Abstract
We provide a list of equivalent conditions under which an additive operator acting on a space of smooth functions on a compact real interval is a multiple of the derivation.
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1 Introduction
By \(\mathbb {R}\) we denote the set of reals, \(\mathbb {Q}\) are rationals, \(\mathbb {Z}\) are integers, \(\mathbb {N}=\{1, 2, \ldots \}\) and \(\mathbb {N}_0=\mathbb {N}\cup \{0 \}\). If \(I\subseteq \mathbb {R}\) is an interval and \(k\in \mathbb {N}_0\), then \(C^k(I)\) is the space of real-valued functions on I that are k-times continuously differentiable on the interior of I. If \(k=0\), then we write simply C(I). The space \(C^k(I)\) is furnished with the standard pointwise algebraic operations and hence it is a real commutative algebra.
Definition
(e.g. Kuczma [12, page 391]) Assume that Q is a commutative ring and P is a subring of Q. A function \(f:P \rightarrow Q\) is called derivation if it is additive:
and it satisfies the Leibniz rule:
The following theorem describes derivations over fields of characteristic zero.
Theorem 1
[12, Theorem 14.2.1] Let K be a field of characteristic zero, F be a subfield of K, S be an algebraic base of K over F if it exists, and let \(S = \varnothing \) otherwise. If \(f :F \rightarrow K\) is a derivation, then, for every function \(u:S \rightarrow K\) there exists a unique derivation \(g:K \rightarrow K\) such that \(g=f\) on F and \(g=u\) on S.
From this theorem it follows in particular that nonzero derivations \(f:\mathbb {R}\rightarrow \mathbb {R}\) exist. It is well known they are discontinuous and very irregular mappings. For an exhaustive discussion of the notion of derivation and related functional equations the reader is referred to Gselmann [5, 6], Gselmann, Kiss, Vincze [7] and the references therein. Recently Ebanks [2, 3] studied derivations and derivations of higher order on rings.
The “model” example of a derivation is the operator of derivative on the space \(C^k(I)\) for \(k>0\) . Indeed, if we define \(T:C^k(I)\rightarrow C(I)\) as \(T(f) = f'\) for \(f \in C^k(I)\), then clearly \( C^k(I)\) is a subring of C(I), T is additive and it satisfies the Leibniz rule:
Crucial results about equation (3) on the space \(C^k(I)\) are due to H. König and V. Milman. We refer the reader to their recent monograph [11]. They studied several operator equations and inequalities that are related to derivatives on the spaces of smooth functions. Later on, we will utilize their elegant result [11, Theorem 3.1] regarding (3). Briefly, if I is an open set, then the general solution of (3) for all \( f, g \in C^k(I)\) is of the form
for some continuous functions \(c, d\in C(I)\), if \(k>0\), and
if \(k=0\) (in formulas (4) and (5) the convention that \(0\cdot \ln 0 = 0\) is adopted). Note that no additivity is assumed.
It is a natural question to characterize real-to-real derivations among additive functions with the aid of a relation which is weaker than (2). In particular, the very first article published in the first volume of Aequationes Mathematicae by Nishiyama and Horinouchi [14] addresses this question. The authors studied the following relations, each of which is a direct consequence of (2) alone and together with (1) implies (2):
and
where \(a\ne 1\) and n, m are integers such that \(am = n\ne 0\). Further similar results, as well as some generalizations, are due to Jurkat [8], Kannappan and Kurepa [9, 10], Kurepa [13], among others. Ebanks [4] generalized and extended these results to arbitrary fields. A recent paper by Amou [1] provides some n-dimensional generalizations of the results of [8,9,10, 13].
This paper provides versions of the above-mentioned results for operators \(T:C^k(I)\rightarrow C(I)\). Therefore, we seek conditions which are equivalent to (3).
2 Main results
Throughout this section let us fix \(k\in \mathbb {N}_0\) and an interval \(I\subseteq \mathbb {R}\). We will study conditions upon an additive operator \(T:C^k(I)\rightarrow C(I)\) which yield analogues to Eqs. (6), (714) and (8). Therefore, we will focus on the following operator relations:
Our first theorem is a simple observation that some reasonings concerning derivations from the real-to-real case can be extended to arbitrary commutative rings without substantial changes. We adopted parts of the proof of [12, Theorem 14.3.1].
