Interpretations of the Asymptotic Conditions

Abstract

In this chapter, we study some important properties of the sets $\mathcal {R}^p_{\mathrm {S}}$ and $\mathcal {D}^p_{\mathrm {S}}$ and provide interpretations of the asymptotic condition that defines the set $\mathcal {R}^p_{\mathrm {S}}$.

In this chapter, we study some important properties of the sets $\mathcal {R}^p_{\mathrm {S}}$ and $\mathcal {D}^p_{\mathrm {S}}$ and provide interpretations of the asymptotic condition that defines the set $\mathcal {R}^p_{\mathrm {S}}$.

We also investigate the sets $\mathcal {R}^p_{\mathrm {S}}\cap \mathcal {K}^p$ and $\mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$ and show that they actually coincide and are independent of S (and hence we can remove this subscript). We also provide an interpretation of this common set $\mathcal {D}^p\cap \mathcal {K}^p$ and present some of its properties that will be very useful in the next chapters. In particular, we show that the intersection set $\mathcal {C}^p\cap \mathcal {D}^p\cap \mathcal {K}^p$ is precisely the set of functions $g\in \mathcal {C}^p$ for which g ^(p) eventually increases or decreases to zero (see Theorem 4.14).

4.1 Some Properties of the Sets $\mathcal {R}^p_{\mathrm {S}}$ and $\mathcal {D}^p_{\mathrm {S}}$

Although the definition of the set $\mathcal {R}^p_{\mathrm {S}}$ seems rather technical (see Definition 1.9), the following proposition shows that this set can be nicely characterized in terms of interpolating polynomials. We omit the proof for it follows immediately from (2.11) and (2.12).

Proposition 4.1

Let $p\in \mathbb {N}$ . A function $g\colon \mathbb {R}_+\to \mathbb {R}$ lies in $\mathcal {R}^p_{\mathrm {S}}$ if and only if for each x > 0 such that $x^{ \underline {p}}\neq 0$ , we have that

$$\displaystyle \begin{aligned} g[a,a+1,\ldots,a+p-1,a+x] ~\to ~0\qquad \mathit{\mbox{as }}a\to_{\mathrm{S}}\infty. \end{aligned}$$

When $\mathrm {S}=\mathbb {R}$ (resp. $\mathrm {S}=\mathbb {N}$ ), this latter condition means that g asymptotically coincides with its interpolating polynomial whose nodes are any p points equally spaced by 1 (resp. any p consecutive integers).

Interestingly, from (3.2) and (3.3) we can also immediately derive the following alternative characterization of the set $\mathcal {R}^p_{\mathbb {N}}$. For any function $f\colon \mathbb {R}_+\to \mathbb {R}$, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} f\in\mathcal{R}^0_{\mathbb{N}} & \Leftrightarrow &\displaystyle f(x) ~=~ -\sum_{k=0}^{\infty}\Delta f(x+k){\,},\qquad x>0{\,};\\ f\in\mathcal{R}^{p+1}_{\mathbb{N}} & \Leftrightarrow &\displaystyle f(x) ~=~ f(1)+\lim_{n\to\infty}f^p_n[\Delta f](x){\,},\qquad x>0{\,}. \end{array} \end{aligned} $$

(Note that we have already used these equivalences in the proofs of the uniqueness Theorems 3.1 and 3.12 and Proposition 3.9.)

We now present a proposition that reveals some interesting inclusions among the sets $\mathcal {R}^p_{\mathrm {S}}$ and $\mathcal {D}^p_{\mathrm {S}}$. In particular, it shows that just as the sets $\mathcal {D}_{\mathrm {S}}^0, \mathcal {D}_{\mathrm {S}}^1, \mathcal {D}_{\mathrm {S}}^2, \ldots$ are increasingly nested, so are the sets $\mathcal {R}_{\mathrm {S}}^0, \mathcal {R}_{\mathrm {S}}^1, \mathcal {R}_{\mathrm {S}}^2, \ldots$, and hence each of these families defines a filtration.

Proposition 4.2

For any $p\in \mathbb {N}$ and any $\mathrm {S}\in \{\mathbb {N},\mathbb {R}\}$ , the sets $\mathcal {R}^p_{\mathrm {S}}$ and $\mathcal {D}^p_{\mathrm {S}}$ are real linear spaces that satisfy the identity

$$\displaystyle \begin{aligned} \mathcal{R}^p_{\mathrm{S}} ~=~ \mathcal{R}^{p+1}_{\mathrm{S}}\cap\mathcal{D}^p_{\mathrm{S}} \end{aligned} $$

(4.1)

and the strict inclusions

$$\displaystyle \begin{aligned} \mathcal{R}^p_{\mathrm{S}} ~\varsubsetneq ~\mathcal{R}^{p+1}_{\mathrm{S}}\qquad \mathit{\mbox{and}}\qquad \mathcal{D}^p_{\mathrm{S}} ~\varsubsetneq ~\mathcal{D}^{p+1}_{\mathrm{S}}. \end{aligned}$$

