The article covers important mathematical properties of uniformly almost-periodic functions and provides proofs of the corresponding theorems. It is shown that a uniformly almost-periodic function is bounded on the entire real axis and, as a consequence, the square of such a function also possesses the uniformly almost-periodic property. It is also proved that these functions are uniformly continuous on the entire real axis. Furthermore, theorems are established demonstrating that the sum and product of uniformly almost-periodic functions retain the same property. The limit of a uniformly convergent sequence of such functions is also shown to be a uniformly almost-periodic function. If the derivative of an almost-periodic function is uniformly continuous on the entire real axis, then its derivative is also uniformly almost-periodic. Finally, it is proved that if an indefinite integral of a uniformly almost-periodic function is bounded, then the integral itself is a uniformly almost-periodic function.
Almost-periodic functions constitute an important class of functions that generalize the classical concept of periodicity
A continuous function f : R → C is called uniformly almost-periodic if, for every ε > 0, the set of real numbers τ, called ε-almost-periods, such that |f(x + τ) − f(x)| < ε for all x ∈ R. is relatively dense. The corresponding number ℓ = ℓ(ε) is called an inclusion length. This property is a natural generalization of periodicity and gives rise to a rich collection of structural properties
Since Bohr’s work, several wider classes have been introduced, namely the Stepanov, Weyl and Besicovitch classes
tions
Theorem 1. A uniformly almost-periodic function is bounded on the entire real axis.
Proof : Let f(x) be a uniformly almost-periodic function and let ε = 1. Since the set of 1-almost-periods is relatively dense, there is a number l = l(1) such that every interval of length l contains a 1-almost-period. Denote by M0 the maximum of the continuous function |f(x)| on the closed interval [0, l].
Let x0 be an arbitrary point of the real axis. Choose an almost-period τ = τ(1) in the interval (−x0, −x0 + l). Then 0 < x0 + τ < l. Consequently,
|f(x0)| = |f(x0) − f(x0 + τ) + f(x0 + τ)| ≤ |f(x0) − f(x0 + τ)| + |f(x0 + τ)| < 1 + M0 = M.
Thus, f(x) is bounded on the entire real axis. □ 183 Mammadov and Agaliyeva | K. Multidiscip. Int. Sci. J. 3(3) (2026)181-187
Therefore, for every ε/(2M) > 0, an ε/(2M)-almost-period of f(x) is an ε-almost-period of f 2(x). Hence f 2(x) is uniformly almost-periodic.
Theorem 2. A uniformly almost-periodic function is uniformly continuous on the entire real axis.
Proof : Let ε > 0 be arbitrary and let l = l(ε/3) be an inclusion length for the ε/3-almost-periods. The function f(x) is uniformly continuous on the closed interval [−1, 1 + l]. Therefore there exists a number δ = δ(ε/3) with 0 < δ ≤ 1 such that, for arbitrary y1, y2 ∈ [−1, 1 + l], the condition |y2 − y1| < δ implies |f(y2) − f(y1)| < ε/3.
Now let x1 and x2 be arbitrary real numbers with |x2 − x1| < δ. Let τ be an ε/3-almost-period of f(x) lying in the interval (−x1, −x1 + l). Then 0 < x1 + τ < l, and since |x2 − x1| < δ ≤ 1, the point x2 + τ also belongs to [−1, 1 + l]. Hence |f(x2) − f(x1)| ≤ |f(x2) − f(x2 + τ)| + |f(x2 + τ) − f(x1 + τ)| + |f(x1 + τ) − f(x1)| ε ε ε < + + = ε.
3 3 3 Since ε is arbitrary, f(x) is uniformly continuous on the entire real axis. □
A continuous function f(x) is called a normal function if the family {f(x + h) : −∞ < h < ∞} is relatively compact with respect to uniform convergence on the entire real axis, that is, every sequence f(x + hn) has a subsequence which converges uniformly on R.
Theorem 3 (Bochner’s criterion). A continuous function f(x) is uniformly almost-periodic if and only if it is a normal function.
This theorem was proved by Bochner
Theorem 4. The sum and product of uniformly almost-periodic functions are also uniformly almost-periodic.