Theorem 2
Assume that Q is a commutative ring, P is a subring of Q and \(T:P\rightarrow Q\) is an additive operator. Then, the following conditions are pairwise equivalent:
-
(i)
T satisfies \(T(f^2) = 2f \cdot T(f)\) for all \(f \in P\),
-
(ii)
T satisfies \(T(f\cdot g) = f\cdot T(g) + g \cdot T(f)\) for all \(f, g \in P\),
-
(iii)
T satisfies \(T(f^{n}) = nf^{n-1}\cdot T(f)\) for all \(f \in P\) and \(n \in \mathbb {N}\).
Proof
\(\mathrm{(i)} \Rightarrow \mathrm{(ii)}\). Fix arbitrarily \(f, g \in P\). By (9) we get
Since T is additive,
Using (9) again, after reductions we obtain (3).
\(\mathrm{(ii)} \Rightarrow \mathrm{(iii)}\). If \(n=1\), then (11) reduces to an identity. Assume that (11) holds for some \(n \in \mathbb {N}\) and all \(f \in P\). Then, by (3) and the induction hypothesis we have
\(\mathrm{(iii)} \Rightarrow \mathrm{(i)}\). Take \(n=2\). \(\square \)
The next corollary will be utilized later on.
Corollary 1
Assume that \(T:C^k(I)\rightarrow C(I)\) is an additive operator. Then, the following conditions are pairwise equivalent:
-
(i)
T satisfies \(T(f^2) = 2f \cdot T(f)\) for all \(f \in C^k(I)\),
-
(ii)
T satisfies \(T(f\cdot g) = f\cdot T(g) + g \cdot T(f)\) for all \(f, g \in C^k(I)\),
-
(iii)
T satisfies \(T(f^{n}) = nf^{n-1}\cdot T(f)\) for all \(f \in C^k(I)\) and \(n \in \mathbb {N}\).
Our next result characterizes the Leibniz rule (3) on a domain restricted to functions separated from zero. Thus, we can consider conditions (10) and (11) for negative n, which involve the function 1/f. The situation is a bit more complicated, but Theorem 3 below has a mainly technical role.
Theorem 3
Assume that \(T:C^k(I)\rightarrow C(I)\) is an additive operator and \(\varepsilon _1\in (0,1)\), \(\varepsilon _2\in (0,1)\) and \(c\in (1, +\infty ]\) are constants. Consider the following conditions:
-
(i)
T satisfies \(T(f) = -f^{2}\cdot T\left( \frac{1}{f}\right) \) for all \(f \in C^k(I)\), \(c>f>\varepsilon _1\),
-
(ii)
T satisfies \(T(f^2) = 2f \cdot T(f)\) for all \(f \in C^k(I)\), \(f>\varepsilon _2\),
-
(iii)
T satisfies \(T(f\cdot g) = f\cdot T(g) + g \cdot T(f)\) for all \(f, g \in C^k(I)\), \(f>\varepsilon _2\), \(g>\varepsilon _2\),
-
(iv)
T satisfies \(T(f^{n}) = nf^{n-1}\cdot T(f)\) for all \(n \in \mathbb {Z}\) and all \(f \in C^k(I)\) such that \(\varepsilon _2<f<1/\varepsilon _2\), and \(f^{n-1}>\varepsilon _2\) if \(n>0\) and \(f^{n+1}>\varepsilon _2\) if \(n<0\).
Then: (i) with \(c=+\infty \) implies (ii) with \(\varepsilon _2> \sqrt{\varepsilon _1}\), (ii) and (iii) are equivalent, (iii) implies (iv), (iv) implies (i) with \(\varepsilon _1=\varepsilon _2\) and \(c= 1/\varepsilon _2\).