When p ≥ 1 we also have

$$\displaystyle \begin{aligned} \mathcal{R}^p_{\mathrm{S}} ~\varsubsetneq ~\mathcal{D}^p_{\mathrm{S}}. \end{aligned}$$

Finally, when p = 0 we have

$$\displaystyle \begin{aligned} \mathcal{D}^0_{\mathbb{R}} ~=~ \mathcal{R}^0_{\mathbb{R}} ~\varsubsetneq ~\mathcal{R}^0_{\mathbb{N}} ~\varsubsetneq ~\mathcal{D}^0_{\mathbb{N}}{\,}. \end{aligned}$$

Proof

It is clear that the sets $\mathcal {R}^p_{\mathrm {S}}$ and $\mathcal {D}^p_{\mathrm {S}}$ are closed under linear combinations; hence they are real linear spaces. Identity (4.1) then follows immediately from (3.4) and (3.5). This identity also shows that $\mathcal {R}^p_{\mathrm {S}}\subset \mathcal {R}^{p+1}_{\mathrm {S}}$. As already observed, we also have $\mathcal {D}^p_{\mathrm {S}}\subset \mathcal {D}^{p+1}_{\mathrm {S}}$ trivially. Now, identity (2.11) shows that the polynomial function x↦x ^p lies in $\mathcal {R}^{p+1}_{\mathrm {S}}\setminus \mathcal {R}^p_{\mathrm {S}}$ and we can easily see that it lies also in $\mathcal {D}^{p+1}_{\mathrm {S}}\setminus \mathcal {D}^p_{\mathrm {S}}$. The inclusion $\mathcal {R}^p_{\mathrm {S}}\subset \mathcal {D}^p_{\mathrm {S}}$ follows from (4.1) and we can easily see that the 1-periodic function $x\mapsto \sin {}(2\pi x)$ lies in $\mathcal {D}^p_{\mathrm {S}}\setminus \mathcal {R}^p_{\mathrm {S}}$ for any $p\in \mathbb {N}^*$ as well as in $\mathcal {D}^0_{\mathbb {N}}\setminus \mathcal {R}^0_{\mathbb {N}}$. Finally, let us now show that $\mathcal {R}^0_{\mathbb {R}} \varsubsetneq \mathcal {R}^0_{\mathbb {N}}$. Using bump functions for instance, we can easily construct a smooth function $f\colon \mathbb {R}_+\to \mathbb {R}$ such that for any $n\in \mathbb {N}^*$, we have f = 0 on the interval $[n-1,n-\frac {1}{n}]$ and $f(n-\frac {1}{2n})=1$. Such a function clearly lies in $\mathcal {R}^0_{\mathbb {N}}$. However, it does not vanish at infinity, i.e., it does not lie in $\mathcal {R}^0_{\mathbb {R}}$. □

We now present an important result that will be used repeatedly as we continue. It actually follows from the second of the following straightforward identities

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_{a+1}^p[f](x) - \rho_a^p[f](x) & =&\displaystyle \rho_a^p[\Delta f](x){\,},{} \end{array} \end{aligned} $$

(4.2)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_{a}^{p+1}[f](x+1) - \rho_a^{p+1}[f](x) & =&\displaystyle \rho_a^p[\Delta f](x){\,}.{} \end{array} \end{aligned} $$

(4.3)

Proposition 4.3

Let $j,p\in \mathbb {N}$ be such that j ≤ p. The following assertions hold.

• (a) If $f\in \mathcal {R}^p_{\mathrm {S}}$ , then $\Delta ^jf\in \mathcal {R}^{p-j}_{\mathrm {S}}$.

• (b) $f\in \mathcal {D}^p_{\mathrm {S}}$ if and only if $\Delta ^jf\in \mathcal {D}^{p-j}_{\mathrm {S}}$.

Proof

If f lies in $\mathcal {R}^{p+1}_{\mathrm {S}}$, then Δf lies in $\mathcal {R}^p_{\mathrm {S}}$ by (4.3). On the other hand, it is clear that f lies in $\mathcal {D}^{p+1}_{\mathrm {S}}$ if and only if Δf lies in $\mathcal {D}^p_{\mathrm {S}}$. □

It is easy to see that a function $f\colon \mathbb {R}_+\to \mathbb {R}$ whose difference Δf lies in $\mathcal {R}^p_{\mathrm {S}}$ for some $p\in \mathbb {N}$ need not lie in $\mathcal {R}^{p+1}_{\mathrm {S}}$. For instance, the function $f\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation $f(x) = \sin {}(2\pi x)$ for x > 0 does not lie in $\mathcal {R}^1_{\mathrm {S}}$ but its difference Δf = 0 lies in $\mathcal {R}^0_{\mathrm {S}}$. However, we will see in Corollary 4.10 that, if $f\in \mathcal {K}^{p-1}$, then the implication in assertion (a) of Proposition 4.3 becomes an equivalence.

