Analytic Theory of Finite Asymptotic Expansions in the Real Domain . Part IIB : Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives

Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) ( ) ( ) ( ) ( ) ( ) n n n f x a x a x o x 1 1 = + + + φ φ φ , x x0 → , n 3 ≥ where the asymptotic scale ( ) ( ) ( ) n x x x 1 2 φ φ φ , x x0 → , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n − 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.

, x x 0 → , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x 0 .The main result states that to each scale of this type it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n − 1 times in the two special senses characterized in Part II-A.A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.real domain.
• §8 contains the main result in the paper: to each Chebyshev asymptotic scale , , , n x x x φ φ φ it remains associated an important class of functions enjoying the property that an asymptotic expansion according to this scale, if valid, is automatically formally differentiable 1 n − times in the two special senses characterized in § §4,5 in Part II-A.Under the regularity assumptions of the factorizational theory, this class is characterized by an nth-order differential inequality whereas in the yet-to-be-developed geometric theory it will be the class of generalized convex functions as in the special case of polynomial expansions ( [2], §4).
• In §9, discussing formal application of standard derivatives to an asymptotic expansion, we characterize the existence of certain polynomial expansions at an endpoint where derivatives may fail to exist and such that the growth-order estimates of the remainders of the differentiated expansions follow unexpected algebraic rules.• §10 contains the proofs and §11 contains a few remarks about our theory.
Whereas the results in Part II-A show that "formal differentiation of asymptotic expansions" is usually admissible only if suitable operators linked to the given scale are used, the results in this Part II-B shed further light on this classical problem by exhibiting a meaningful and not too special case wherein suitable formal differentiations are automatically admissible and by showing that standard derivatives are admissible in very special cases only and that they may yield formulas algebraically skew from a classical viewpoint.
We continue the numbering of sections and formulas in [1], but we adopt an independent numbering of the references in the bibliography.In order to agree with some classical terminology about the matter in this paper, it is convenient to specify the signs of certain Wronskians, so we list the fundamental properties of the scale we shall use taken from ( [1], Def.2.1 and Prop.2.3): , , 1 ; , ; , , 0 on , , 1 ; , , , 0 on , , 1 ; 1 0 0 on , , namely sign 1 , 1 .( ) , and we are supposing 3 n ≥ as the two-term theory has been thoroughly studied in [3].Operators k L and k M are defined in formulas (3.1) to (3.4) in Part II-A; properties of the k L 's are reported in the first few lemmas in §4 and properties of the k M 's are to be found in Proposition 3.1 with the signs specified by (3.19), due to our present assumption (7.3).We recall the acronym C.F. for "canonical factorization" ( [1], Prop.2.1).

