ON THE DIFFERENCES AND SUMS OF PERIODIC MEASURABLE FUNCTIONS

BY TAMÁS KELETI ^{1}

1. Introduction

Let f : R → R be a real function and let α_{1}, …, α_{n} be given real numbers. We say that f = f_{1} + … + f_{n} is an (α_{1}, …, α_{n})-decomposition of f if f_{i} is periodic modulo α_{i} for every i = 1, …, n. We say that f = f_{1}+…+f_{n}+p is an (α_{1}, …, α_{n})-quasi-decomposition of f if p is a polynomial of degree < n and f_{i} is periodic modulo α_{i} for every i = 1, …, n. (The decomposition or the quasi-decomposition is said to be continuous or measurable if all the functions f_{i} are continuous or measurable resp.) If f has an (α_{1}, …, α_{n})-decomposition or an (α_{1}, …, α_{n})-quasi-decomposition, then it is easy to see that Δ_{α_{1}} … Δ_{α_{n}}f = 0, where Δ_{α} is the difference operator with step α, that is Δ_{α}f(x) = f(x + α) − f(x). A class F of real functions is said to have the decomposition property if, for every f ∈ F and α_{1}, …, α_{n} ∈ R, the equation Δ_{α_{1}} … Δ_{α_{n}}f = 0 implies that f has an (α_{1}, …, α_{n})- decomposition with functions in F.

These notions were introduced in [2] by M. Laczkovich and Sz. Révész. They proved in [2] and [3] that a series of important classes of real functions have the decomposition property (e.g. L_{∞}(R), the class of bounded Lipschitz functions and the class of bounded continuous functions). On the other hand it is easy to see that the class of all R → R functions and the class of all continuous functions do not have the decomposition property. (f(x) = x is a counter-example for both classes.) It was asked in [3] whether it is true that if f is continuous and has a measurable (α_{1}, …, α_{n})-decomposition then f also has a continuous (α_{1}, …, α_{n})-decomposition or at least a continuous (α_{1}, …, α_{n})-quasi-decomposition.

If f is uniformly continuous, then it is an easy consequence of a result in [3], that the answer is affirmative. However, as we shall prove below, in the general case the answer is negative: we construct two measurable periodic functions such that their sums are continuous and unbounded.

A set Λ ⊂ R is called pseudo-Dirichlet if there exists a sequence of integers 0 < q_{1} < q_{2} < … and a sequence of positive reals ε_{1}, ε_{2}, … converging to zero such that, for any α ∈ Λ, if n is big enough then

| sin q_{n}πα| ≤ ε_{n} (n ≥ N(x)).

This notion together with several other notion of thinness in harmonic analysis is discussed in the survey [1] of L. Bukovskỳ, N. N. Kholshchevnikova and M. Repickỳ. It is known, for example, that every countable set and some nonempty perfect sets are pseudo-Dirichlet.

For any pseudo-Dirichlet set Λ ⊂ R we construct a modulo 1 periodic real function h ∈ L_{2} \ L_{∞} for which the function Δ_{α}h(x) = h(x + α) − h(x) is continuous for any α ∈ Λ.

1 This research was supported by the OTKA grant F 019468 (Here and in the sequel by L_{2} we mean the class of those measurable functions f : R → R for which f^{2} has a finite integral on any finite interval. The class L_{∞} consists of those real functions that are almost everywhere equal to a bounded function.) Adding a proper modulo α periodic function g_{α}(x) for each α ∈ Λ to this function h we will get continuous functions g_{α} + h for all α. As we will see later these sums cannot be bounded. In this way we can construct simultaneously many pairs of measurable periodic functions such that their sum is continuous and unbounded.

2. Some facts

In this section we prove some easy facts and corollaries that do not seem to have been stated previously. We will use the following known (and easy to prove) fact ([3]):

Proposition 0. No non-constant polynomial (more generally no function f with lim_{x→∞} |f| = ∞) can be the sum of finitely many periodic measurable functions.

Proposition 1. Let α_{1}, …, α_{n} be real numbers such that α_{i}/α_{j} is irrational for every 1 ≤ i < j ≤ n. Then a function f : R → R can have at most one measurable (α_{1}, …, α_{n})-decomposition, apart from additive constants and changes on a null-set.

Proof. Clearly it is enough to prove that if

(1)  f_{1} + … + f_{n} = 0  almost everywhere,

where f_{j} is measurable and periodic modulo α_{j} for all j, then f_{n} is constant almost everywhere. We prove it by induction. For n = 1 it is obvious. Let n ≥ 2. Since f_{1} is periodic modulo α_{1} the equation (1) implies

Δ_{α_{1}}f_{2} + … + Δ_{α_{1}}f_{n} = 0 almost everywhere.