Remark 4.4

In view of Proposition 4.3(b), it is natural to wonder whether there exists a set $\mathcal {D}$ of functions from $\mathbb {R}_+$ to $\mathbb {R}$ having the property that $f\in \mathcal {D}^0_{\mathrm {S}}$ if and only if $\Delta f\in \mathcal {D}$. However, such a set does not exist. Indeed, identities (3.1) and (3.2) show that if f lies in $\mathcal {D}^0_{\mathrm {S}}$, then necessarily Δf lies in $\mathcal {D}^{-1}_{\mathrm {S}}$. Conversely, for any $g\in \mathcal {D}^{-1}_{\mathrm {S}}$, there are infinitely many functions $f\colon \mathbb {R}_+\to \mathbb {R}$ that satisfy Δf = g but that do not lie in $\mathcal {D}^0_{\mathrm {S}}$. $\lozenge$

It is clear that, for any $p\in \mathbb {N}$, if both functions h and g − h lie in the space $\mathcal {R}^p_{\mathrm {S}}$, then so does the function g. For instance, if $g\colon \mathbb {R}_+\to \mathbb {R}$ has the asymptotic property that

$$\displaystyle \begin{aligned} g(x)-P(x) ~\to ~0\qquad \mbox{as }x\to_{\mathrm{S}}\infty \end{aligned}$$

for some polynomial P of degree less than or equal to p − 1, then g must lie in $\mathcal {R}^p_{\mathrm {S}}$. Indeed, P clearly lies in $\mathcal {R}^p_{\mathrm {S}}$ and we also have that g − P lies in $\mathcal {R}^0_{\mathrm {S}}$ (which is included in $\mathcal {R}^p_{\mathrm {S}}$ by Proposition 4.2). Thus, the space $\mathcal {R}^p_{\mathrm {S}}$ contains not only every polynomial of degree less than or equal to p − 1 but also every function that behaves asymptotically like a polynomial of degree less than or equal to p − 1. To give another illustration of the property above, we observe for instance that both functions $\ln x$ and $H_x-\ln x$ (the latter tends to Euler’s constant γ as x →_S ∞) lie in $\mathcal {R}^1_{\mathbb {R}}$ and hence so does the function H _x, which means that, for each a ≥ 0,

$$\displaystyle \begin{aligned} H_{x+a}-H_x ~\to ~0\qquad \mbox{as }x\to\infty \end{aligned}$$

(which, a priori, is a not completely trivial result).

These examples illustrate the fact that the spaces

$$\displaystyle \begin{aligned} \mathcal{R}^{\infty}_{\mathrm{S}} ~=~ \bigcup_{p\geq 0}\mathcal{R}^p_{\mathrm{S}}\qquad \mbox{and}\qquad \mathcal{D}^{\infty}_{\mathrm{S}}=\bigcup_{p\geq 0}\mathcal{D}^p_{\mathrm{S}} \end{aligned}$$

are very rich and contain a huge variety of functions, including not only all the functions that have polynomial behaviors at infinity as discussed above, and in particular all the rational functions, but also many other functions. We observe, however, that they do not contain any strictly increasing exponential function. For instance, if g(x) = 2^x, then Δ^p g(x) = 2^x for any $p\in \mathbb {N}$, and this function does not vanish at infinity. Actually, such exponential functions grow asymptotically much faster than polynomial functions and may remain eventually p-convex even after adding nonconstant 1-periodic functions. For instance, both functions 2^x and $2^x+\sin {}(2\pi x)$ are eventually p-convex for any $p\in \mathbb {N}$.

Remark 4.5

Using (1.7) and (4.1), we also obtain $\mathcal {R}^p_{\mathrm {S}} = \mathcal {R}^{\infty }_{\mathrm {S}}\cap \mathcal {D}^p_{\mathrm {S}}$ for any $p\in \mathbb {N}$. $\lozenge$

4.2 The Intersection Sets $\mathcal {R}^p_{\mathrm {S}}\cap \mathcal {K}^p$ and $\mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$

Let us now consider the set $\mathcal {K}^p$ and its subsets $\mathcal {R}^p_{\mathrm {S}}\cap \mathcal {K}^p$ and $\mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$. As these sets will be used repeatedly throughout this book, it is important to study their basic properties. In this section, we present a number of results about these sets that will be very useful in the subsequent chapters.

Let us first observe that the set $\mathcal {K}^p$ is not a linear space. For instance, using Lemma 2.6 we can see that both functions

$$\displaystyle \begin{aligned} f(x) ~=~ x^{p+1}+\sin x\qquad \mbox{and}\qquad g(x) ~=~ x^{p+1} \end{aligned}$$

lie in $\mathcal {K}^p$ but f − g does not. We also have that Δf does not lie in $\mathcal {K}^p$ (because D ^p Δf =  ΔD ^p f does not lie in $\mathcal {K}^0$), which shows that $\mathcal {K}^p$ is not closed under the operator Δ.