Absolute Convergence and Solutions of Differential Inequalities
The theory developed in Part II-A becomes particularly simple when the involved improper integrals are absolutely convergent and still more expressive for a function f satisfying the nth-order differential inequality ( ) [ [ 1 , , 0 0 a.e.on , .
Under the assumptions (7.1) and (7.3) this is a subclass of the so-called "generalized convex functions with re-spect to the system ( ) 1 , , n φ φ ".The nice result stated in the next theorem claims that: if such a function admits of an asymptotic expansion (7.7) then this expansion is automatically differentiable ( ) 3) The following set of asymptotic expansions holds true: ( ) ( ) ( ) ( ) 1; see (4.31).
To this list we may obviously add the other properties in Theorem 5.1 and if this is the case the remainder ( ) 0 R x of the expansion in (8.3) admits of both representations: whence it follows that The above equivalence "1) ⇔ 2)" simply means that, under condition (8.1), a relation " ( ) In addition to the equivalence "3) ⇔ 4)" there is another remarkable circumstance wherein the two types of formal differentiations are simultaneously admissible namely when the convergence of the pertinent improper integrals is absolute.( ) ( ) Hence each of these three conditions implies both sets of asymptotic expansions (4.31) and (5.5)-(5.6)(here the signs of the Wronskians are immaterial).
An indirect brief proof of the equivalence "(8.11) ⇔ (8.12)" can be based on Theorem 8.1, but it also follows from the following remarkable relation valid for any signs of the Wronskians in (7.3), (7.4): ; ; , 0 ; (which last relations are written in (4.28) in an expanded form); ; To the foregoing list we may obviously add property 2) or property 5) in Theorem 4. .As soon as one of these integrals converges to a real number as 0 x x → then we may apply the theorem with a greater value of i.And in the case of divergence, under the present assumption of one-signedness, it is possible to infer from condition (8.20) Under the assumptions in Theorem 8.3 let 3 n ≥ , 1 2 i n ≤ ≤ − and suppose that all the integrals appearing in the "O"-estimates in (8.17) diverge as Then the estimates in (8.17) for ( )  can be replaced by: ( ) In the present context the above estimates are by no means obvious or "natural": they have been obtained by adapting the standard calculations in the proof of the Abel-Dirichlet's test for convergence of weighted improper integrals (Lemma 10.1 below).As a simple check of their validity we reobtain classical estimates for the derivatives of nth-order convex functions, and to be consistent with the meaning of n in the present series of papers, namely "n = dimension of the Chebyshev system ( ) 1 , , n φ φ ", we state the result for convex functions of order 1 n − according to a standard terminology.
Then the following asymptotic relations hold true as Here the asymptotic scale is: ; and 1 i q ≡ .The special case 0 i = asserts that if an ( ) The case ( ) ( )  shows that the estimates in Theorem 8.4 are the best possible, generally speaking.
The estimates in Corollary 8.5 also follow from old results by Landau, Hardy and Littlewood about differen-tiation of asymptotic relations involving real powers, under assumptions of monotonicity on the derivatives, results that were discussed in [4] and then extended in [5] to asymptotic expansions in real powers.The special case 0 i = has also been obtained independently by Popoviciu ([6], p. 28).Specialization of Theorems 8.3-8.4  to  An important remark.In Theorem 8.1 the two types of formal differentibility 1, 2, , 1 n − times are equivalent facts whereas it is not so for a generic f such that ( )  changes sign on each deleted left neighborhood of 0 x .This has been proved for polynomial expansions [2] and for real-power expansions [5] in an indirect way by expressing the two sets of differentiated expansions as suitable sets of expansions involving the standard operators d d k k x ; the new sets of expansions made evident that what we called "weak formal diffe- rentiability", linked to the C.F. of type (I), is indeed a weaker property than what we called "strong formal differentiability", linked to a C.F. of type (II).This will be also proved true in Part II-C, §15, for a special class of expansions including the real-power case.The same circumstance occurs for a general two-term expansion ( [3]; Remarks, p. 261) but is not a self-evident fact.In each of these three cases direct proofs could be also provided working on the corresponding integral conditions.Hence in these cases the locutions of "weak or strong formal differentiation" are legitimate.But in the general theory for 3 n ≥ we face a nontrivial situation and state Open problem.For 3 n ≥ consider the two types of formal differentiability characterized in Theorems 4.5 and 5.1.Investigate whether or not the property in Theorem 5.1 always implies the one in Theorem 4.5, the two properties being equivalent in the case of absolute convergence described in Theorem 8.2.
We shall not dwell on this marginal aspect of the theory though it leaves unsolved whether or not we may use representation formula (14.38), in alternative to (15.12)-(15.13),under condition (15.10).

