E-sequences and the Stone–Weierstrass Theorem

Journal of Number Theory 133 (2013) 1525–1536

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Journal of Number Theory www.elsevier.com/locate/jnt

E-sequences and the Stone–Weierstrass Theorem L. Klingler a , M. Marshall b,∗ a b

Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-6498, United States College of Business, Western Governors University, Salt Lake City, UT 84107, United States

a r t i c l e

i n f o

Article history: Received 26 October 2011 Revised 1 October 2012 Accepted 1 October 2012 Available online 23 December 2012 Communicated by David Goss MSC: primary 13F20 secondary 13B25, 13A18 Keywords: Integer-valued polynomials Discrete valuation domain Even and odd continuous functions Stone–Weierstrass Theorem

a b s t r a c t For discrete valuation domain V with finite residue field, we define a variation of the V.W.D.W.O. sequence of Cahen and Chabert in order to construct V -bases for the algebra of even integervalued polynomials on V and the module of odd integer-valued polynomials on V . Using these bases, we prove a version of the Stone–Weierstrass Theorem for V , namely, that every even (respectively odd) continuous function on the completion Vˆ can be approximated by means of even (respectively odd) integer-valued polynomials on V . Using these approximations, we give series expansions for all even (respectively odd) continuous functions on Vˆ , analogous to results of Mahler for the p-adic integers. © 2012 Elsevier Inc. All rights reserved.

1. Introduction For an integral domain D with field of fractions K , the set of integer-valued polynomials on D is containing D [ X ]. Interest in defined to be Int( D ) = { f ∈ K [ X ] | f ( D ) ⊆ D }; it is in fact  X  a D-algebra X ( X −1)···( X −n+1) Int(Z) is classical, with the definition of the polynomials n = dating back to Newton n! and Gregory in the seventeenth century [2]. In 1919 Polyá [9] and Ostrowski [8] studied free bases for the D-module Int( D ) for the case where D is the ring of algebraic integers in a number field K , and in 1936 Skolem [12] studied the ring structure of Int(Z). Renewed interest in rings of integer-valued polynomials began in the 1980’s after Brizolis [1] showed that the ring Int(Z) is a Prüfer domain of Krull dimension two. For an extensive bibliography on integer-valued polynomials through 1996, see [2].

*

Corresponding author. E-mail addresses: [email protected] (L. Klingler), [email protected] (M. Marshall).

0022-314X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jnt.2012.10.003

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The Stone–Weierstrass Theorem for discrete valuation domains with finite residue field says that continuous integer-valued functions can be approximated by integer-valued polynomials. Dieudonné [3] proved a version of this theorem for the p-adic integers, later generalized by Kaplansky [5] and ˆ p using series of the Mahler [6]. Mahler gave an explicit expansion of functions continuous on Z

X

basis polynomials n (e.g., Example 1 below). These results were generalized to arbitrary discrete valuation domains with finite residue fields by Cahen and Chabert [2], using what they call V.W.D.W.O. sequences (Definition 1 below). Among the many interesting exercises in Polyá and Szegö’s classic Problems and Theorems in Analysis, II [10], 88 and 89 of the number section claim that the sequences of polynomials  X +n−theory  X +problems   n−1 1 E n = nX 2n−1 (with E 0 = 1) and O n = 2n−1 form Z-bases for the even polynomials and the odd

ˆ p -bases for the even polynopolynomials, respectively, in Int(Z). Indeed, these same sequences form Z