The following corollary shows that $\mathcal {K}^p$ is actually the union of two convex cones. This result is an immediate consequence of Proposition 2.4.

Corollary 4.6

For any $p\in \mathbb {N}$ , the sets $\mathcal {K}^p_+$ and $\mathcal {K}^p_-$ are convex cones. These cones are opposite in the sense that f lies in $\mathcal {K}^p_+$ if and only if − f lies in $\mathcal {K}^p_-$ . Moreover, the intersection $\mathcal {K}^p_+\cap \mathcal {K}^p_-$ is the real linear space of all the real functions on $\mathbb {R}_+$ that are eventually polynomials of degree less than or equal to p.

It is now clear that $\mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$ is also the union of two opposite convex cones that is not a linear space. For instance, both functions

$$\displaystyle \begin{aligned} f(x) ~=~ 2\ln x+\frac{\sin x}{x^2}\qquad \mbox{and}\qquad g(x) ~=~ 2\ln x \end{aligned}$$

lie in $\mathcal {D}^1_{\mathrm {S}}\cap \mathcal {K}^1$ (use, e.g., Theorem 4.14(b) below) but f − g does not.

Now, the following proposition shows that, just as the sets $\mathcal {C}^0, \mathcal {C}^1, \mathcal {C}^2, \ldots$ are decreasingly nested, so are the sets $\mathcal {K}^{-1}{\!}, \mathcal {K}^0, \mathcal {K}^1, \ldots$. Thus, this latter family defines a descending filtration and we can therefore introduce the intersection set

$$\displaystyle \begin{aligned} \mathcal{K}^{\infty} ~=~ \bigcap_{p\geq 0}\mathcal{K}^p.{} \end{aligned}$$

Proposition 4.7

For any integer p ≥−1, we have $\mathcal {K}^{p+1}\varsubsetneq \mathcal {K}^p$.

Proof

Let f lie in $\mathcal {K}^{p+1}$ for some integer p ≥−1. Suppose for instance that f lies in $\mathcal {K}^{p+1}_+$ and let I be an unbounded subinterval of $\mathbb {R}_+$ on which f is (p + 1)-convex. Let $\mathcal {I}_{p+2}$ denote the set of tuples of I ^p+2 whose components are pairwise distinct. By Lemma 2.5, it follows that the restriction of the map

$$\displaystyle \begin{aligned} (z_0,\ldots,z_{p+1}) ~\mapsto ~f[z_0,\ldots,z_{p+1}] \end{aligned}$$

to $\mathcal {I}_{p+2}$ is increasing in each place. If f does not lie in $\mathcal {K}^p_-$, then there are p + 2 points x ₀ < ⋯ < x _p+1 in I such that f[x ₀, …, x _p+1] > 0. But then, f is p-convex on the interval (x _p+1, ∞), and hence it lies in $\mathcal {K}^p_+$, which establishes the inclusion. To see that the inclusion is strict, using Lemma 2.6 we just observe that the function $f\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation

$$\displaystyle \begin{aligned} f(x) ~=~ x^{p+1}+\sin x\qquad \mbox{for }x>0 \end{aligned}$$

lies in $\mathcal {K}^p\setminus \mathcal {K}^{p+1}$. □

Interestingly, Proposition 4.7 shows that the assumption that g lies in $\mathcal {K}^p$, which occurs in many statements (e.g., in Theorem 3.6), can be given equivalently by the condition that g lies in $\cup _{q\geq p}\mathcal {K}^q$.

We now present two useful propositions. The first one is very important: it shows that the sets $\mathcal {R}^p_{\mathrm {S}}\cap \mathcal {K}^p$ and $\mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$ coincide and are actually independent of S.

Proposition 4.8

For any $p\in \mathbb {N}$ , we have

$$\displaystyle \begin{aligned} \mathcal{R}^p_{\mathbb{R}}\cap\mathcal{K}^p ~=~ \mathcal{D}^p_{\mathbb{R}}\cap\mathcal{K}^p ~=~ \mathcal{R}^p_{\mathbb{N}}\cap\mathcal{K}^p ~=~ \mathcal{D}^p_{\mathbb{N}}\cap\mathcal{K}^p. \end{aligned}$$

Proof

We already know that $\mathcal {R}^p_{\mathrm {S}}\subset \mathcal {D}^p_{\mathrm {S}}$ (cf. Proposition 4.2) and $\mathcal {D}^p_{\mathbb {R}}\subset \mathcal {D}^p_{\mathbb {N}}$. Moreover, we have that $\mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p\subset \mathcal {R}^p_{\mathrm {S}}$ by Theorem 3.6. It remains to show that $\mathcal {D}^p_{\mathbb {N}}\cap \mathcal {K}^p\subset \mathcal {D}^p_{\mathbb {R}}$. Let g lie in $\mathcal {D}^p_{\mathbb {N}}\cap \mathcal {K}^p$. Suppose for instance that g lies in $\mathcal {K}^p_+$ and let a > 0 be so that g is p-convex on [a, ∞). By Lemma 2.5, Δ^p g is increasing on [a, ∞). Thus, for any x ≥ a + 1, we have

$$\displaystyle \begin{aligned} \Delta^pg(\lfloor x\rfloor) ~\leq ~ \Delta^pg(x) ~\leq ~ \Delta^pg(\lceil x\rceil). \end{aligned}$$