Asymptotically-Admissible Operators
Before investigating cases wherein standard derivatives d d k k x are formally applicable to an asymptotic ex- pansion it is good to give a rigorous definition of the involved concept, cursorily treated in ( [1], §3) and ( [7], §3), with a few examples.
Definition 9.1 (Asymptotically-admissible operators).Let  be a linear operator acting between two linear spaces of real-or complex-valued functions of one real variable, ( ) ( ) ( ) without any further regularity assumptions.
(I) (A definition valid in special cases but highlighting the concept). is said to be asymptotically admissi- ble with respect to a given asymptotic expansion if its formal application to both sides of (9.2) yields a new asymptotic expansion This implicitly implies that 1 f ∈  and that the operator  changes the asymptotic scale (9.1) into a new asymptotic scale Put in these terms the definition is well-posed if none of the functions is the zero element 0 of 2  which means the function identically zero on some neighborhood In general, to avoid inconsistencies, the definition must be modified as follows.
(II) (A general definition).First, if ( ) , then the concept in question is not defined.If this is not the case then we put and say that  is asymptotically admissible with respect to (9.2) if ( ) ( ) ( ) after suppression of all the zero terms.An alternative locution for an asymptotically-admissible  is "  is formally applicable to the asymptotic expansion (9.2)"; and the validity of (9.6) may be expressed by saying that "  preserves the asymptotic hierarchy in (9.1)".
A first group of examples clarifies the necessity of specifying "after suppression of all the zero terms".In each of the following three examples the standard operator of differentiation d dx is asymptotically admissible according to Definition 9.1 only if all the identically-zero terms have been suppressed. x .
That the standard operator of differentiation does not preserve asymptotic hierarchies is quite elementary but a second group of examples shows that it may not preserve asymptotic hierarchies even when acting on an n-tuple forming a Chebyshev asymptotic scale (signs apart) on a neighborhooh of 0 x which, in the examples below, is taken as +∞ .
sin cos ; : e ; , ; 2 none of the two limits : lim , lim , does exist in because of : 1 cos .

4) Example of a Chebyshev asymptotic scale (
) ′ is an asymptotic scale as well but there exists a function f such that has no asymptotic expansion of type ; and there exists a function g such that ( ) , , has an incomplete asymptotic expansion, say , , with .

Asymptotic Admissibility of Standard Derivatives
Let us ask the question: What are the natural scales granting the asymptotic admissibility of the standard operators d d k k x ?Some of the above examples show that these operators do not automatically turn a given Cheby- shev asymptotic scale into an asymptotic scale and, so, a rash answer to our question might suggest a scale is again an asymptotic scale but this is glaringly disproved even for the familiar scale at +∞ , , by a function such as ( ) ( ) , n α ≥ ; here we have that f, but not f ′ , admits of an asymptotic expansion with respect to the mentioned scale.

Of course a special case occurs when one of the sets of operators either
x ; then our operator, which we denote by the special symbol n D φ , admits of the factorization for some function 0 φ > and of class Here the operators d d k k x are to be applied not to the function f whose expansion is given but to the ratio f φ .From formulas (2.7), (2.8) in Part II-A, we get the C.F.'s of n D φ .Lemma 9.1. .
The reader must not think that we are now filling a few pages with trivialities about Taylor's formula; as a matter of fact if we apply our theory to the operator n D φ in the case 0 x = +∞ we obtain the results about asymptotic parabolas for the function f φ whose theory is thoroughly studied in [2].But for 0 x ∈  the first factorizational appproach characterizes a set of asymptotic expansions wherein (quite surprisingly) the estimates of the remainders in the differentiated expansions may follow algebraic rules different from those valid both in the case 0 x = +∞ and in the case of the standard Taylor's formula; and the second factorizational appproach gives Taylor's formula as a "limit" of Taylor's formulas which is a classical elementary result to be commented on in our context.In this last case one must pay attention to the fact that formal application of the standard derivative is in general permissible only a number of times related to the growth-order of the remainder in the given asymptotic expansion.See also "examples and a final comment" at the end of this section.
Theorem 9.2.Let: whereas the one associated to factorization (9.20) is the standard derivative ; Consider now a generic polynomial of order .