Proof : Let f(x) and g(x) be uniformly almost-periodic functions. By Theorem 3 the families {f(x + h)} and {g(x + h)} are relatively compact, and by Theorem 1 both functions are bounded, say |f|, |g| ≤ M. Consider an arbitrary sequence hn. First choose a subsequence h′ n such that f(x + h′ n) converges uniformly to some φ. Then, from h′ n, choose a further subsequence h′′ n such that g(x + h′′ n) converges uniformly to some ψ. Consequently f(x + h′′ n) + g(x + h′′ n) converges uniformly to φ + ψ. Moreover, |f g(x + h′′ n) − φψ| ≤ M |g(x + h′′ n) − ψ| + |ψ| |f(x + h′′ n) − φ| → 0 uniformly on R, because |ψ| ≤ M. Therefore the families {f(x + h) + g(x + h)} and {f(x + h)g(x + h)} are relatively compact, and by Theorem 3 the functions f(x) + g(x) and f(x)g(x) are uniformly almost-periodic. □
Theorem 5. Let f1(x), f2(x), . . . be a sequence of uniformly almost-periodic functions which converges uniformly on R, and let f(x) = limn→∞ fn(x). Then f(x) is uniformly almost-periodic.
Proof : The limit f is continuous, being a uniform limit of continuous functions. For an arbitrary positive number ε there exists an integer N = N(ε) such that
sup |f(x) − fN (x)| < .
Let τ be an ε/3-almost-period of the uniformly almost-periodic function fN (x). Then |f(x + τ) − f(x)| ≤ |f(x + τ) − fN (x + τ)| + |fN (x + τ) − fN (x)| + |fN (x) − f(x)| < ε.
Thus every ε/3-almost-period of fN is an ε-almost-period of f, and f(x) is uniformly almost-periodic. □ Theorem 6. Let f(x) be a uniformly almost-periodic function which is differentiable on R, and suppose that f ′(x) is uniformly continuous on R. Then f ′(x) is uniformly almost-periodic.
Proof : For h = 0 ̸ put f(x + h) − f(x) gh(x) = .
Every ε-almost-period of f is also an ε-almost-period of the translate f(x + h), so f(x + h) is uniformly almost-periodic. By Theorem 4 (and since multiplication by a constant preserves almost periodicity), gh(x) is uniformly almost-periodic for each h = 0 ̸ . The derivative f ′ is continuous, hence ∫︂ h ′ 1 (︁ ′ ′ )︁ gh(x) − f (x) = f (x + s) − f (x) ds.
Let ω(η) = sup{|f ′(u) − f ′(v)| : |u − v| ≤ η} be the modulus of continuity of f ′; by assumption ω(η) → 0 as η → 0. Then supx |gh(x) − f ′(x)| ≤ ω(|h|) → 0 as h → 0. Thus g1/n → f ′ uniformly on R, and Theorem 5 shows that f ′ is uniformly almost-periodic. □
For a periodic function f(x), a necessary and sufficient condition for its indefinite integral to be periodic is that the constant term (the mean value) in the Fourier series of f(x) be equal to zero
A similar question arises for uniformly almost-periodic functions. Such functions also possess Fourier series, and 1 ∫︁ T the mean value limT →∞ 2T −T f(x) dx exists
On the other hand, if the indefinite integral F of a uniformly almost-periodic function is itself uniformly almost-periodic, then F must be bounded on the entire real axis by Theorem 1. The following theorem shows that this necessary condition is also sufficient.
Theorem 7. If the indefinite integral of a uniformly almost-periodic function is bounded, then it is also a uniformly almost-periodic function.
Proof : By assumption, ∫︂ x F(x) = f(t) dt + C
is a bounded function. Denote its least upper bound and greatest lower bound by G and g, respectively. If G = g, then F is constant and there is nothing to prove, so let G > g. It suffices to consider ε with 0 < ε < 3(G − g), because an ε′-almost-period with ε′ < ε is also an ε-almost-period. We shall show that there exists a positive number ε1 = ε1[ε, f] such that every ε1-almost-period of f(x) is an ε-almost-period of F(x).