Letting x →∞ and using the squeeze theorem, we obtain that g lies in $\mathcal {D}^p_{\mathbb {R}}$. □

Proposition 4.9

If $f\in \mathcal {K}^p$ for some $p\in \mathbb {N}$ , then the following assertions are equivalent:

$$\displaystyle \begin{aligned} \begin{array}{rlcrlcrlcrl} (\mathrm{i}) & f\in\mathcal{R}^{p+1}_{\mathrm{S}}, & & (\mathrm{ii}) & f\in\mathcal{D}^{p+1}_{\mathrm{S}}, & & (\mathrm{iii}) & \Delta f\in\mathcal{R}^p_{\mathrm{S}}{\,}, & & (\mathrm{iv}) & \Delta f\in\mathcal{D}^p_{\mathrm{S}}{\,}. \end{array} \end{aligned}$$

Proof

By Proposition 4.2, we clearly have that (i) implies (ii) and that (iii) implies (iv). By Proposition 4.3, we also have that (i) implies (iii) and that (ii) implies (iv). Finally, by Theorem 3.1, we have that (iv) implies (i). □

Combining Proposition 4.3 with Propositions 4.7 and 4.9, we immediately obtain the following corollary, which naturally complements Proposition 4.3.

Corollary 4.10

Let $j,p\in \mathbb {N}$ be such that j ≤ p. If $f\in \mathcal {K}^{p-1}$ , then we have $f\in \mathcal {R}^p_{\mathrm {S}}$ if and only if $\Delta ^jf\in \mathcal {R}^{p-j}_{\mathrm {S}}$.

Due to Proposition 4.8, we will henceforth write $\mathcal {D}^p\cap \mathcal {K}^p$ instead of $\mathcal {D}^p_{\mathrm {S}}\cap \mathcal {K}^p$. In view of (3.10), we will also write $\mathcal {D}^{-1}\cap \mathcal {K}^0$ instead of $\mathcal {D}^{-1}_{\mathrm {S}}\cap \mathcal {K}^0$.

Since the set $\mathcal {D}^p\cap \mathcal {K}^p$ is clearly a central object of our theory (cf. our existence Theorem 3.6), it is important to investigate its properties. In this respect, we have the following two propositions.

Proposition 4.11

Let $j,p\in \mathbb {N}$ be such that j ≤ p. The following assertions hold.

• (a) If $g\in \mathcal {K}^p_+$ , then $\Delta ^j g\in \mathcal {K}^{p-j}_+$ . More precisely, for any unbounded open interval I of $\mathbb {R}_+$ , if g is p-convex on I, then Δ ^j g is (p − j)-convex on I.

• (b) If $g\in \mathcal {D}^p\cap \mathcal {K}^p_+$ , then $\Delta ^j g\in \mathcal {D}^{p-j}\cap \mathcal {K}^{p-j}_+$.

Proof

This result immediately follows from Lemma 2.6(b) and Proposition 4.3. □

Proposition 4.12

Let $j,p\in \mathbb {N}$ be such that j ≤ p and let $g\in \mathcal {C}^j$ . The following assertions hold.

• (a) $g\in \mathcal {K}^p_+$ if and only if $g^{(j)}\in \mathcal {K}^{p-j}_+$ . More precisely, for any unbounded open interval I of $\mathbb {R}_+$ , we have that g is p-convex on I if and only if g ^(j) is (p − j)-convex on I.

• (b) $g\in \mathcal {D}^p\cap \mathcal {K}^p_+$ if and only if $g^{(j)}\in \mathcal {D}^{p-j}\cap \mathcal {K}^{p-j}_+$.

Proof

Assertion (a) follows from assertions (c) and (d) of Lemma 2.6. To see that assertion (b) holds, it is enough to show that, for any p ≥ 1, we have $g\in \mathcal {D}^p\cap \mathcal {K}^p_+$ if and only if $g'\in \mathcal {D}^{p-1}\cap \mathcal {K}^{p-1}_+$.

Suppose first that g lies in $\mathcal {D}^p\cap \mathcal {K}^p_+$. Then g′ lies in $\mathcal {K}^{p-1}_+$ by assertion (a). Let x > 1 be so that g is p-convex on [x − 1, ∞). Then Δ^p−1 g′ is increasing on [x − 1, ∞) by assertion (a) and Proposition 4.11(a). By the mean value theorem, there exist $\xi ^1_x,\xi ^2_x\in (0,1)$ such that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Delta^pg(x-1) ~=~ \Delta^{p-1}g'(x-1+\xi^1_x) & \leq &\displaystyle \Delta^{p-1}g'(x)\\ & \leq &\displaystyle \Delta^{p-1}g'(x+\xi^2_x) ~=~ \Delta^pg(x). \end{array} \end{aligned} $$

Letting x →∞, we see that g′ lies in $\mathcal {D}^{p-1}_{\mathbb {R}}$ by the squeeze theorem.