As concerns the bound k i
≤ when comparing with Theorems 5.1-5.2notice that, in the present setting, the i φ 's of the general theory are given by ( ) ( ) which for 1 i n = − must be read as On account of the hypothesis , , condition (9.30) or (9.31) is equivalent to condition which is equivalent to condition Hence relations (9.29) are nothing but Taylor's formula of order i of f φ at 0 x together with the standard differentiated relations up to order i obtained as the "limit" of the Taylor's formula at the point ξ as (II) (A polynomial expansion at an endpoint where derivatives may fail to exist).The following are equivalent properties for a fixed The set of aymptotic expansions as .
4) The set of aymptotic expansions as where the two estimates coincide for The nondifferentiated expansion is written differently in (9.34) than in (9.35) to correctly apply the operators k L as defined in (9.26).Notice that for 1 i n = − the remainder in the second relation in (9.34) is ( ) ( ) We must comment on the above claims.Part (I) is a classical elementary property which may be traced back to Walter and Ford ([9], Lemma II, p. 350), 1911, and a proof is reported in Aumann and Haupt ([10], Ch. 8, §8.9.2.1, pp.235-236) valid under weaker regularity assumptions involving only the existence of the highestorder left derivative of the given function and its limit as 0 x x − → .Put in geometric terms it asserts that: If the osculating parabola of a certain order i at a generic point ξ admits of a limit position as then this last is the left osculating parabola of order i at 0 x .This fact historically is the idea underlying the geometric theory of limit parabolas at +∞ , see ( [2], §1) where the two main results characterize expansions involving remainder-estimates at +∞ either of the form ( ) ( ) ( ) , 0, From an al-gebraic viewpoint the first stronger form follows the same formal rule as in Taylor's formula (9.29) and in relations (9.35) for 0 i = , whereas the second weaker form has no counterpart for 0 x ∈  and 1 i ≥ .The equiva- lence "4) ⇔ 5)" is no trivial fact and let us have a closer look at the set of relations in (9.35) which may seem strange and even incorrect at a first sight.For simplicity we put 0 0 x = .First, the powers appearing in the o-terms decrease with k which amounts to say that we have worse estimates for higher derivatives; and this is a natural phenomenon.Second, if for two functions 1 f , 2 f we have , 0; , 0; then the estimate for ( ) is sharper than that for ( ) Examples.The following simple examples involving oscillatory functions will reassure the doubtful readers (and the author himself was in the number).For 2 n = and 1 i = we have the equivalence an example being provided by which is differentiable at 0 x = but f ′ has no limit as 0 x and is unbounded.For 3 n = and 1, 2 i = we have the following contingencies, assuming It is obvious that in both cases f can be extended so as to be of class 1 C on a complete neighborhood of 0 x = and an example for both contingencies is provided by the function: A final comment.The discussion in this section shows that formal applications of ordinary derivatives to an asymptotic expansion is not admissible generally speaking, and even for the very special asymptotic scale (9.22) the first (but not the second) factorizational approach can give seemingly-unnatural results.It is in principle true that each of the two sets of expansions characterized in Part II-A, § §4,5, can provide asymptotic information (not always meaningful and not necessarily expansions) for the ordinary derivatives; however this is easily achieved for the first-order derivative but is practically unmanageable for higher-order derivatives and yields no theoretical result.It is also true that for expansions in arbitrary real powers as x → +∞ two lemmas of an alge- braic character permit to transform each set of expansions involving the pertinent operators k L or k M into a meaningful set of expansions involving the ordinary derivatives and here again the first factorizational approach yields uncommon results, ([5], Lemma 7.3, p. 196, and Lemma 7.4, p. 201); but as 0 x x → ∈ a possible analogue of Theorem 9.2-(II) for arbitrary powers is complicated by the fact that it is necessary to separate many a case for the exponents.On the contrary, the use of weighted derivatives defined by canonical factorizations yields a coherent and applicable theory.

