Almost Periodic Functions

Motivation and Setting

A function that is not periodic, yet "almost periodic"

Consider $f(t)=\cos t+\cos\sqrt{2}\,t$. The first summand $\cos t$ has period $2\pi$, the second $\cos\sqrt{2}\,t$ has period $2\pi/\sqrt{2}=\pi\sqrt{2}$. Their ratio $\sqrt{2}$ is irrational, so $f$ has no exact period: there is no positive $T$ for which $f(t+T)=f(t)$ holds for all $t$.

Yet when one plots $f$, the waveform appears to repeat itself approximately on infinitely many intervals. This can be verified numerically:

If one chooses $\tau\approx 17$, then $\cos(t+\tau)$ nearly coincides with $\cos t$ (for some integer $n$, $\tau\approx 2\pi n$), while $\sqrt{2}\,\tau$ also lies close to an integer multiple of $2\pi$. Such a $\tau$ exists by Diophantine approximation, and crucially the set of such $\tau$ is evenly distributed along the real line.

Bohr's idea: "ε-almost period" and "relative density"

Harald Bohr (1887–1951) translated this observation into a formal analytic framework in the 1920s. The two key concepts are:

• $\varepsilon$-almost period $\tau$: a real number such that $f(t+\tau)$ agrees with $f(t)$ within uniform error $\varepsilon$ (an "approximate period").
• Relative density: no matter how large an interval $[a,a+L]$, at least one $\varepsilon$-almost period lies inside it.

"For any precision, at least one such $\tau$ lies in every fixed-length interval" — Bohr called functions satisfying these two conditions almost periodic functions.

Position in the broader theory

The class $AP(\mathbb{R})$ of almost periodic functions:

• contains all continuous periodic functions as a special case (for a function of period $T$, every $\tau=nT$ is a $0$-almost period)
• is closed under finite linear combinations of exponentials $e^{i\lambda_k t}$ ($\lambda_k\in\mathbb{R}$, with periods that may be irrationally related) and under uniform limits
• admits a unique generalized Fourier expansion $f(t)\sim\sum_n a_n e^{i\lambda_n t}$ (frequencies $\lambda_n\in\mathbb{R}$, coefficients $a_n$ uniquely determined)

It is therefore the natural extension of Fourier series theory freed from the constraint of integer-multiple frequencies. It also connects to harmonic analysis on locally compact abelian groups, quasi-periodic orbits in dynamical systems, and ergodic theory.

Bohr's Definition (ε-Almost Periods)

What is an $\varepsilon$-almost period?

For a periodic function $f$ with period $T$, the translation $t\mapsto t+T$ recovers $f$ exactly. The $\varepsilon$-almost period weakens this to "recovers $f$ within uniform error $\varepsilon$".

The use of $\sup_t$ is crucial — not "close at some $t$", but "uniformly close across all $t$": shifting by $\tau$ overlays the waveform on itself with uniform error at most $\varepsilon$.

Relative density: "always near"

The mere existence of $\varepsilon$-almost periods is not enough — they must be evenly distributed along the real line. This is captured by relative density.

Note that relative density and topological density are distinct concepts. For example, $\mathbb{Z}$ is relatively dense ($L=1$) but not topologically dense in $\mathbb{R}$ ($\mathbb{Q}$ satisfies both as a special case). On the other hand, a set like $\{2^n:n\in\mathbb{N}\}$ whose gaps grow exponentially is not relatively dense, because for any $L$ there exist sufficiently large windows containing no point.

Definition of a Bohr almost periodic function

Informal restatement:

"For any required precision $\varepsilon$, there is always a translation time $\tau$ that recovers the function within $\varepsilon$ — and such $\tau$ can be found in any window of length $L(\varepsilon)$ on the real line."

For a periodic function with period $T$, every $\tau=nT$ (integer $n$) is automatically an $\varepsilon$-almost period for any $\varepsilon>0$, so $E(\varepsilon,f)\supset T\mathbb{Z}$ is trivially relatively dense ($L=T$ works). Hence periodic functions are a special case of Bohr almost periodic functions.

