Besicovitch and other generalizations of Bohr s almost periodic functions

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1 Besicovitch and other generaizations of Bohr s amost eriodic functions Kevin Nowand We discuss severa casses of amost eriodic functions which generaize the uniformy continuous amost eriodic (a..) functions originay defined by Harad Bohr. We have two goas, which we accomish in two ways from two different sources. The first is to deveo the roer setting for a Riesz-Fischer tye theorem for amost eriodic functions, which we do by foowing [3], and which eads to the definition of the Besicovitch amost eriodic functions. Then we switch in section 5 to showing that Besicovitch functions are naturay occurring, but by defining such functions on the set of nonnegative numbers. For this we foow [2]. This note is just a summary of resuts with few roofs. A good reference on the subte differences between the ethora of definitions of a.. functions is []. Warning: Tabe 2 at the end of section 6 of that aer is an exceent summary of what the authors show, but there is a one arrow that oints u and to the eft that shoud oint down to the right. Bohr amost eriodic functions The cassica eriodic functions on R can be characterized by f(x) = f(x+t) for a x and some eriod t. The first generaization of this condition was due to Bohr (924), who instead required that a function f at a oint x be aroximated by f at x+t to an arbitrary degree of accuracy for a arge set of t. Definition. (amost eriods). A continuous function f : R C is Bohr amost eriodic if for any ε > 0, the set of ε-eriods {τ : f(x+τ) f(x) < ε} is reativey dense in R, i.e., there exists an = (ε) such that every interva of the form [x,x + ] intersects the set of ε-eriods. The set of such functions is denoted AP B (R). We can of course give an equivaent definition based on ε-eriods for functions defined on other saces; in articuar, we wi consider functions in AP B (N) with N the set of nonnegative integers. Exame.2. Periodic functions are in AP B (R). Exame.3. f(x) = sinx+sin 2x is in AP B (R). Proosition.4. If f AP B (R), then f is bounded and uniformy continuous. For amost eriodic functions, we have a mean. Definition.5. Let the mean of f : R C to be the vaue of M {f(x)} := im if this imit exists, which it does for a f AP B (R). 0 f(x)dx, Definition.6. AP B (R) is an inner roduct sace if we define { } f,g := M f(x)g(x). Using this mean and a given f, we can define a function a f (λ) : R C by a f (λ) := M { f(x)e iλx}. Note that the (eriodic) functions e iλx form an orthonorma basis with resect to this mean, such that a f (λ) is zero for a but at most countaby many λ. Definition.7. We define the Fourier series of an a.. function to be the forma sum n a(λ n)e iλnx where the λ n are the nonzero frequencies of f. We write f(x) n a(λ n )e iλnx.

2 Note that we are being coy by writing a.. instead of Bohr a.., as the mean wi exist for more genera casses of amost eriodic functions. It turns out that the structura definition given by Bohr can be reaced with the foowing equivaent statement: Definition.8 (aroximation). A function f is in AP B (R) if there exists an at most countabe sequence of rea numbers (λ n ) and comex numbers (A n ) such that f(x) is the uniform imit of sums of trigonometric oynomias N n= A ne iλnx. Another name for for Bohr s amost eriodic functions is uniformy amost eriodic functions, as they are the cosure of the trigonometric oynomias under the uniform norm, which we denote by. This gives us a second way to define amost eriodic functions as the cosure of the trigonometric oynomias under various norms and seminorms. This uniform aroximation by oynomias aows us to recover a notion of Fourier series, and some hoe of recovering a Riesz-Fischer tye theorem for amost eriodic functions. However, the Riesz-Ficher theorem is a statement about L 2 functions on a comact