possible distributions f . When d = 1 then D rad ( R ) and D ′ rad ( R ) become the spaces of even test functions anddistributions, respectively, and are also denoted as D even ( R ) and D ′ even ( R ) . This was thesituation considered in the previous subsection.Observe that the space D ′ rad (cid:0) R d (cid:1) is naturally isomorphic to the dual space (cid:0) D rad (cid:0) R d (cid:1)(cid:1) ′ , that is to say, if the action of a radial distribution is known in all radial test functions, thenit can be obtained for arbitrary test functions. Indeed, if f ∈ D ′ rad (cid:0) R d (cid:1) and φ ∈ D (cid:0) R d (cid:1) , then(7.16) h f ( x ) , φ ( x ) i = D f ( x ) , e φ ( x ) E , where e φ ∈ D rad ( R ) is given as(7.17) e φ ( x ) = φ o ( | x | ) ,φ o ∈ D even ( R ) being defined as(7.18) φ o ( r ) = 1 ω Z S φ ( rθ ) d σ ( θ ) . Here we denote by S the unit sphere of R d , d σ is the Lebesgue measure in S and ω =2 π d/ / Γ ( d/
2) is the surface area of the sphere.Equations (7.17) and (7.18) define the radial component of a test function.
We can alsodefine the radial component of a distribution f, e f ∈ D ′ rad (cid:0) R d (cid:1) , as(7.19) D e f ( x ) , φ ( x ) E = D f ( x ) , e φ ( x ) E . The distributional analog of (7.18) is not well defined, however [4, 7].We say that a distribution f has a radial distributional point value at x equal to γ if(7.20) e g ( ) = γ , where e g is the radial component of g ( x ) = f ( x + x ) . Similar to Lemma 7.4, we have thefollowing characterization.
Lemma 7.6.
A distribution f ∈ D ′ (cid:0) R d (cid:1) has a radial distributional value γ at x if andonly if (7.21) lim ε → h f ( x + ε x ) , φ rad ( x ) i = γ Z R d φ rad ( x ) d x , for all radial test functions φ rad . Proof. If e g ( ) = γ, thenlim ε → h f ( x + ε x ) , φ rad ( x ) i = lim ε → h e g ( ε x ) , φ rad ( x ) i = γ Z R d φ rad ( x ) d x . On the other hand, if (7.21) holds, then for any test function φ, lim ε → h e g ( ε x ) , φ ( x ) i = lim ε → De g ( ε x ) , e φ ( x ) E = lim ε → D f ( x + ε x ) , e φ ( x ) E = γ Z R d e φ ( x ) d x = γ Z R d φ ( x ) d x . Hence e g ( ) = γ, that is, the radial distributional value of f at x equals γ. (cid:3) Therefore, we obtain the ensuing equivalence for the fact that f ( x ) = γ ( F rad ) . Proposition 7.7.
Let f ∈ D ′ (cid:0) R d (cid:1) . Then the following are equivalent:1. If F rad is the family of standard delta sequences generated by a positive normalizedradial test function then (7.22) f ( x ) = γ ( F rad ) .
2. The radial distributional point value of f exists at x and equals γ. Proof.
Indeed, if (7.22) holds then(7.23) lim n →∞ h f ( x + ε x ) , φ n ( x ) i = γ , for all standard delta sequences { φ n } ∞ n =1 generated by a positive normalized radial testfunction φ rad . Use of Proposition 3.1 yields that(7.24) lim ε → h f ( x + ε x ) , φ rad ( x ) i = γ , for such normalized radial test functions φ rad . This last statement is equivalent to thefact that (7.12) holds for all radial test functions because of Lemma 6.1, and Lemma 7.6yields that, in turn, this is equivalent to the radial distributional point value being equalto γ. (cid:3) We also have the next result, that is obtained from Lemma 6.1.
Proposition 7.8.
The limit lim n →∞ h f ( x + ε x ) , φ n ( x ) i = γ { φ n } exists for all standarddelta sequences { φ n } ∞ n =1 generated by a positive normalized radial test function φ rad if andonly if this limit is a constant γ and f ( x ) = γ ( F rad ) . The family of all positive normalized test functions.
We saw in Subsection7.2 that Sasane’s notion of point values was not equivalent to the standard definition, nor,in the next subsection, is the notion based on the family F rad of standard delta sequencesgenerated by a positive normalized radial test function. Nevertheless, for the family F ofstandard delta sequences generated by a positive normalized test function the point valuedefinition is in fact equivalent to the standard Lojasiewicz definition. Of course, Sasanewas considering the family of standard delta sequences generated by an even positivenormalized test function F sy . Both F sy and F rad are subfamilies of F . We can also considerfamilies larger than F . For instance, we can consider the family F all of all delta sequencesformed with positive normalized test functions. In this next example, we will see that the ISTRIBUTIONAL POINT VALUES AND DELTA SEQUENCES 15 distributional point value f ( x ) = γ is not equivalent to f ( x ) = γ ( F all ) . Later on weshall find an equivalent formulation to f ( x ) = γ ( F all ) . Example 7.9.
Let f be the regular distribution given by f ( x ) = sin(1 /x ). Then f (0) = 0distributionally [11]. Let a n be a positive sequence with a n → f ( a n ) = C > . Forinstance, we could take a n = 1 / (2 πn + π/ . For a fixed n, let { ψ n,m } ∞ m =1 be a sequenceof positive test functions such that ψ n,m → δ ( x − a n ) as m → ∞ . Then as n → ∞ , weobtain a sequence δ n ( x ) = δ ( x − a n ) that converges to δ ( x ) . For each n, let m n be largeenough so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B /n ( a n ) f ( x ) ψ n,m ( x ) d x − f ( a n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C/ ψ n,m ⊂ B /n ( a n ) for m ≥ m n . Then we can define the sequence φ n ( x ) = ψ n,m n ( x ). By Lemma 5.1, this is a delta sequence and we have h f ( x ) , φ n ( x ) i > C/ n and so lim n →∞ h f ( x ) , φ n ( x ) i cannot be equal to 0 . The next lemma will be useful momentarily.
Lemma 7.10. If { φ n } ∞ n =1 is a delta sequence of positive test functions then (7.25) lim n →∞ k φ n k L ( B \ U ) = 0 , where B and U are both neighborhoods of the origin.Proof. Choose ψ ∈ D (cid:0) R d (cid:1) such that ψ ≥ , (7.26) ψ ( x ) = 1 , x ∈ B \ U , ψ ( ) = 0 , which is possible because / ∈ B \ U .
Then k φ n k L ( B \ U ) = Z B \ U | φ n ( x ) | d x = Z B \ U φ n ( x ) d x = Z B \ U ψ ( x ) φ n ( x ) d x ≤ Z B ψ ( x ) φ n ( x ) d x → ψ ( ) = 0 , as n → ∞ . (cid:3) We are now ready to prove the main result of this section.
Proposition 7.11.
Suppose f ∈ D ′ (cid:0) R d (cid:1) and x ∈ R d . If (7.27) lim n →∞ h f ( x + x ) , φ n ( x ) i = γ , for all positive delta sequences { φ n } , then the following two conditions hold: (1) There is an r ∗ > such that f | B r ∗ ( x ) ∈ L ∞ ( B r ∗ ( x )) . (2) lim r → (cid:13)(cid:13)(cid:13) f | B r ( x ) − γχ B r ( x ) (cid:13)(cid:13)(cid:13) ∞ = 0 . Here χ B r ( x ) is the characteristic function of the ball B r ( x ) . Conversely, if (1) and(2) are satisfied, then (7.27) holds for all positive delta sequences with support containedin B r ∗ ( ) . Proof.
Suppose that (7.27) holds. Notice that (1) follows from Lemma 6.2. To see that(2) is true, suppose instead that(7.28) lim sup r → (cid:13)(cid:13)(cid:13) f | B r ( x ) − γχ B r ( x ) (cid:13)(cid:13)(cid:13) ∞ = C > . Let r n be a decreasing sequence of positive numbers with r n < r ∗ and r n → . For each n , there is a positive normalized test function supported in B r n ( ) , say φ n , such that(7.29) |h f ( x + x ) , φ n ( x ) i − γ | > C . By Lemma 5.1, { φ n } ∞ n =1 forms a delta sequence and so lim n →∞ h f ( x + x ) , φ n ( x ) i = γ, whichcontradicts (7.29).For the converse, let { ψ n } be a delta sequence of positive normalized test functionssupported in B r ∗ ( ) . Since by (1) f is a regular distribution in B r ∗ ( ) we have(7.30) h f ( x + x ) , ψ n ( x ) i − γ = Z B r ∗ ( ) ( f ( x + x ) − γ ) ψ n ( x ) d x . Let ε > . By condition (2), we can find an open neighborhood V of the origin that iscontained in B r ∗ ( ) such that if W = x + V, then k f − γ k L ∞ ( W ) < ε and for this V we canfind n such that k ψ n k L ( B r ∗ ( ) \ V ) < ε if n ≥ n . If M is the constant k f − γ k L ∞ ( B r ∗ ( x )) , then we have |h f ( x + x ) , ψ n ( x ) i − γ | = (cid:12)(cid:12)(cid:12)(cid:12)Z W ( f ( x ) − γ ) ψ n ( x − x ) d x + Z B r ∗ ( ) \ V ( f ( x + x ) − γ ) ψ n ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k f − γ k L ∞ ( W ) k ψ n k L ( V ) + k f − γ k L ∞ ( B r ∗ ( x ) \ W ) k ψ n k L ( B r ∗ ( ) \ V ) < ε + M ε , and consequently lim n →∞ h f ( x + x ) , ψ n ( x ) i = γ. (cid:3) References [1] Antosik, P., Mikusi´nski, J. and Sikorski, R.,
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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
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