Bochner's Characterization

The "translation orbit" object

Bohr's definition is combinatorial in nature, focusing on the relative density of the set of $\varepsilon$-almost periods. Bochner adopted a more topological standpoint, looking not at the function $f$ itself but at the family of all translates of $f$.

For each real $\tau$, define $f_\tau:\mathbb{R}\to\mathbb{C}$ by $f_\tau(t):=f(t+\tau)$. Visually, $f_\tau$ is the graph of $f$ shifted left by $\tau$. The set

is called the translation orbit of $f$. It is a subset of the Banach space $L^\infty(\mathbb{R})$ equipped with the uniform norm $\|\cdot\|_\infty$.

Main theorem: compactness ≡ almost periodicity

Intuition for the equivalence

Why are "relative density of $\varepsilon$-almost periods" and "compactness of the translation orbit" the same thing? The intuition in both directions:

• (1)⇒(2): If $f$ is Bohr almost periodic, then for every $\varepsilon>0$ one can choose finitely many $\tau_1,\dots,\tau_N$ from $E(\varepsilon,f)$ so that every $\tau\in\mathbb{R}$ satisfies $\|f_\tau-f_{\tau_k}\|_\infty<\varepsilon$ for some $\tau_k$ (relative density provides a finite $\varepsilon$-net). Thus $\{f_{\tau_k}\}$ forms a finite $\varepsilon$-net for $\mathcal{O}(f)$, yielding relative compactness.
• (2)⇒(1): If $\mathcal{O}(f)$ is compact, then for every $\varepsilon>0$ a finite $\varepsilon$-net $\{f_{\tau_1},\dots,f_{\tau_N}\}$ exists. In any interval $[a,a+L]$ (with $L$ the maximum gap among $\{\tau_k\}$), the translate $f_a$ is within $\varepsilon$ of some $f_{\tau_k}$, so $\tau=\tau_k-a$ is an $\varepsilon$-almost period (by translation invariance). Hence $E(\varepsilon,f)$ is relatively dense.

In short: "compactness ≡ existence of finite $\varepsilon$-nets ≡ even distribution of $\varepsilon$-almost periods" — the same fact viewed from two angles.

Advantages of Bochner's formulation

The topological recasting yields several benefits:

• Easy abstraction: replacing $\mathbb{R}$ by any locally compact abelian group $G$ leaves the definition intact. Bohr's $\varepsilon$-almost-period approach requires a new notion of "relatively dense" for each group, whereas compactness is a universal concept.
• Direct use of group structure: translation is a group action, so invariance and continuity arguments become natural.
• Connection to function-space theory: the Banach algebra and $C^*$-algebra structure of $AP(\mathbb{R})$, and its isomorphism with the Bohr compactification, follow as corollaries.

Uniform Approximation by Trigonometric Polynomials

"Generalized trigonometric polynomial"

An ordinary Fourier-series partial sum $\sum_{n=-N}^N c_n e^{int}$ restricts the frequencies to integers $n$. Allowing real frequencies $\lambda_k\in\mathbb{R}$ instead gives a generalized trigonometric polynomial:

This is a "finite linear combination of exponentials" — each $e^{i\lambda_k t}$ has period $2\pi/\lambda_k$, so $P$ is a finite linear combination of periodic functions.