interva, and we are thus far deaing with uniformy continuous functions, so it aears we are not in the correct setting yet. Before moving on, we note that there is a third equivaent definition to the above two. Definition.9 (normaity). A continuous function f : R C is uniformy norma if for every sequence (h n ) of rea numbers, the set of transates {f(x+h n )} contains a uniformy convergent subsequence. In other words, the given set of transates is re-comact. This definition is due to Bochner, and is a normaity tye definition. In a comete metric sace, recomactness is equivaent to tota boundedness. Definition.0. A set in a metric sace is caed totay bounded if for every ε > 0 there exists a finite number of ε-bas covering the set. We do not a riori have a comete metric sace, as the functions are ony required to be continuous, but if we assume that they are bounded (as they end u being) then we do. Theorem.. The definitions.,.8, and.9 are equivaent. The function sace AP B (R) is an agebra. 2 Steanov amost eriodic functions We have the above three equivaent definitions of the Bohr a.. functions. These functions are a uniformy continuous, which is a very strong statement. To reax this, and eave the continuous functions entirey, we can generaize the three definitions. Steanov was the first to rovide such a generaization. We wi switch to the aroximation by trigonemtric oynoimas being out main source of a definition, as we are keeing an eye for a Riesz-Fischer theoroem. Definition 2. (aroximation). Let < and > 0. We define the sace AP S (R) of Steanov (Steanoff) amost eriodic functions to be the cosure of the trigonmetric oynomias under the norm f S ( := su x R x+ x f(t) dt Note. These functions necessariy are in L oc (R), which are ony defined u to sets of Lebesgue measure zero, such that imits are ony unique on casses of functions. Proosition 2.2. The Steanov norms induce equivaent tooogies, in the sense that for any, there exist C,C > 0 deending on and such that C f S f S C f S. We tyicay take =, in which case we write S in ace of S. As a consequence of Höder s inequaity, ) / 2

3 Proosition 2.3. Let < < and f AP S (R). Then which imies AP S (R) AP S (R). Even more simy, f S f S, Proosition 2.4. Let < and f AP B (R). Then which imies AP B (R) AP S (R). The other two definitions are the foowing: f S f, Definition 2.5 (amost eriods). Let <. A function f L oc (R) is in AP S(R) if for every ε > 0 the set of a Steanov ε-amost eriods τ satisfying is reativey dense in R. ( x+ ) / f(t+τ) f(t) S = su f(t+τ) f(t) dt x R x Definition 2.6 (normaity). Let <. A function f L oc (R) is in AP S (R) if the set {f(x+τ)} t R is re-comact. We have been soy in saying that these are a definitions of the same sace AP S (R), but for the Steanov a.. functions, we continue our good uck and have the foowing theorem. Theorem 2.7. The definitions 2., 2.5, and 2.6 are equivaent. The function sace AP S (R) is an agebra for a <. Exame 2.8. Let E be set of cosed intervas of the form [2k,2k +] where k is any integer and et O be the comement of these intervas. Defined { x E f(x) = 0 e O. Ceary f is not Bohr amost eriodic, as f is not continuous. However, it is in AP S for any, as for any ε > 0, we can take the reativey dense set of integers as the eriods. We aso have the theorem due to Bochner. Theorem 2.9 (Bochner s Theorem). If f AP S (R) is uniformy continuous, then f AP B (R). 3 Wey amost eriodic functions Though the ε-amost eriodic, aroximation by trigonometric oynomias, and normaity/ re-comactness definitions have to this oint coincided, we wi no onger have that uxury. However, we are not yet at the oint where we have the Riesz-Fischer theorem. We kee using trigonometric oynomias as our main definition. The definition of the Wey a.. functions is based on the foowing observation. Lemma 3.. The imit im f S imit is infinite. exists, in the sense that if the norm is infinite for any > 0 then the 3

4 Proof. We consider ony the = case. This roof bois down to the fact that a vaues of give rise to the same tooogy. Note that if f S is infinite for some > 0, then it is infinite for a > 0. Thus it suffices to consider the case where a such are finite. Let 0 be greater than zero and et n be the ositive integer such that (n ) 0 < n 0. Note that n 0 x+n0 x f(x) dx = n Therefore f Sn0 f S0. We cacuate ( x+0 f(x) dx+ + ) x+n0 f(x) dx f S0. 0 x 0 x+(n ) 0 Therefore as desired. f S n 0 f Sn0 + 0 im su f S iminf f S 0 = iminf f S, 0 f Sn0 + 0 f S0. () With this in hand, we make a definition. Definition 3.2 (aroximation). Let <. We define the sace of Wey amost eriodic functions, denoted AP W (R), to be the set of functions on R which can be aroximated arbitrariy we by triogonometric oynomias under the Wey seminorm f W := im f S. Exame 3.3. There exist functions f on R which are stricty ositive but satisfy f W = 0. Thus the uniqueness of eements in the cosure is u to functions which may differ on sets of even infinite measure. The second definition is in terms of Steanov ε-amost eriods. Definition 3.4 (amost eriods). Let <. A function f L oc (R) is in AP W(R) if for every ε > 0 there exists an = (ε) such that that there is a reativey dense set {τ} of Steanov ε-amost eriods satisfying f(t+τ) f(t) S < ε. It is cear that AP S (R) AP W (R). Remark. In the above definition, we are not using the Wey seminorm but rather the Steanov seminorm, and the is aowed to vary with ε, which is not the case for the S functions. This turns out to be crucia, as if we insist on defining the ε-amost eriods in terms of the Wey seminorm, we find a stricty arger sace. Theorem 3.5. The definitions 3.2 and 3.4 are equivaent. 4 Besicovitch eriodic functions Finay, we arrive at the saces which give us a Riesz-Fischer theorem. We sti do not have a three definitions, however, but we do have two equivaent definitions, as with the Wey a.. functions. Definition 4.. Let <. We define the sace AP B (R) of Besicovitch amost eriodic functions as the functions on R which can be aroximated arbitrariy we by trigonometric oynomias under the seminorm f B = imsu ( 2 f(x) dx) /. 4

5 Note. As with the Wey seminorm, there are functions which are nonzero on a of R and nonetheess have zero Besicovitch seminorm. It is cear from the definition that f B f W, such that AP W (R) AP B (R). Proosition 4.2. Let <. Then and a the incusions are strict. AP B (R) AP S (R) AP W (R) AP B (R), We wish to find another characterization based on ε-eriods, but this is not easy to do. It turns out that reativey dense sets are too broad to describee the eements of AP B (R). Definition 4.3. A subset E of R is caed satisfactoriy uniform if there exists a a ositive number such that the ratio of the maximum number of terms of E in an interva [x,x+] to the minimum number of terms of E in such an interva is ess than 2. Exame 4.4. Every satisfactoriy uniform set is reativey dense. Exame 4.5. The set {,2,...,} { n} is reativey dense but not satisfactoriy uniform. n= Definition 4.6. Let < and f L (R). Then f AP B (R) if for any ε > 0, there corresonds a satisfactoriy uniform set < τ < t 0 < t < such that and for every c > 0, im su ( 2 [ im su n f(x+τ) f(x) B < ε 2n+ n i= n c x+c Then we have the equivaence of definitions that we desire. Theorem 4.7. The definitions 4. and 4.6 are equivaent. x f(t+τ i ) f(t) dt Finay, we are in a osition where we have a Riesz-Fischer theorem. ] dx) / < ε. Theorem 4.8. To any (generaized) Fourier series n A ne iλnx such that n A n 2 converges there corresonds a function f AP B 2(R) with this as its Fourier series. 5 Amost eriodicity on the nonnegative integers We now shift focus from amost eriodic functions on R to amost eriodic functions on N = {0,,...}. Note that we incude zero when we use the symbo N. One way to define AP G (N) for G one of B, S, W, and B, is to reace the integras with sums over the integers. However, a different aroach is taken in [2]. We begin by definining AP B (N), the Bohr amost eriodic functions, in the same way as originay, by using the reative density of the ε-eriods with resect to the ( ) norm. As before, eements in AP B (N) are contained in λ( ). Definition 5.. Let f : N C be such that f ( ). Then we define W(f) to be the set of functions {f(x+a) : a N}, i.e., the set of transates of f. Proosition 5.2. Let f AP B (N). Then W(f) is reativey comact (totay bounded) in ( ). Proof. To show that W(f) is totay comact in the metric sace ( ), it suffices to find a finite number of ε-bas which cover the set. Let ε > 0 be fixed, and et = (ε) be such that every interva of ength at east λ in N contains an ε-amost eriod of f. Let J = {j,j +,...,j +}. We caim that W(f) is contained in the set of + ε-bas, each centered at f(x+k) for k J. Consider f(x+n) for some n N. If k J, we 5

6 are done. If k < j, the consider the set {k n : k J}. This is an interva of ength + in N and hence must contain an ε-amost eriod. Write f(x+n) f(x+k) = f(x+n) f(x+n+(k n)). Then for some k n, this wi be ess than ε for a x. Thus f(x+n) is contained in some ε-ba centered at f(x+k) for k J, as required. If n > j +, then we do the same trick but reace k n with n k. With this roosition in hand, we can now define new casses of amost eriodic functions on N. Definition 5.3. We say that f : N C with f ( ) is Wey amost eriodic if the set of transates W(f) is reativey comact in ( ). This set is denoted by AP W (N). From roosition 5.2, we have that AP B (N) AP W (N). Exame 5.4. The converse is fase, such that the incusion above is strict, as can be seen by ooking at the function { 0 x = 0, f(x) = x > 0.. The set of transates of f consists of f and the constant function, such that W(f) is reativey comact. However, f is not Bohr amost eriodic, as can be seen for any ε <. Note. AP W (Z) = AP B (Z). Fréchet gave the foowing characterization of amost eriodic funtions which gives rise to many exames. We state the resut without roof. Theorem 5.5 (Fréchet, 94). A function f ( ) is in AP W (N) if and ony if f admits a (unique) decomosition f = +w wehre AP B (N) and im n w(n) = 0. This definition of AP W (N) is is sometimes caed W -norma. Definition 5.6. The Eberein or weaky amost eriodic functions on N are bounded functions such that W(f) is weaky reativey comact. We denote this set by AP w (N). Eberein characterized the weaky amost eriodic functions in a theorem simiar that that of Fréchet. Theorem 5.7 (Eberein, 956). f ( ) is weaky amost eriodic on N if and ony if f admits a (unique) decomosition f = +w where AP B (N) and w satisfies with the imit is uniform in x. n im ω(x+j) = 0, n n j=0 Exame 5.8. Let w : N C be defined as a sequence of ones and and zeros such that the number of zeros between consecutive ones is increasing. Then w(n) does not tend to zero as n tends to infinity, but the average about does tend to zero uniformy in x. Coroary 5.9. AP W (N) AP w (N), and the incusion is roer. The characterization theorems of Fréchet and Eberein imy that AP W (N) and AP w (N) are agebras. Finay, we return to the Besicovitch amost eriodic functions. Instead of using tooogica roerties of transates to define these functions, we return to a definition based on the cosure under a seminorm. Definition 5.0. Let X be the set of sequences of the form (z k ) where z C, z =, and k = 0,,... A sequence is a trigonometric oynomia if it is a inear combination of a finite number of sequences in X. 6

7 Definition 5.. Let <. We say that f is Besicovitch amost eriodic on N if f is in the cosure of the trigonometric oynomias under the seminorm defined by f = imsu n n n k=0 f(k). In this case we write f AP B (N). We say that f is bounded Besicovitch if f AP B (N) ( ). It turns out that AP B (N) ( ) = AP B (N) ( ) for <. The idea behind the roof wi be exained beow, but no roof wi be given. For any function f, we can define its mean M {f} to be n M {f} := im f(j). n n The mean does not necessariy exist or is finite. Using this mean, we can define a function a : R C by j=0 a f (λ) = M { fe iλx}. For any f AP B (N) ( ), this function is we-defined for a λ R. If we take f to be the urey eriodic e iλx, then { a f (λ λ = λ, ) = 0 λ λ. It foows that for any f AP B (N) ( ), a f (λ) is nonzero for an at most countabe set of λ. (This reies on standard inner-roduct sace resuts, which we goss over.) Definition 5.2. Let f AP B (N) ( ) and et (λ n ) be the sequence of rea numbers such that a f (λ) 0. Then we write f(x) a f (λ n )e iλnx, n and ca the right hand side of the above the (generaized) Fourier series for f. Definition 5.3. Let f and (λ n ) be as in the revoius definition. Then a set of (at moust) countabe rea numbers B f is caed a base of f if B f forms a basis for (λ n ) over Q, i.e., B f consists of numbers such that a finite subsets of B f are ineary indeendent over Q and every λ n can be written as a finite inear combination of eements in B f with coefficients in Q. Let K n (t) be the Fejér kerne defined as Then a Bochner-Fejér kerne K n (t) = λ <n ( λ ) e iλt. n K(k) = K n (b k) k nm (b m k) is a finite roduct of m Fejér kernes with the b i ineary indeendent over Q. We take k N. Then, using the base B f for a function f in AP B (N) L ( ), it is ossibe to define a sequence of Bochner-Fejér kernes K f n with the foowing roerties. (i) σ f n(x) := M { f(x+j)k f n(j) } is a trigonometric oynomia; (ii) σ f n f for a n; (iii) σ f n f for a n; (iv) σ f n f 0 as n. Using these roerties and Höder s inequaity, we have that for <. AP B (N) ( ) = AP B (N) ( ) 7

8 6 Reation to Ergodic Theory We now show that the bounded Besicovitch functions are naturay occurring in the context of ergodic theory and rove a reated theorem about ointwise convergence of a weighted sequence of oerators acting on a function. Before stating the theorem which indicates that Bounded besicovitch functions occur naturay, we need a definition. Definition 6.. Let T be a bimeasurabe, measure-reserving, ergodic, bijection on a robabiity sace (Ω,A,µ). Let U T be the induced transformation on functions, i.e., U T f(x) = f(tx). Then T has discrete sectrum if L 2 (Ω,µ) has an orthonorma basis of eigenfunctions of T. The ergodic functions T can be characterized as foows. Theorem 6.2. Let T be as in the definition above. Then the foowing are equivaent: (i) T has discrete sectrum. (ii) If g L (Ω), then for amost every ω Ω, the sequence (g(t n ω)) for n N is bounded Besicovitch. Proof. (i) (ii). Let {f i } be an orthonorma set of eigenfunctions for U T. Let g L (Ω). This imies g L (Ω), since (Ω,A,µ) is a robabiity sace. Then there exists a sequence h n in the inear san of {f i } such that h n g inl.linearsanmeansfinite inear combinations ofthef i.bytheindividua (Birkhoff s?) ergodictheorem, for each n N, there exists a set Ω n of measure such that ω Ω n imies n im g(t j ω) h n (T j ω) = k k k=0 Ω g h n dµ. Let Ω g be the intersection of the Ω n, and note that this set has fu measure. If we consider the sequences v = (g(t j ω)) and h(n) = (h n (T j ω)). Then, as n goes to infinity, k v h(n) := im g(t j ω) h n (T j ω) = k k j=0 Ω g h n dµ 0. It is caimed in [2] that h(n) is ceary a trigonometric oynomia, such that g is in the cosure of these oynomias by the above. g is bounded by assumtion. Thus g AP B (N) ( ), as caimed. We eave the roof of the reverse direction to the interested reader, who may consut [2]. References [] J. Andres, A.M. Bersani, and R.F. Grande, Hierarchy of amost-eriodic functions saces, Rend. Mat. A. (2006). [2] A. Beow and V. Losert, The weighted ointwise ergodic theorem and the individua ergodic theorem aong subsequences, T. Am. Math. Soc. (985). [3] A.S. Besicovitch, Amost eriodic functions, Cambridge University Press,