Choose two fixed numbers x1 and x2 satisfying
F(x1) < g + , F(x2) > G − .
(︁ ε )︁ Then F(x2) − F(x1) > G − g − ε/3 > 0, so x1 ̸= x2. Let d = |x1 − x2| > 0 and m = min(x1, x2). Let l0 = l 6d be an inclusion length for the 6εd -almost-periods of f, and put L0 = l0 + d.
Step 1. We show that every interval (a, a + L0) contains points z1 and z2 such that
F(z1) < g + , F(z2) > G − . (1)
Choose an 6εd -almost-period τ such that m + τ ∈ (a, a + l0). Then both numbers z1 = x1 + τ and z2 = x2 + τ lie in [m + τ, m + τ + d] ⊂ (a, a + L0), and
By the meaning of G and g, the inequality F(z2) − F(z1) > G − g − ε/2 can hold only if both inequalities (1) are satisfied.
185 Mammadov and Agaliyeva | K. Multidiscip. Int. Sci. J. 3(3) (2026) 181-187
Step 2. Put ε1 = ε/(2L0) and let τ be an ε1-almost-period of f(x). We show separately that for every real x
F(x + τ) − F(x) > −ε, (2)
F(x + τ) − F(x) < ε. (3) To prove (2), choose by Step 1 a point z1 ∈ (x, x + L0) with F(z1) < g + ε/2. Then ∫︂ z1 F(x + τ) − F(x) = F(z1 + τ) − F(z1) + {f(y) − f(y + τ)} dy (︂ ε)︂ x ε ε > g − g + − L0 ε1 = − − = −ε.
2 2 2 To prove (3), choose a point z2 ∈ (x, x + L0) with F(z2) > G − ε/2. Then ∫︂ z2 F(x + τ) − F(x) = F(z2 + τ) − F(z2) + {f(y) − f(y + τ)} dy (︂ ε)︂ x < G − G − + L0 ε1 = ε.
Hence |F(x + τ) − F(x)| < ε for all x ∈ R. Since the ε1-almost-periods of f form a relatively dense set, so do the ε-almost-periods of the continuous function F, and F is uniformly almost-periodic. □
Corollary 1 (Bohr’s criterion). The indefinite integral of a uniformly almost-periodic function f is uniformly almost-periodic if and only if it is bounded on R.
Necessity follows from Theorem 1, sufficiency from Theorem 7.
Remark. Theorems 4, 5 and 6, together with the fact that translates and complex conjugates of uniformly almost-periodic functions are again uniformly almost-periodic, show that these functions form a translation-invariant closed subalgebra of the Banach algebra of bounded continuous functions on R with the supremum norm. By Bohr’s approximation theorem it coincides with the uniform closure of the set of trigonometric polynomials
Remark. The properties established above remain valid, with essentially the same proofs, for functions with values in a Banach space, except for Theorem 7: for infinite-dimensional spaces, boundedness of the integral is in general not sufficient and additional assumptions on the space (for instance, the absence of subspaces isomorphic to c0) are required
Remark. In applications, Theorems 6 and 7 are used to prove the existence of almost-periodic solutions of ordinary and abstract differential equations, where bounded solutions on R are shown to be almost-periodic
In this article, we have presented self-contained proofs of several fundamental properties of uniformly almost-periodic functions: boundedness, uniform continuity, closedness under sums, products, squares and uniform limits, almost periodicity of uniformly continuous derivatives, and almost periodicity of bounded indefinite integrals. These properties show that the class of uniformly almost-periodic functions is stable under the basic operations of analysis, which makes it a convenient framework for the study of differential and integral equations and of harmonic analysis on the real line. As directions for further research we indicated generalizations to Banach-space-valued functions and to the wider classes of Stepanov, Weyl, Besicovitch and almost-automorphic functions.
The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.
F. Mammadov: methodology, software, and editing. S. Agaliyeva: conceptualization. J.Aliyev: writing and editing. All authors have read and agreed to the published version of the manuscript.
The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions.
The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.