Conversely, suppose that g′ lies in $\mathcal {D}^{p-1}\cap \mathcal {K}^{p-1}_+$. Then g lies in $\mathcal {K}^p_+$ by assertion (a). Let x > 0 be so that g′ is (p − 1)-convex on [x, ∞) and let t ∈ (x, x + 1). Then Δ^p−1 g′ is increasing on [x, ∞) by Proposition 4.11(a), and hence we have

$$\displaystyle \begin{aligned} \Delta^{p-1}g'(x) ~\leq ~ \Delta^{p-1}g'(t) ~\leq ~\Delta^{p-1}g'(x+1). \end{aligned}$$

Integrating on t ∈ (x, x + 1), we obtain

$$\displaystyle \begin{aligned} \Delta^{p-1}g'(x) ~\leq ~ \Delta^pg(x) ~\leq ~\Delta^{p-1}g'(x+1). \end{aligned}$$

Letting x →∞, we see that g lies in $\mathcal {D}^p_{\mathbb {R}}$. □

Remark 4.13

If a function $f\colon \mathbb {R}_+\to \mathbb {R}$ is such that Δf lies in $\mathcal {K}^p$ for some $p\in \mathbb {N}$, then f need not lie in $\mathcal {K}^{p+1}$, even if Δf lies in $\mathcal {D}^p\cap \mathcal {K}^p$. For instance, the function $f\colon \mathbb {R}_+\to \mathbb {R}$ defined by the equation

$$\displaystyle \begin{aligned} f(x) ~=~ \frac{1}{2^x}\left(1+\frac{1}{3}\sin x\right)\qquad \mbox{for }x>0 \end{aligned}$$

lies in $\mathcal {K}^0_-\setminus \mathcal {K}^1$. Indeed, 2^x f′(x) is 2π-periodic and negative while 2^x f″(x) is 2π-periodic and change in sign from $x=\frac {\pi }{6}$ to x = π. However, the function Δf lies in $\mathcal {D}^0\cap \mathcal {K}^0_+$ for 2^x Δf′(x) is 2π-periodic and positive. This example shows that the implications in Proposition 4.11 cannot be equivalences. $\lozenge$

If a function $g\colon \mathbb {R}_+\to \mathbb {R}$ lies in $\mathcal {D}^p\cap \mathcal {K}^p$ for some $p\in \mathbb {N}$, then by Proposition 4.11 the function Δ^p g lies in $\mathcal {D}^0\cap \mathcal {K}^0$, i.e., Δ^p g eventually increases or decreases to zero. However, a function $g\colon \mathbb {R}_+\to \mathbb {R}$ that satisfies this latter property need not lie in $\mathcal {D}^p\cap \mathcal {K}^p$, unless g lies in $\mathcal {K}^p$ or p = 0. The example introduced in Remark 4.13 illustrates this phenomenon when p = 1. On the other hand, when g lies in $\mathcal {C}^p$, by Proposition 4.12 we have that g lies in $\mathcal {D}^p\cap \mathcal {K}^p$ if and only if g ^(p) lies in $\mathcal {D}^0\cap \mathcal {K}^0$.

We gather these important observations in the following theorem.

Theorem 4.14

Let $p\in \mathbb {N}$ . The following assertions hold.

• (a) Let $g\in \mathcal {K}^p_+$ (resp. $\mathcal {K}^p_-$ ). Then g lies in $\mathcal {D}^p\cap \mathcal {K}^p_+$ (resp. $\mathcal {D}^p\cap \mathcal {K}^p_-$ ) if and only if Δ ^p g eventually increases (resp. decreases) to zero.

• (b) Let $g\in \mathcal {C}^p$ . Then g lies in $\mathcal {D}^p\cap \mathcal {K}^p_+$ (resp. $\mathcal {D}^p\cap \mathcal {K}^p_-$ ) if and only if g ^(p) eventually increases (resp. decreases) to zero.

Proof

Assertion (a) immediately follows from Propositions 4.3 and 4.11. Assertion (b) immediately follows from Proposition 4.12. □

Remark 4.15

It is not difficult to see that the function $g(x)=\frac {1}{x}\sin x^2$ vanishes at infinity while its derivative does not. Theorem 4.14(b) shows that if g lies in $\mathcal {C}^q\cap \mathcal {D}^p\cap \mathcal {K}^q$ for some $p,q\in \mathbb {N}$ such that p ≤ q, then all the functions g ^(p), g ^(p+1), …, g ^(q) vanish at infinity. $\lozenge$

Propositions 4.11 and 4.12 do not provide any information on the functions Δg and g′ when g lies in $\mathcal {D}^0\cap \mathcal {K}^0$ and $\mathcal {C}^1\cap \mathcal {D}^0\cap \mathcal {K}^0$, respectively. The following proposition fills this gap under the additional assumptions that Δg and g′ lie in $\mathcal {K}^0$, respectively.

Proposition 4.16

The following assertions hold.

• (a) If g lies in $\mathcal {D}^0\cap \mathcal {K}^0_-$ and is such that Δg lies in $\mathcal {K}^0$ , then Δg lies in $\mathcal {D}^{-1}\cap \mathcal {K}^0_+$.

• (b) If g lies in $\mathcal {C}^1\cap \mathcal {D}^0\cap \mathcal {K}^0_-$ and is such that g′ lies in $\mathcal {K}^0$ (or equivalently, g lies in $\mathcal {K}^1$ ), then g′ lies in $\mathcal {C}^0\cap \mathcal {D}^{-1}\cap \mathcal {K}^0_+$.

Proof

Let us first prove assertion (a). Since g is eventually decreasing, Δg must be eventually negative. But since Δg also lies in $\mathcal {D}^0\cap \mathcal {K}^0$, it must be eventually increasing to zero. On the other hand, since g lies in $\mathcal {D}^0$, Δg must lie in $\mathcal {D}^{-1}_{\mathbb {N}}$. This proves assertion (a).

Let us now prove assertion (b). Since g is eventually decreasing, g′ must be eventually negative. Since g′ lies in $\mathcal {K}^0$ (and hence g lies in $\mathcal {K}^1$ by Lemma 2.6), we have that g lies in $\mathcal {D}^1\cap \mathcal {K}^1$ (since $\mathcal {D}^0_{\mathrm {S}}\subset \mathcal {D}^1_{\mathrm {S}}$). Proposition 4.12 then tells us that g′ lies in $\mathcal {D}^0\cap \mathcal {K}^0$, and hence it must be eventually increasing to zero.

It remains to show that g′ lies in $\mathcal {D}^{-1}_{\mathbb {N}}$. Let x > 1 be so that g is decreasing and g′ is increasing on I _x = [x − 1, ∞). By the mean value theorem, for any integer k ≥ x there exist ξ _k ∈ (0, 1) such that

$$\displaystyle \begin{aligned} \Delta g(k-1) ~=~ g'(k-1+\xi_k) ~\leq ~ g'(k). \end{aligned}$$

For any integers m, n such that x ≤ m ≤ n, we then have

$$\displaystyle \begin{aligned} g(n-1)-g(m-1) ~=~ \sum_{k=m}^{n-1}\Delta g(k-1) ~\leq ~\sum_{k=m}^{n-1}g'(k) ~\leq ~ 0. \end{aligned}$$

Letting $n\to _{\mathbb {N}}\infty$, we can see that g′ lies in $\mathcal {D}^{-1}_{\mathbb {N}}$. □

Remark 4.17

The assumption that Δg lies in $\mathcal {K}^0$ cannot be ignored in Proposition 4.16(a). Indeed, take for instance the function g =  Δf, where f is the function defined in Remark 4.13. We have seen that this function lies in $\mathcal {D}^0\cap \mathcal {K}^0$. However, it is not difficult to see that Δg does not lie in $\mathcal {K}^0$. Similarly, the assumption that g′ lies in $\mathcal {K}^0$ cannot be ignored in Proposition 4.16(b). Indeed, one can show that the same function g has the property that g′ does not lie in $\mathcal {K}^0$. To give another example, one can show that the function

$$\displaystyle \begin{aligned} g(x) ~=~ \frac{1}{x^3}(x+\sin x) \end{aligned}$$

lies in $\mathcal {D}^0\cap \mathcal {K}^0$ whereas its derivative g′ does not lie in $\mathcal {K}^0$. $\lozenge$

We also have the following two corollaries, in which the symbols $\mathcal {R}$ and $\mathcal {D}$ can be used interchangeably.

Corollary 4.18

Let g lie in $\mathcal {K}^p_+$ (resp. $\mathcal {K}^p_-$ ) for some $p\in \mathbb {N}$ . Then g lies in $\mathcal {D}^p_{\mathrm {S}}$ if and only if there exists a solution $f\colon \mathbb {R}_+\to \mathbb {R}$ to the equation Δf = g that lies in $\mathcal {D}^{p+1}_{\mathrm {S}}\cap \mathcal {K}^p_-$ (resp. $\mathcal {D}^{p+1}_{\mathrm {S}}\cap \mathcal {K}^p_+$ ).

Proof

The $\mathcal {D}$-version immediately follows from Theorem 3.6 and Proposition 4.3(b). The $\mathcal {R}$-version then follows from Proposition 4.9 and Proposition 4.3(a). □

Corollary 4.19

For any $p\in \mathbb {N}$ , we have that

$$\displaystyle \begin{aligned} \mathcal{D}^p\cap\mathcal{K}^p_+ ~\subset ~\mathcal{K}^{p-1}_-\qquad \mathit{\mbox{and}}\qquad \mathcal{D}^p\cap\mathcal{K}^p_- ~\subset ~\mathcal{K}^{p-1}_+. \end{aligned}$$

More precisely, if g lies in $\mathcal {D}^p\cap \mathcal {K}^p$ and is p-convex (resp. p-concave) on an unbounded interval of $\mathbb {R}_+$ , then on this interval it is also (p − 1)-concave (resp. (p − 1)-convex).

Proof

Let g lie in $\mathcal {D}^p\cap \mathcal {K}^p_+$. Then the function $f\colon \mathbb {R}_+\to \mathbb {R}$ defined in the existence Theorem 3.6 is p-concave on any unbounded subinterval of $\mathbb {R}_+$ on which g is p-convex. By Lemma 2.6(b), the function g =  Δf is also (p − 1)-concave on this interval. □

We end this chapter by providing a characterization of the set $\mathcal {R}^p\cap \mathcal {K}^p=\mathcal {D}^p\cap \mathcal {K}^p$ in terms of interpolating polynomials. We also give a corollary that will be very useful in the subsequent chapters.

Proposition 4.20

Let g lie in $\mathcal {K}^p$ for some $p\in \mathbb {N}$ . Then we have that g lies in $\mathcal {D}^p_{\mathrm {S}}$ if and only if for any pairwise distinct x ₀, …, x _p > 0, we have that

$$\displaystyle \begin{aligned} g[a+x_0,\ldots,a+x_p]\to 0\qquad \mathit{\mbox{as }}a\to_{\mathrm{S}}\infty. \end{aligned}$$

This latter condition means that g asymptotically coincides with its interpolating polynomial with any p nodes.

Proof

(Necessity) Suppose for instance that g lies in $\mathcal {D}^p\cap \mathcal {K}^p_+$. By Corollary 4.19, it also lies in $\mathcal {K}^{p-1}_-$. Let x ₀, …, x _p > 0 be any pairwise distinct points and let a > 0 be so that g is p-convex and (p − 1)-concave on [a, ∞). Then the map

$$\displaystyle \begin{aligned} x ~ \mapsto ~ g[x+x_0,\ldots,x+x_p] \end{aligned}$$

is nonpositive on [a, ∞) and, by Lemma 2.5, it is also increasing on [a, ∞). By (2.8), we then have

$$\displaystyle \begin{aligned} \frac{1}{p!}\,\Delta^pg(a) ~=~ g[a,a+1,\ldots,a+p] ~\leq ~ g[a+p+x_0,\ldots,a+p+x_p] ~\leq ~ 0, \end{aligned}$$

where the left side increases to zero as a →_S ∞.

(Sufficiency) This immediately follows from Propositions 4.1 and 4.8. □

Corollary 4.21

Let g lie in $\mathcal {K}^p_+$ (resp. $\mathcal {K}^p_-$ ) for some $p\in \mathbb {N}$ , let a > 0 and b ≥ 0, and let $h\colon \mathbb {R}_+\to \mathbb {R}$ be defined by the equation h(x) = g(ax + b) for x > 0. Then

• (a) h lies in $\mathcal {K}^p_+$ (resp. $\mathcal {K}^p_-$ );

• (b) if g lies in $\mathcal {D}^p\cap \mathcal {K}^p$ , then h lies in $\mathcal {D}^p\cap \mathcal {K}^p_+$ (resp. $\mathcal {D}^p\cap \mathcal {K}^p_-$ ).

Proof

The result is trivial if p = 0. So let us assume that p ≥ 1 and for instance that g is p-convex on [s, ∞) for some s > 0. Using (2.4), we can easily show that for any pairwise distinct points x ₀, …, x _p > 0 we have

$$\displaystyle \begin{aligned} h[x_0,\ldots,x_p] ~=~ a^p{\,}g[ax_0+b,\ldots,ax_p+b]. \end{aligned}$$

This immediately shows that h is p-convex on $[\frac {1}{a}(s-b),\infty )$ and hence that assertion (a) holds. Now, suppose that g lies in $\mathcal {D}^p\cap \mathcal {K}^p_+$. Then h lies in $\mathcal {K}^p_+$ by assertion (a). Moreover, for any pairwise distinct x ₀, …, x _p > 0, by Proposition 4.20 we have that

$$\displaystyle \begin{aligned} h[n+x_0,\ldots,n+x_p] ~=~ a^p{\,}g[an+ax_0+b,\ldots,an+ax_p+b] ~\to ~0 \end{aligned}$$

as $n\to _{\mathbb {N}}\infty$. Hence h also lies in $\mathcal {D}^p\cap \mathcal {K}^p_+$ by Proposition 4.20. This establishes assertion (b). □