Proofs
Proof of Theorem 8.1.The only thing to be proved is the inference "1) ⇒ 5) ∧ 6)", the other properties being included in Theorems 4.5 and 5.1.We use a procedure already used in ( [2], p. 193) and in ( [5], p. 213).From representation in (4.15) we get (using the simplified notation By the assumption (8.2) the left-hand side has a finite limit as 0 x x → , and for the right-hand side we have: after applying L'Hospital's rule ( ) times (which is legitimate as all the denominators diverge to +∞ ).By the positivity of the integrand this last limit exists in  and coincides with the limit of the left-hand side in (10.1) hence it must be a real number and (4.15) can take the form: with suitable constants 2 , , n c c .From this we get: Here again the left-hand side has a finite limit as 0 x x → whereas the limit of the right-hand side, by (4.1), equals: after applying L'Hospital's rule ( ) 2 n − times.Hence this last limit, which exists in  , must be a real number and (10.3) can be rewritten as: with suitable constants 3 , , n c c .It is now clear how this procedure works and by induction one can prove the validity of representation: with a suitable constant n c .As a last step we observe that (8.2) implies: and (10.6) in turn implies: By the positivity of the integrand this last relation implies (8.6) and the first representation in (8.8) for ( ) 0 R x .To prove (8.7) we apply the same ideas starting from representation (5.1) and dividing by 1 φ ; recalling that 1 0 and (5.1) can be rewritten as with suitable constants 2 , , n c c .From this we get Evaluating the limit of the right-hand side by L'Hospital's rule and using formula in (2.31), ( ) and this last limit, which exists in  , must be a real number.This means that ( ) ( ) and (10.11) can be rewritten as with suitable constants 3 , , n c c .For the clarity's sake we make explicit the steps in this second part of our proof.Assume by induction that the following two conditions hold true: ( ) ( ) for some ,1 2 i i n ≤ ≤ − , and suitable constants 1 , , Dividing both sides of (10.16) by 1 i φ + and taking account of (8.2) we infer that the limit of the quantity .17) exists in  .Applying L'Hospital's rule i times to evaluate this limit we get the new limit ( ) ( ) which, by the positivity of the integrand, exists in  hence it must be a real number.We infer that condition (10.15) holds true with i replaced by 1 i + and this implies representation (10.16) with i replaced by 1 i + and suitable constants 2 , , . By this inductive procedure we arrive at representation: with some constant n c .Dividing by n φ and using (2.2) we may now conclude that ( ) ( ) ( ) and if we try to evaluate the limit of the ratio on the left applying L'Hospital's rule ( ) .on , ; a.e. on [ [ 0 , T x and Theorem 8.1 implies the equivalence between (8.11) and (8.12).Now we prove relation (8.13) recalling that all the involved functions and Wronskians are strictly one-signed on the interval.The symbol of asymptotic equivalence is referred to 0 x x − → of course, and, whenever used, is granted by the hierarchies of the Wronskians in (2.14) and the divergence of the involved integrals.We report a classical differentiation formula in Proposition 2.4: ) valid for any ordered ( ) , , , i i g g g + at any point where the required derivatives exist.Now for 2,3 n = we have: ( ) where in the last but one passage we have applied formula (10.22) to the ordered triplet ( ) ) Now, for a fixed 3 n ≥ , we use the procedure in (10.24) to prove by induction on i that ( ) ( ) , , , As in (10.24) we can prove that (10.26) is true for 2 i = ; assuming it to be true with i replaced by 1 i − we have: ( ) that is (10.26).Using this relation for 1 i n = − and the expression in (2.43) for 1 n p we, finally, that is (8.13).
Proof of Theorem 8.3.The only thing to prove is the O-estimates in (8.17).From representation (5.
for some constant c, whence the estimate for  follows.And so on for the other estimates.Proof of Theorem 8.4 is contained in the following lemma only valid under the stated one-signedness restrictions.
Lemma 10.1 (Growth-order estimates for iterated improper integrals with nonnegative integrands).Assumptions: [ [ ( ) we want to prove ( ) ( ) ( ) For the sake of simplicity, we write out the calculations for 4 H using (10.36), which reads ( ) ( ) ( ) , and integrating by parts in the integral appearing in (10.36):In part (II) relations (9.34) are those given in Theorems 4.4-4.5 specialized to our case and the only thing to be proved is the equivalence "3) ⇔ 4)" contained in the next lemma where we simplify all formulas by assuming 0 0 x = and 1 φ ≡ .