Main theorem: Bohr's approximation theorem

$AP(\mathbb{R})$ as a closure

Recasting in function-space terms, $AP(\mathbb{R})$ is characterized as the following closed subspace:

Reading the formula piece by piece

The expression is dense in symbols; let us unpack it from the inside out:

1. $\{e^{i\lambda t}\,:\,\lambda\in\mathbb{R}\}$ — the set of all complex exponentials with real frequency $\lambda$. This is the "raw material" — an infinite family of functions ($\lambda$ ranges continuously over all reals).
2. $\operatorname{span}_{\mathbb{C}}\{\cdots\}$ — the set of all finite linear combinations with complex coefficients: every function of the form $\sum_{k=1}^N c_k e^{i\lambda_k t}$ for some finite $N$, $c_k\in\mathbb{C}$, $\lambda_k\in\mathbb{R}$. These are precisely the generalized trigonometric polynomials.
3. $\overline{\,\cdots\,}^{\,\|\cdot\|_\infty}$ — the closure in the uniform (sup) norm: add all limits of sequences of generalized trigonometric polynomials that converge uniformly. With $\|f-g\|_\infty=\sup_t|f(t)-g(t)|$, this is the collection of all functions that can be approached arbitrarily closely.

Putting it all together, the formula asserts:

"A Bohr almost periodic function is exactly a function that can be uniformly approximated by finite linear combinations of complex exponentials $e^{i\lambda t}$ with arbitrary real frequencies."

Equivalently, for any $f\in AP(\mathbb{R})$, suitable choices of frequencies $\lambda_1,\dots,\lambda_N$ and coefficients $c_1,\dots,c_N$ make $\sup_t|f(t)-\sum c_k e^{i\lambda_k t}|$ as small as desired; conversely, every function so approximable is almost periodic.

Contrast with the periodic case

This is a natural extension of the classical result for periodic functions:

• Periodic version (Stone-Weierstrass / Fejér): a continuous $2\pi$-periodic function is a uniform limit of trigonometric polynomials $\sum c_k e^{ikt}$ with integer frequencies $\lambda_k=k\in\mathbb{Z}$: $C(\mathbb{T})=\overline{\operatorname{span}_{\mathbb{C}}\{e^{ikt}:k\in\mathbb{Z}\}}^{\,\|\cdot\|_\infty}$.
• Almost periodic version (Bohr): a Bohr almost periodic function is a uniform limit of generalized trigonometric polynomials with any real frequencies.

The only difference is the set of allowed frequencies: from the discrete lattice $\mathbb{Z}$ to all of $\mathbb{R}$. Restricting to integer multiples gives periodic theory; unrestricting yields almost periodic theory.

Mean Value and Bohr-Fourier Coefficients

Why a "mean value" is needed

Fourier coefficients of a periodic function are defined by integrating over one period: $\frac{1}{T}\int_0^T f\cdot e^{-int/T}\,dt$. An almost periodic function has no fixed period, so we use the long-time average as $T\to\infty$:

Computation principle: the exponential case

The most basic case is $f(t)=e^{i\mu t}$ ($\mu\in\mathbb{R}$):

• If $\mu=0$, then $e^0=1$ and $M\{1\}=1$.
• If $\mu\ne 0$, then $\frac{1}{2T}\int_{-T}^T e^{i\mu t}\,dt=\frac{\sin(\mu T)}{\mu T}\to 0$ as $T\to\infty$.

In short:

This is the foundational principle of almost periodic analysis: "exponentials of distinct frequencies vanish under long-time averaging". It is the continuous analogue of the orthogonality relation $\frac{1}{2\pi}\int_0^{2\pi}e^{i(m-n)t}\,dt=\delta_{mn}$ for ordinary Fourier coefficients.

For a finite generalized trigonometric polynomial $P(t)=\sum c_k e^{i\lambda_k t}$, linearity gives $M\{P\}=c_{k_0}$, where $\lambda_{k_0}=0$ is the zero-frequency coefficient (zero if no such term exists). For general $f\in AP(\mathbb{R})$, one uses Bohr's approximation theorem to write $f$ as a uniform limit of generalized trigonometric polynomials and uses continuity of $M$.

Definition of Bohr-Fourier coefficients

The idea: "demodulate $f$ by frequency $\lambda$, then take the time average". This is the natural generalization of the ordinary Fourier coefficient $c_n=\frac{1}{2\pi}\int_0^{2\pi}f(t)e^{-int}\,dt$.

Why the spectrum is at most countable

Intuition: By Parseval's identity (or the weaker Bessel inequality), $\sum_\lambda |a(\lambda;f)|^2 \le M\{|f|^2\} < \infty$. Since the sum is finite, for each $\varepsilon>0$ only finitely many $\lambda$ satisfy $|a(\lambda;f)|>\varepsilon$. Taking $\varepsilon=1/n$ over $n\in\mathbb{N}$, $\Lambda(f)=\bigcup_n\{\lambda:|a(\lambda;f)|>1/n\}$ is a countable union of finite sets, hence countable.

Bohr-Fourier Expansion and Parseval's Identity

The formal Bohr-Fourier expansion

Enumerate the spectrum $\Lambda(f)=\{\lambda_1,\lambda_2,\dots\}$ and let $a_n:=a(\lambda_n;f)$. Then $f$ is associated with the formal series

The "$\sim$" sign is used because the partial sums $S_N(t)=\sum_{n=1}^N a_n e^{i\lambda_n t}$ need not converge uniformly in general. Even for classical Fourier series of continuous periodic functions, partial sums are not guaranteed to converge uniformly (Gibbs phenomenon at discontinuities), and the Fejér average (Cesàro means of partial sums) is needed to obtain uniform convergence (Fejér's theorem). The same situation occurs for almost periodic functions: only after passing through the Bochner-Fejér average (below) is $f$ recovered as a uniform limit.

Energy conservation: Parseval's identity

This is the almost periodic counterpart of the ordinary Parseval identity $\frac{1}{2\pi}\int_0^{2\pi}|f|^2\,dt=\sum_n|c_n|^2$. In words:

• The left side $M\{|f|^2\}$ is the "time-averaged energy" (power) of $f$.
• The right side is the sum of squared amplitudes $|a(\lambda;f)|^2$ over all frequencies.
• Thus the power of $f$ is completely distributed across a countable set of spectral components.

Uniqueness

In other words, the family of Bohr-Fourier coefficients $\{a(\lambda;f)\}_\lambda$ uniquely determines $f$ — the almost periodic analogue of uniqueness for ordinary Fourier coefficients.

Recovering $f$ via Bochner-Fejér summation

To pass from the formal series $\sum a_n e^{i\lambda_n t}$ back to $f$, one uses the Bochner-Fejér polynomials

where $K_n^{(N)}$ are appropriately chosen Cesàro-type weights (constructed by Bochner). As $N$ increases, the polynomial absorbs more and more Bohr-Fourier coefficients. The explicit form of the weights involves a multiple Cesàro construction and is too intricate to detail here; see e.g. Corduneanu §2.4.

This is the natural extension of Fejér's theorem for periodic functions (Cesàro averages of Fourier sums converge uniformly for continuous periodic functions). The Bochner-Fejér theorem provides the explicit construction behind Bohr's approximation theorem.

Stepanov, Weyl, and Besicovitch Classes

Why extensions are needed

The Bohr class $AP(\mathbb{R})$ consists of uniformly continuous functions. For applications, however, several important function families exhibit a more relaxed almost periodicity:

• Discontinuous step functions can have an "almost periodic jump pattern" that Bohr's framework cannot accommodate (it requires continuity).
• Functions whose behavior shifts slowly at infinity — e.g. $f(t)+g(t)$ where $g(t)\to 0$ as $|t|\to\infty$: the $f$ part is almost periodic, but the presence of $g$ breaks Bohr's relative-density condition.

To handle these, broader classes containing $AP(\mathbb{R})$ have been introduced. They differ in the norm used to measure translation error.

The three extensions

Each class is obtained by changing the error norm in the definition of an $\varepsilon$-almost period. Fix $p\ge 1$:

• Stepanov class $S^p(\mathbb{R})$: use the local $L^p$ norm $$\|g\|_{S^p}\;=\;\sup_{a\in\mathbb{R}}\left(\int_a^{a+1}|g(t)|^p\,dt\right)^{1/p}.$$ A function whose translation difference $g=f(\cdot+\tau)-f(\cdot)$ satisfies $\|g\|_{S^p}<\varepsilon$ for a relatively dense set of $\tau$ is called Stepanov almost periodic. Feature: errors are measured by an $L^p$ average over each unit interval $[a,a+1]$, so large pointwise oscillations are tolerated provided they average out. This allows discontinuous almost periodic functions such as step functions.
• Weyl class $W^p(\mathbb{R})$: let the window width grow to infinity: $$\|g\|_{W^p}\;=\;\lim_{\ell\to\infty}\sup_{a\in\mathbb{R}}\left(\dfrac{1}{\ell}\int_a^{a+\ell}|g(t)|^p\,dt\right)^{1/p}.$$ Feature: even more relaxed than Stepanov. Functions whose error shrinks under long-time averaging are admitted.
• Besicovitch class $B^p(\mathbb{R})$: a global average: $$\|g\|_{B^p}\;=\;\limsup_{T\to\infty}\left(\dfrac{1}{2T}\int_{-T}^{T}|g(t)|^p\,dt\right)^{1/p}.$$ Feature: the largest class. It has deep ties to stationary stochastic processes and ergodic theory.

Inclusion relations and properties

The classes form a strict chain of inclusions:

Importantly, every class supports:

• a well-defined long-time mean $M\{f\}$,
• Bohr-Fourier coefficients $a(\lambda;f)=M\{f(t)e^{-i\lambda t}\}$,
• at most countable spectrum,
• an appropriate Parseval-type identity.

These extensions are widely used in the asymptotic analysis of differential equations with almost periodic coefficients, the spectral analysis of stationary stochastic processes, and the study of ergodic averages.

Applications

Dynamical systems and quasi-periodic orbits

In multi-degree-of-freedom Hamiltonian systems, KAM theory preserves orbits described by Bohr almost periodic functions $x(t)=F(\omega_1 t,\dots,\omega_k t)$, where $\omega_1,\dots,\omega_k$ are rationally independent frequencies and $F$ is a continuous function on the torus $\mathbb{T}^k$. These are called quasi-periodic orbits. The Bohr-Fourier expansion is the basic tool for their spectral analysis.

Abstract harmonic analysis and the Bohr compactification

$AP(\mathbb{R})$ is isometrically isomorphic to $C(b\mathbb{R})$ — continuous functions on the Bohr compactification $b\mathbb{R}$, constructed as the Pontryagin dual of the dual group $\widehat{\mathbb{R}}=\mathbb{R}$ equipped with the discrete topology. Combined with Pontryagin duality, this is the gateway to treating almost periodic functions uniformly within harmonic analysis.

Partial differential equations and almost periodic solutions

For linear and quasi-linear PDEs whose coefficients depend almost periodically on time (or space), questions of existence, uniqueness, and asymptotic behavior of solutions are studied within the Besicovitch or Stepanov classes. The systematic treatment by Levitan-Zhikov is representative.

Ergodic theory

For a measure-preserving transformation $T$, the spectral theory of $AP$-functions corresponds to the characterization of pure point spectrum components: an ergodic system with pure point spectrum is isomorphic to an almost periodic dynamical system (von Neumann-Halmos).

References

• H. Bohr, Almost Periodic Functions, Chelsea, 1947 (German original 1933) — Bohr's own systematic introduction.
• C. Corduneanu, Almost Periodic Functions, 2nd English ed., Chelsea, 1989 — a standard textbook with proofs.
• B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, 1982 — emphasis on applications to differential equations.
• A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, 1932 (Dover reprint 1955) — introduction of the Besicovitch class.
• Almost periodic function — Wikipedia

Related topics

• $L^2$ spaces and Hilbert spaces — foundation for generalized Fourier expansions
• Introduction to harmonic analysis — Fourier analysis on locally compact groups
• Definition of Fourier Series — the periodic starting point

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