Proof
\(\mathrm{(i)}\Rightarrow \mathrm{(ii)}\). First, note that by applying (10) for \(f=1\) and using the rational homogeneity of T we get that T vanishes on each constant function equal to a rational number. Observe that for an arbitrary rational \(\delta >0\) (which will be chosen later) the identity
holds for \(f \in C^k(I)\) such that \(f>\delta \). Next, if \(\varepsilon _1>0\) is given and \(\varepsilon _2> \sqrt{\varepsilon _1}\), then we will find some rational \(\delta >0\) such that \(\varepsilon _2>\varepsilon _1+\delta \) and \(\varepsilon _2^2>\varepsilon _1+\delta ^2\). Consequently, if \(f \in C^k(I)\) and \(f>\varepsilon _2\), then \(f\pm \delta >\varepsilon _1\) and \(f^2-\delta ^2 >\varepsilon _1\). Using (i) three times together with (12) and the additivity of T we obtain
\(\mathrm{(ii)}\Leftrightarrow \mathrm{(iii)}\). Analogously as in Theorem 2 for \(f>\varepsilon _2\) and \(g>\varepsilon _2\). \(\mathrm{(iii)}\Rightarrow \mathrm{(iv)}\). If \(n=1\), then (11) is trivially satisfied. Assume that f, n and \(\varepsilon _2\) satisfy the assumptions of (iv). For \(n>1\) we proceed like in Theorem 2. If \(n=0\), then (iv) reduces to \(T(1)=0\), which follows from (iii). If \(n=-1\), then for \(1/\varepsilon _2>f>\varepsilon _2\) we have
Assume that \(n<-1\). By downward induction, one can check that for \(f^{n+1}>\varepsilon _2\) we have from (3)
\((iv)\Rightarrow (i)\). Take \(n=-1\). \(\square \)
If we assume additionally that interval I is compact, then the situation clarifies considerably.
Theorem 4
Assume that I is compact and \(T:C^k(I)\rightarrow C(I)\) is an additive operator. Then, the following conditions are pairwise equivalent:
-
(i)
T satisfies \(T(f\cdot g) = f\cdot T(g) + g \cdot T(f)\) for all \(f, g \in C^k(I)\),
-
(ii)
T satisfies \(T(f\cdot g) = f\cdot T(g) + g \cdot T(f)\) for all \(f, g \in C^k(I), f>0\), \(g>0\),
-
(iii)
T satisfies \(T(f^2) = 2f \cdot T(f)\) for all \(f \in C^k(I)\),
-
(iv)
T satisfies \(T(f^2) = 2f \cdot T(f)\) for all \(f \in C^k(I)\), \(f>0\),
-
(v)
T satisfies \(T(f) = -f^{2}\cdot T\left( \frac{1}{f}\right) \) for all \(f \in C^k(I)\), \(f>0\),
-
(vi)
T satisfies \(T(f^{n}) = nf^{n-1}\cdot T(f)\) for all \(f \in C^k(I)\) and \(n \in \mathbb {N}\),
-
(vii)
T satisfies \(T(f^{n}) = nf^{n-1}\cdot T(f)\) for all \(f \in C^k(I)\), \(f>0\) and \(n \in \mathbb {N}\).
Proof
This statement is a consequence of Corollary 1 and Theorem 3. Since I is compact, f attains its global extrema. Thus, we will find some rational \(r, q \in \mathbb {Q}\) such that \(1/2<rf+q<2\). Moreover, as it was already observed in the proof of Theorem 3, each of the conditions of Theorem 4 implies that \(T(1)=0\) and then T vanishes on constant functions equal to a rational number. Consequently, we have \(T(rf+q)=rT(f) + T(q) = rT(f)\) and therefore Theorem 3 applies to the conditions (ii), (iv), (v) and (vii) with appropriately chosen \(\varepsilon _1\) and \(\varepsilon _2\). The remaining conditions are equivalent by Corollary 1. Therefore, we are done if we prove for example the implication \(\mathrm{(iv)}\Rightarrow \mathrm{(iii)}\).
Fix \(f\in C^k(I)\) arbitrarily and choose \(r, q \in \mathbb {Q}\) such that \(1/2<rf+q<2\). By (iv) we get
Then using additivity we obtain
and after reduction
i.e. condition (iii). \(\square \)
One can join Corollary 1 and Theorem 4 with the mentioned result of H. König and V. Milman to obtain a corollary.
Corollary 2
Under the assumptions of Corollary 1 or Theorem 4, if \(k>0\), then each of the conditions listed there is equivalent to the following one:
-
(x)
there exists some \(d\in C(I)\) such that \(T(f) = d\cdot f'\) for all \(f \in C^k(I)\)
and if \(k=0\), then \(T=0\) is the only additive operator that fulfils any of the equivalent conditions.
Proof
Consider \(f(x) = x\) on I and denote \({\tilde{d}}:= T(f)\in C(I)\). Next, note that by [11, Theorem 3.1] the formulas (4) and (5), respectively hold on the interior of I with some \(c, d \in C(\textrm{int} I)\). The additivity of T implies that \(c=0\). Therefore \({\tilde{d}}\) is a continuous extension of d to the whole interval I. \(\square \)
3 Final remarks
Remark
The inequalities between f, g and constants \(\varepsilon _1\) and \(\varepsilon _2\) in Theorem 3 are not optimal. This however was not our goal since the role of this result is auxiliary only. Similarly, the inequality \(f>0\) in some of the conditions of Theorem 4 can be equivalently replaced by an estimate from above or from below by any other fixed constant.
Moreover, in the proof of Theorem 4 we showed more than is stated. Namely, it is equivalently enough to assume, instead of \(f>0\), that f is bilaterally bounded by two rational numbers, like 1/2 and 2. However, since this generalization is only apparent and easy, we do not include it in the formulation of the theorem.
Example 1
Assume that \(\varphi :(1, \infty ) \rightarrow \mathbb {R}\) is a smooth mapping that satisfies the equation
Such mappings exist in abundance. In fact, every map \(\varphi _0\) defined on (1, 2] can be uniquely extended to a solution of (13). Next, let \(d:(e, \infty ) \rightarrow \mathbb {R}\) be defined as
It is easy to see that
and
in general, unless \(\varphi \) is additive. Define \(T:C^1((e, \infty ))\rightarrow C((e, \infty ))\) as follows:
One can see that T satisfies (9) for all \(f, g \in C((e, \infty ))\), but fails to satisfy the Leibniz rule (3). Thus, the assumption of additivity in all our results is essential. Observe also that T has the property that it vanishes on constant functions equal to a rational. This fact, as a consequence of additivity, was frequently used in the proofs of our Theorems 3 and 4. Therefore, the additivity assumption cannot be relaxed to this property.
Example 2
Assume that I is an interval and T is given by the formula
Then T satisfies (3) for all \(f, g \in C^2(I)\) such that \(f>0\) and \(g>0\). This observation is a particular case of the second part of [11, Corollary 3.4]. Clearly, T is not additive. Moreover, T cannot be extended in such a way that it satisfies (3) on the whole space \(C^2(I)\).
The following examples show that if the domain of operator T is changed, then the conditions discussed in our results are no longer equivalent and various situations are possible.
Example 3
Let \(\mathcal {S}\) be the space of all functions \(f \in C^1((0, \infty ))\) which satisfy the functional equation
Note that \(\mathcal {S}\) is not closed under multiplication. Moreover, each function \(f_0: (0, 1] \rightarrow \mathbb {R}\) can be uniquely extended to a solution of (714). Therefore, \(\mathcal {S}\) is an infinite-dimensional subspace of \(C^1((0, \infty ))\). Define \(T:C^1((0, \infty )) \rightarrow C^1((0, \infty ))\) by the formula
It is easy to check that T is additive and satisfies (3) for \(f, g \in \mathcal {S}\). Thus, there are more solutions of (3) if the domain of T is restricted to a particular subspace of \(C^k(I)\).
Example 4
Let P[x] be the space of all real polynomials of variable x. By \(\deg (f)\) we denote the degree of a polynomial \(f\in P[x]\). Define \(T:P[x] \rightarrow P[x]\) by
Then T is not additive, it satisfies (3) and there exists no extension of T to the whole space \(C^k(\mathbb {R})\) which is a solution of (3).
Example 5
Let
Note that \(\mathcal {S}\) is closed under multiplication but it is not a linear space. Next, let a double sequence \(\varphi \) on \(\mathbb {Z}\) of natural numbers be defined as follows: \(\varphi (0)=0\), \(\varphi (k)\) is arbitrary but \(\ne k\) if k is odd, and if \(k=2^n \cdot m\) with some \(n \in \mathbb {N}\) and odd \(m\in \mathbb {Z}\), then
Note that we have
Define \(T:\mathcal {S}\rightarrow C((0, \infty ))\) by
if \(f(x) = x^k\) for \( x \in (0, \infty )\). One can see that if f is of this form, then by (15)
for all \(x \in (0, \infty )\), i.e. T satisfies (9).
Moreover, one can see that (10) is equivalent to the equality
Therefore, we can construct a sequence \(\varphi \) such that T defined by (16) satisfies (10) as well as another sequence \(\varphi '\) for which T does not satisfy (10). Finally, (3) is not true on \(\mathcal {S}\). Indeed, note that if (3) is satisfied by T given by (16), then:
which does not hold.
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Fechner, W., Świątczak, A. Characterizations of derivations on spaces of smooth functions. Aequat. Math. 97, 777–786 (2023). https://doi.org/10.1007/s00010-022-00934-x
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DOI: https://doi.org/10.1007/s00010-022-00934-x