Now using the induction assumption we get that Δ_{α_{1}}f_{n} = c almost everywhere. Then the function g(x) = f_{n}(x) − (c/α_{1})x is measurable and periodic modulo α_{1} except a set of measure zero. Thus (c/α_{1})x is the sum of two measurable periodic functions, which implies, using Proposition 0, that c = 0. Therefore the measurable function f_{n} has two incommensurable periods (except a set of measure zero), which implies that f_{n} is constant almost everywhere. This completes the proof.

Proposition 2. If f = f_{1} + … + f_{n}, f ∈ L_{∞}, the functions f_{i} are measurable and periodic modulo α_{i} (i = 1, …, n), where α_{i}/α_{j} is irrational for every i ≠ j, then the functions f_{1}, …, f_{n} are also in L_{∞}.

Proof. It is proved in [2] that the class of functions L_{∞} has the decomposition property. Using Proposition 1, this implies Proposition 2.

Proposition 3. If f has a continuous (α_{1}, …, α_{n})-quasi-decomposition and a measurable (α_{1}, …, α_{n})-decomposition, then f has a continuous (α_{1}, …, α_{n})-decomposition, too.

Proof. Suppose that f_{1} + … + f_{n} = f = g_{1} + … + g_{n} + p, where the functions f_{i} are measurable, the functions g_{i} are continuous and p is a polynomial. Then p = (f_{1} − g_{1}) + … + (f_{n} − g_{n}), so using Proposition 0, p must be constant, so g_{1} + … + g_{n−1} + (g_{n} + p) is a continuous (α_{1}, …, α_{n})-decomposition of f.

Theorem 0. If a uniformly continuous function f has a measurable (α_{1}, …, α_{n})- decomposition, then it also has a continuous (α_{1}, …, α_{n})-decomposition.

Proof. It is proved in [3] (4.2.Thm.) that a function f has a continuous (α_{1}, …, α_{n})- quasi-decomposition with linear p if and only if Δ_{α_{1}} … Δ_{α_{n}}f = 0 and f is uniformly continuous. By the previous proposition this implies the theorem.

Remark 1. By Proposition 1, this theorem implies that if the sum of finitely many measurable periodic functions with pairwise incommensurable periods is uniformly continuous, then these functions must be continuous apart from changes on a null-set.

Remark 2. These results do not remain true if we omit the word ‘uniformly’. This is one of the main results of this paper. For bounded continuous functions, however, these results become true again, since the class of bounded continuous functions has the decomposition property.

3. Some lemmas

Lemma 1. If the integers 0 < q_{1} < q_{2} < … satisfy the condition

(∗)  q_{N} ∑_{n=N+1}^{∞} \frac{1}{nq_{n}} ≤ C  ( for a fixed C for any N)

then the modulo 1 periodic L_{2} function h with Fourier series

h(x) ∼ ∑_{n=1}^{∞} \frac{cos(2πq_{n}x)}{n}

cannot be in L_{∞}.

Proof. For a fixed N we use the following notation:

x_{N} = \frac{1}{6q_{N}} ,  S_{N} = ∑_{n=1}^{N} \frac{1}{n} ,  m_{N}(x) = h(x) − ∑_{n=1}^{N} \frac{cos(2πq_{n}x)}{n} ∼ ∑_{n=N+1}^{∞} \frac{cos(2πq_{n}x)}{n}.

Then for any x ∈ [0, x_{N}]

h(x) = ∑_{n=1}^{N} \frac{cos(2πq_{n}x)}{n} + m_{N}(x) ≥ \frac{1}{2}S_{N} + m_{N}(x).

On the other hand, using the term by term integrability of Fourier series,

|∫_{0}^{x_{N}} m_{N}(x)dx| = | ∑_{n=N+1}^{∞} ∫_{0}^{x_{N}} \frac{cos(2πq_{n}x)}{n}| = | ∑_{n=N+1}^{∞} \frac{sin(2πq_{n}x_{N})}{2πq_{n}n}| ≤ ∑_{n=N+1}^{∞} \frac{1}{6q_{n}n}.

Therefore the average of m_{N}(x) on the interval [0, x_{N}] is

|\frac{∫_{0}^{x_{N}} m_{N}(x)dx}{x_{N}} | ≤ q_{N} ∑_{n=N+1}^{∞} \frac{1}{q_{n}n} ≤ C.

Thus on a subset H_{N} of [0, x_{N}] with positive Lebesgue measure m_{N}(x) ≥ −C, which implies that

h(x) ≥ \frac{1}{2}S_{N} + m_{N}(x) ≥ \frac{1}{2}S_{N} − C (x ∈ H_{N}).

On the other hand S_{N} → ∞, which means that h ∉ L_{∞}.

Lemma 2. Let A be an additive subgroup of R and let S be a dense union of translated copies of A. Suppose that we have a function \overline{h} : R → R and continuous functions l_{a} : R → R for all a ∈ A such that Δ_{a}\overline{h}|S = l_{a}|S for any a ∈ A.

Then there exists a function h : R → R such that h|S = \overline{h}|S and Δ_{a}h = l_{a} for every a ∈ A.

Proof. Choose a point from all those translated copies of A that is not in S. Denote this set by B. Then every x ∉ S can be uniquely written in the form x = β + b where β ∈ B and b ∈ A. For such an x ∉ S let h(x) = l_{b}(β). For x ∈ S let h(x) = \overline{h}(x).

Now we need to prove that Δ_{a}h(x) = l_{a}(x) for all a ∈ A and x ∈ R. We know that this equality holds if x ∈ S. If x ∉ S then x = β +b (β ∈ B, b ∈ A), so using the definition of h:

Δ_{a}h(x) = Δ_{a}h(β + b) = h(β + b + a) − h(β + b) = l_{b+a}(β) − l_{b}(β).

So it is enough to prove that

(2)  l_{a}(β + b) = l_{b+a}(β) − l_{b}(β)

for any β ∈ B and a, b ∈ A.

Fix a and b. If β is not in B but in S then (2) holds. Indeed, in this case β + b is also in S so l_{a}(β + b) = Δ_{a}\overline{h}(β + b), l_{b+a}(β) = Δ_{b+a}\overline{h}(β) and l_{b}(β) = Δ_{b}\overline{h}(β) and a trivial calculation shows that Δ_{a}\overline{h}(β + b) = Δ_{b+a}\overline{h}(β) − Δ_{b}\overline{h}(β).

Therefore (2) holds on a dense subset of R. On the other hand both sides of this equation are continuous, so this implies that (2) holds on the whole real line, thus specially it also holds on B, which completes the proof of this lemma.

4. The main results

Theorem 1. For any pseudo-Dirichlet set Λ (e.g. for any countable Λ and also for some nonempty perfect Λ) there exists a modulo 1 periodic function h ∈ L_{2} \ L_{∞} for which Δ_{α}h is continuous for any α ∈ Λ.

Proof. Take a sequence q_{1} < q_{2} < … and a sequence ε_{n} → 0 witnessing the pseudo-Dirichlet property of Λ. Then any subsequences of (q_{n}, ε_{n}) also witnessing the pseudo-Dirichlet property of Λ, so choosing a subsequence increasing quickly enough in q_{n} and decreasing quickly enough in ε_{n} we can assume that (q_{n}) satisfies the condition (*) of Lemma 1 and also ε_{n} < 1/n^{2}. Let A denote the set of all α-s for which there exists a K such that | sin q_{n}πα| ≤ K/n for every n. It is easy to check that A is an additive subgroup of R and Λ ⊂ A. Let h̃ ∈ L_{2} be a modulo 1 periodic complex valued function with the Fourier series

h̃(x) ∼ ∑_{n=1}^{∞} \frac{1}{n}e^{2πiq_{n}x}.

Let \overline{h} = Re h̃. Then \overline{h}(x) ∼ ∑_{n=1}^{∞} \frac{cos(2πq_{n}x)}{n}, so according to Lemma 1, \overline{h} ∈ L_{2} \ L_{∞}.

Let α ∈ A and let K be a real number for which | sin q_{n}πα| ≤ K/n (n = 1, 2, ….) The Fourier series of h̃(x + α) is

h̃(x + α) ∼ ∑_{n=1}^{∞}(\frac{1}{n}e^{2πiq_{n}α})e^{2πiq_{n}x},

so

(3)  Δ_{α}h̃(x) ∼ ∑_{n=1}^{∞} \frac{1}{n}(e^{2πiq_{n}α} − 1)e^{2πiq_{n}x}.

On the other hand

|\frac{1}{n}(e^{2πiq_{n}α} − 1)e^{2πiq_{n}x}| = \frac{1}{n}2| sin q_{n}πα| < 2K/n^{2}.

Thus the right-hand side of (3) is uniformly convergent, so denoting it by l̃_{α}(x), the function l̃_{α}(x) is continuous on R.

Let S be the set of points where the averages of the partial sums of the Fourier series (the Fejér means) of h̃ converge to h̃(x). According to Lebesgue’s theorem S contains the Lebesgue points of h̃, so its complement is a null-set. Changing h̃ on this null-set we can make h̃(x) to be equal to the limit of the Fejér means in each points where it exists, so we can assume that S is also the set of points where the Fejér means converge.

Since the Fejér means of l̃_{α}(x) converge to l̃_{α}(x) everywhere, the Fejér means of h̃(x) and h̃(x + α) converge simultaneously, thus x ∈ S if and only if x + α ∈ S. Therefore S is a dense union of translated copies of A. If x ∈ S then, according to (3), Δ_{α}h̃(x) and l̃_{α}(x) are the limits of the averages of the partial sums of the same Fourier series, thus Δ_{α}h̃(x) = l̃_{α}(x). Therefore denoting the real part of l̃_{α} by l_{α} we get

Δ_{α}\overline{h}(x) = l_{α}(x) (x ∈ S, α ∈ A).

Now applying Lemma 2, there exists a function h(x) on R such that h|S = \overline{h}|S and

Δ_{α}h(x) = l_{α}(x) (x ∈ R, α ∈ A).

In particular Δ_{1}h(x) = l_{1}(x) = 0, which implies that h is periodic modulo 1; for any α ∈ Λ ⊂ A we get Δ_{α}h = l_{α}, which shows that Δ_{α}h is continuous for any α ∈ Λ. Since h and \overline{h} are equal almost everywhere and \overline{h} ∈ L_{2} \ L_{∞} we get h ∈ L_{2} \ L_{∞}. This completes the proof.

Theorem 2. For any pseudo-Dirichlet set Λ ⊂ R (e.g. for any countable Λ and also for some nonempty perfect Λ) there exists a modulo 1 periodic function h ∈ L_{2} \ L_{∞} and there are modulo α periodic functions g_{α} ∈ L_{2} \ L_{∞} for all α ∈ Λ such that g_{α} + h is continuous for all α ∈ Λ.

Proof. Let h ∈ L_{2} \ L_{∞} be a modulo 1 periodic function with continuous Δ_{α}h for α ∈ Λ. Since Δ_{α}h is continuous there exists a continuous function f_{α} for each α ∈ Λ such that Δ_{α}f_{α}=Δ_{α}h. (After defining f_{α} arbitrarily in [0, α] such that f_{α}(α)−f_{α}(0) = Δ_{α}h(0), the equation f_{α}(x+α)−f_{α}(x) = Δ_{α}h(x) gives a continuous extension.) Then the functions g_{α} = f_{α} − h (α ∈ Λ) are periodic modulo α, are in L_{2} \ L_{∞} and the functions g_{α} + h = f_{α} are continuous.

Theorem 3. The sum of two periodic measurable functions can be continuous and unbounded.

Proof. Using Theorem 2 for Λ = {α}, where α is irrational, we get functions h, g ∈ L_{2} \ L_{∞} periodic modulo 1 and α resp. such that f = g + h is continuous. But according to Proposition 2, f cannot be bounded.

Corollary. There exists a continuous function f : R → R that has a measurable (α_{1}, …, α_{n})-decomposition but does not have a continuous (α_{1}, …, α_{n})-decomposition, nor has a continuous (α_{1}, …, α_{n})-quasi-decomposition.

Proof. The unbounded continuous sum of the two periodic measurable functions constructed above clearly has a measurable (α_{1}, α_{2})-decomposition but does not have a continuous (α_{1}, α_{2})-decomposition (since otherwise it would be bounded). On the other hand, by Proposition 3, this function cannot have an (α_{1}, …, α_{n})-quasi-decomposition either.

Acknowledgment. I would like to thank my advisor, Professor Miklós Laczkovich for helpful discussions.

References.

[1] L. Bukovskỳ, N. N. Kholshchevnikova and M. Repickỳ, Thin sets of harmonic analysis and infinite combinatorics, Real Analysis Exchange, 20 No. 2 (1994-1995), 454-509.

[2] M. Laczkovich and Sz. Révész, Decompositions into the sum of periodic functions belonging to a given Banach space, Acta Math. Hung., 55 (3-4) (1990), 353-363.

[3] M. Laczkovich and Sz. Révész, Periodic decompositions of continuous functions, Acta Math. Hung., 54 (3-4) (1989), 329-341.

Eötvös Loránd University Department of Analysis Budapest, Múzeum krt. 6-8. H-1088, Hungary e-mail: elek@cs.elte.hu