Lemma 10.2. Let f be of class
1 n AC − on some deleted neighborhood of 0 x = and let ( ) the following set of asymptotic relations as 0 x → : is equivalent to the set of asymptotic relations, as 0 x → , involving only standard derivatives: where the two estimates coincide for .
It is easily proved by induction that the expanded expressions of the k L 's are ( ) ( ) Second step.From the second equality in (10.45) we infer having used (10.47), and from these last estimates we get Suppose now that the relations in (10.43) for the derivatives have been proved true for < − , and let us prove the corresponding relations for ( ) we use the first estimate in (10.42) for ( )  and the first estimates in (10.43) for the derivatives ( ) ( ) j f x so getting from (10.51) For the remaining values of i we must use the second estimate in (10.42) for ( )  and the suitable estimates in (10.43) for ( ) ( ) j f x so getting from (10.51) Now inside the sum each ( ) j f must be replaced by one of the two estimates in (10.43), assumed to be true, and we have two possible contingencies Finally from (10.51) we get the sought-for estimates for ( ) The proof of "(10.42)⇒ (10.43)" is over.The converse implication is checked at once replacing the estimates in (10.43) into the sum (10.45) expressing and that the inference "⇒" can be easily proved when a C.F. of the operator : Lu u′′′ = of type (I) at +∞ is used, see ( [2], proof of Th.4.2).Suppose now to use the following factorization (which is no C.F. at +∞ ) ( ) and the related representation obtained from the results in §8 based on the use of C.F.'s.However, in this case the equivalence in (11.8) can be also obtained using the non C.F. in (11.7) if one starts from the corresponding integral representation for f and reapplies the same procedure used in the proof of Theorem 8.1.

On the Use of Integral Representations Inferred from Factorizations
We start noticing that the convergence of an iterated integral ( ) ( ) with an appropriate representation of the remainder.But, by the initial remark, such an integral condition is almost useless for general results as well as for practical applications if the i T 's are fixed a priori and distinct from 0 x ; it may be well fulfilled by some very special f with an ocillatory Lf, but it cannot be satisfied by any f such that 0 Lf > no matter how small its order of growth at 0 x : in fact if the improper integral  x , this simply means that condition (11.15) may be replaced by a stronger condition yielding additional asymptotic information.
linear ordinary differential operator of type (2.1) 1,2 , acting on the space integral conditions are equivalent:

1 
forming an asymptotic scale at 0 x , possibly 0x − or 0 x + : (The continuity property of Taylor's formula).The following are equivalent properties for a fixed and the remainder in the second relation in (9.35) is ( ) and for 1 n i k n − < ≤ − the two estimates coincide.This fact simply says that the estimates in (9.35) for different values of i are consistent.What may seem unnatural, on the contrary, is the gap of two units between the exponents inside the last o-term in (9.35) corresponding to consecutive values of k.

( ) 5 H
x and so on.Proof of Theorem 9.2.Part (I) is nothing but Theorems 5.1 and 5.2 applied to the present case wherein f is replaced by f φ and ( ) between (9.32) and (9.33) is an elementary fact under the tacit assumption ( ) ( ) are not needed in our calculations.Let us now prove "(10.42) ⇒ (10.43)".First step.From the first equality in (10.45) we get

(
as well.In each case the whole sum in (10.53) is ( ) number, then the iterated integral in (11.13) diverges.Moral.Working with the C.F. of type (I) at 0 x , the sole integral conditions which can be used for sufficiently general results are those appearing in Theorem 4.4 in Part II-A.The situation is technically different when working with a C.F. of type (II).Referring to an integral representation of type .13) Using Theorems 4.4 and 5.2, we can also get the analogues of Theorems 8.1-8.2 for incomplete asymptotic expansions and here is a concise statement, all asymptotic relations referring to order of integrations in(8.10)whereas the equivalence between (8.11) and (8.12) is by no means an obvious fact.A concise proof based on Theorem 8.1 is as follows; putting which exists in  and must be a finite number.This is condition (8.7) which allows the second representation in (8.8) for the remainder in(8.3).The proof is over.Proof of Theorem 8.2.The equivalence between (8.10) and (8.11) easily follows from Fubini's theorem by interchanging the ; Now the integral on the left is convergent by hypothesis and the first integral on the right is convergent by (10.34), hence the second integral on the right converges as well and we get the equality

11. Some Remarks on Factorizational Theory 11.1. On the Use of Non-Canonical Factorizations
.57) We show by two examples that the use of non-C.F.'s is unreliable to construct a general theory.Let us refer, e.g., to the characterizations of an asymptotic expansion for a generalized convex function.
Assuming the validity of the expansion in (11.1), we try to find a necessary integral condition involving f ′′′ .From (11.3), we get Hence, in this example the used factorization does not allow to characterize the expansion at hand.
T t f t t in the case of conditional convergence, may depend in an unpredictable way on the choice of 1 T as in the following two elementary examples: