Orthonormal systems in spaces of number theoretical functions

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Introduction
For a function f : N → C, we define · α by f α := lim sup x→∞ 1 x n x f (n) Let L α := {f : N → C: f α < ∞} be the linear space of functions on N with bounded seminorm f α . By L α we denote the quotient space L α modulo null-functions (i.e., functions f with f α = 0). For α 1, the norm space L α is complete [7]. Let A be an algebra of subsets of N. Then if the limit exists. Then δ is finitely additive on A, that is, δ is a content on A.
We say that an arithmetical function f possesses an (arithmetical) exists. If every A ∈ A possesses an asymptotic density, then every f ∈ L * 1 (A) possesses a mean value. Further, we define an inner product on L * 2 (A) by This product is well-defined. For this, let f, g ∈ L * 2 (A). If ε > 0, then there exist s 1 , Then and fḡ ∈ L * 1 (A). Since L * 2 (A) is complete, the space L * 2 (A) is a Banach space. Therefore the space L * 2 (A) is a Hilbert space with the inner product defined above.
In this paper, we investigate examples of Hilbert spaces L * 2 (A) together with associated (complete) orthonormal systems. Remark 2. The described construction of L * α (A) was the starting point of an integration theory by Indlekofer (see [4,5]).
Embedding N, endowed with the discrete topology, in the compact space βN, the Stone-Čech compactification of N, we get:Ā is an algebra in βN (for details, see [4,5]).
Let δ be a content on A, that is, δ : A → R 0 is finitely additive, and defineδ onĀ bȳ Thenδ is a pseudo-measure onĀ and can be extended to a measure on σ(Ā), which we also denote byδ. This leads to the measure space (βN, σ(Ā),δ). Note that the following relations of the characteristic functions imply that the characteristic function of a set A ∈ A is a finite linear combination of products 1 Ap 1 · · · 1 Ap r . Thus the asymptotic density Obviously, for every prime p, In the same way, we obtain

By induction this leads to
Putting h * n := (ϕ(n)) −1/2 h n (μ 2 (n) = 1), we have shown the following: Theorem 1. The set {h * n : n square-free} is a complete orthonormal system for L * 2 (A 0 ). Remark 3. We easily to see that the function h n : N → Z satisfies h n1n2 = h n1 · h n2 if (n 1 , n 2 ) = 1 and μ(n 1 ) = μ(n 2 ) = 1, that is, n = 1, or n is a product of an even number of different primes.
Remark 4. Every f ∈ E(A 0 ) can be written as a linear combination of multiplicative g j such that g j (p l ) = 1 for all p k j and l ∈ N, since g = 1 − 1 Ap is multiplicative.

Almost even functions
For primes p and k = 0, 1, 2, . . . , let A p k := {n ∈ N: p k | n} be the set of natural numbers divisible by p k . Let A 1 be the algebra generated by the sets {A p k }. Then, for all A p k , the asymptotic density δ(A p k ) exists and equals 1/p k , and, as before, the asymptotic density δ(A) exists for all A ∈ A 1 .
Schwarz and Spilker [8, Chap. VI] considered the space B of even functions and characterized the sets of α-almost even functions (see also [3]).
It is well known that E(A 1 ) equals B and L α (A 1 ) is exactly the space of α-almost even functions (see [5]).
Remark 5. Every f ∈ E(A 1 ) can be written as a linear combination of multiplicative functions g j such that g j (p l ) = 1 for all p k j and l ∈ N.
Remark 6. Let f : N → C be a multiplicative function such that |f | 1. Then the following statements hold.
Define h n = 1 for n = 1 and Putting where ϕ is Euler's function, it is easy to show (see above) that {h * n } is an orthonormal system. We conclude by the following: Theorem 2. The set {h * n : n ∈ N} is a complete orthonormal system for L * 2 (A 1 ). Remark 7. The functions h n appear in a very natural way. It is not difficult to show (see [8, pp. 16-17]) that h n is just the Ramanujan sum c n for every n.

Limit periodic functions
Let A 2 be the algebra generated by all residue classes A a,r := {n ∈ N: n ≡ a mod r}, 1 a r, r ∈ N.
Here again the asymptotic density δ is a finite additive function on A 2 . Then we have the following lemma.    with characteristic functions f + , f 0 , and f − , respectively. Obviously,

Almost periodic functions
We define the algebra A 4 to be the algebra generated by the sets  Then f * is multiplicative. Since . Every h n can be written as a finite linear combination of Theorem 5. Let f : N → R be multiplicative with |f | 1. Then f ∈ L * α (A 4 ) for all α 1.
Next, we construct an orthonormal system for the space L * 2 (A 4 ).
Observe, that in this case, by (ii) of Remark 6, p, f (p)=0 1/p < ∞. Obviously, ∼ is an equivalence relation on R 0 . Now choose a representative from each residue class that takes only the values ±1 and denote this set by F 1 . Then F 1 forms an orthonormal system. For this, let f, g ∈ F 1 and observe that p, f (p) =g(p) 1/p = ∞. Then, by (ii) of Remark 6, M (fḡ) = f, g = 0. Furthermore, for f ∈ F 1 , we have f 2 = 1 and f, f = M (f 2 ) = 1.
This shows that F 1 is an orthonormal system in L * 2 (A 4 ). Consider, for f ∈ F 1 , the system where h * n is the normalized function (2.1).
Theorem 6. F 2 is a complete orthonormal system for L * 2 (A 4 ).
Proof. First, we show that F 2 is an orthonormal system. For this, let h * n f = h * n g. This holds if and only if f = g and n,ñ are arbitrary or f = g and n =ñ. Assume that f = g and n,ñ are arbitrary. Then where h * is (see Remark 6) a finite linear combination of multiplicative functions g j with |g j | = 1 and g j (p) = 1 for p > k j . Therefore So we obtain h * n f, h * n g = 0 if f = g. In the case f = g and n =n, For the proof of the completeness of F 2 , let g ∈ L * (A 4 ). Then g can be approximated in the · 2 norm by some g * ∈ E(A 4 ), Note that 1 A for A ∈ A 4 is a finite linear combination of products of multiplicative functions f taking only the values {−1, 0, 1}.
Therefore it suffices to prove that each real-valued multiplicative function f with values f (N) ⊂ {−1, 0, 1} can be approximated by a linear combination of functions from F 2 . Choose g ∈ F 1 that is equivalent to f . Then f = hg where h = f g, since g 2 = 1. Then and h ∈ L * 2 (A 1 ). Thus h can be approximated by a linear combination of functions h 1 , . . . , h m , that is, for This ends the proof of the completeness of F 2 .

q-ary almost even functions
First, we introduce q-multiplicative functions. Let q 2 be an integer, and let A = {0, 1, . . . , q − 1}. The q-ary expansion of some n ∈ N 0 is defined as the unique sequence ε 0 (n), ε 1 (n), . . . for which The numbers ε 0 (n), ε 1 (n), . . . are called the digits in the q-ary expansion of n. In fact, ε r (n) = 0 if r > log n/ log q. A function f : N 0 → C is called q-multiplicative if f (0) = 1 and for every n ∈ N 0 , Let the algebra A 5 be generated by the sets A j,a := {n ∈ N: ε j (n) = a}, j ∈ N 0 , a ∈ A. Every A ∈ A 5 possesses an asymptotic density δ(A).
Let L * 1 (A 5 ) be the · 1 -closure of E(A 5 ). Here E(A 5 ) is called the space of q-ary even functions. Then L * 1 (A 5 ) is called the space of q-ary almost even functions.
Remark 8. Let f be a real-valued q-multiplicative function of modulus 1. Then the mean values M (|f |) and M (f ) always exist (see [6]). Especially, we have: (ii) If a∈A f aq j = 0 for all j ∈ N 0 and ∞ j=0 a∈A then M (f ) = 0.
As an immediate consequence, we have the following: This ends Remark 8. Let L * 2 (A 5 ) be the · 2 -closure of E(A 5 ). Then we define a complete orthonormal system for the space L * 2 (A 5 ). Now from each equivalence class we choose a representative that is = 0 for all n ∈ N. We denote this set of representatives by F 3 . We consider F 4 := {h a0,...,ar f : f ∈ F 3 , a j ∈ A, j = 0, . . . , r, r ∈ N} and show the following: Theorem 8. F 4 is a complete orthonormal system for L * 2 (A 6 ).
To prove the completeness of F 4 , it suffices to show that every q-multiplicative f with M (|f |) = 0 and f (n) ∈ {−1, 0, 1} for all n ∈ N 0 can be approximated by linear combinations of elements of F 4 .
Let f be such a function. Then f = |f | sign f and sign f ∼ g, g ∈ F 3 , and f = |f | sign f g 2 = |f | sign f g g.
Now |f | sign f g is a q-ary even function and can therefore be approximated by a linear combination of some h a0,...,ar . This proves Theorem 8.