ˆ p ) for any rational prime p. This leads naturally mials and the odd polynomials, respectively, in Int(Z to the question of whether one can adapt V.W.D.W.O. sequences to generalize these polynomial sequences { E n }n0 and { O n }n1 to work for arbitrary discrete valuation domains with finite residue fields. This we do in Section 2, introducing the notion of “extendable” sequences (E-sequences) in Definition 2. In Theorem 1 we show that most discrete valuation domains with finite residue fields have an E-sequence (with restrictions only in case the residue field has even order). For discrete valuation domain V with an E-sequence, we construct a polynomial sequence { E n }n0 forming a V -basis for the algebra Inte ( V ) of even integer-valued polynomials on V (Theorem 2) and a polynomial sequence { O n }n1 forming a V -basis for the module Into ( V ) of odd integer-valued polynomials on V (Theorem 3). As an application, we show that the ring Inte ( V ) is not factorial and compute a lower bound on its elasticity (Corollary 2). In Section 3 we use the bases { E n }n0 and { O n }n1 to prove the Stone–Weierstrass Theorem for even continuous functions (Theorem 5) and odd continuous functions (Theorem 7) for discrete valuation rings with E-sequences. Using these approximation theorems, we derive series expansions for even continuous functions (Theorem 6) and odd continuous functions (Theorem 8) on the completion of a discrete valuation domain with E-sequence in terms of the polynomial bases { E n }n0 and { O n }n1 . Finally, decomposing a function into even and odd parts, we combine these series to get an analog of a Fourier series expansion for continuous functions on the completion (Corollary 3). Properties of even and odd functions are used in many areas of mathematics, for example the proofs of the results on the digamma function in [7] and those for results on the L-polynomials in [11] used properties of even and odd functions, respectively. Even and odd functions are an interesting class of functions in their own right; see [4] and [13]. Nonetheless, little is known about even and odd functions defined in abstract settings, such as over discrete valuation domains. In this paper, we show the possibility of expanding even and odd continuous function on the completion Vˆ as a series of even and odd, respectively, integer-valued polynomials. Note that this intuitive result does not automatically follow from the corresponding result for continuous functions and integer-valued polynomials in general. In Example 1, we give a simple example of an even continuous function  X on the 5-adic integers Z5 whose Mahler expansion using the sequence n of polynomials from [2, Theorem II.2.7] is not an expansion by even polynomials. The authors would like to thank the anonymous referee, whose numerous suggestions greatly improved the accuracy and exposition of this paper. 2. E-sequences Throughout this paper, we let V be a discrete valuation domain with field of fractions K , valuation v, maximal ideal m, and finite residue field V /m of order q. We begin by recalling the notion of a very well distributed and well ordered sequence, an analog to the set of natural numbers in the ring of p-adic integers [2, Definition II.2.1]. Definition 1. A sequence {un }n0 of elements of V is said to be a V.W.D.W.O. sequence if, for all non-negative integers n and m, v (un − um ) = v q (n − m), the largest power of q that divides n − m.

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An important fact which we shall find very useful (and noted in [2, Remarks II.2.2]) is that {un }n0 is a V.W.D.W.O. sequence if and only if, for each positive integer k, every subsequence of qk consecutive terms forms a complete set of residues of mk in V . In this section, we look at a special class of V.W.D.W.O sequences, which we call E-sequences, an analog to the set of rational integers in the ring of p-adic integers. E-sequences allow us to focus separately on the even and odd integer-valued polynomials. As we shall see (Theorem 1), E-sequences exist for most, but not all, discrete valuation domains. Using E-sequences, we construct V -bases for the V -algebra Inte ( V ) of even integer-valued polynomials (Theorem 2) and the V -module Into ( V ) of odd integer-valued polynomials (Theorem 3) on the discrete valuation domain V . As an application, we investigate the irreducibility of the basis polynomials for Inte ( V ) and prove that the ring Inte ( V ) is not factorial, in the process obtaining a lower bound on its elasticity (Corollary 2). Definition 2. Let {un }n0 a V.W.D.W.O. sequence of the discrete valuation domain V , and for each positive integer k, let

 ukn

=

−uk−n if n < k, un−k if n  k.

Thus, {unk }n0 is the sequence −uk , . . . , −u 1 , u 0 , u 1 , . . . , uk , . . . . We call {un }n0 an E-sequence of V if {unk }n0 is a V.W.D.W.O. sequence of V for every positive integer k, and we call {unk }n0 the kextension of {un }n0 . Before settling the question of existence, we note that an E-sequence must begin with 0. Proposition 1. If {un }n0 is an E-sequence of the discrete valuation domain V , then u 0 = 0. Proof. Fix a positive integer k. If q is odd, then we can write qk = 2h + 1 for some integer h, so in the h-extension of {un }n0 , the subsequence −u h , . . . , −u 1 , u 0 , u 1 , . . . , u h must be a complete set of residues of mk , from which it follows that u 0 ∈ mk . Suppose instead that q is even, and write qk = 2h for some integer h. Then in the h-extension of {un }n0 , the subsequence −u h , . . . , −u 1 , u 0 , u 1 , . . . u h−1 must be a complete set of residues of mk . If −u h ∈ mk , then u h − (−u h ) = 2u h ∈ 2mk ⊆ mk+1 , contradicting the fact that the subsequence −u h , . . . , −u 1 , u 0 , u 1 , . . . u h (of length qk + 1) must be in distinct k+1 k residue classes of m ∞ . Thus, u 0 ∈ m in this case as well. Therefore, u 0 ∈ k=0 mk = {0} (since V is a Noetherian domain). 2 We characterize those discrete valuation domains which have E-sequences. Theorem 1. The discrete valuation domain V has an E-sequence if and only if the cardinality q of its residue field is odd, or q = 2 and the maximal ideal of V is m = (2). q−1

Proof. Suppose first that q is odd; set q = 2 , and let {u 0 , ±u 1 , . . . , ±u q } be a complete set of residues of m in V , with u 0 = 0 (from Proposition 1). For each index i, 1  i  q , let u −i = −u i , so that our complete set of residues of m in V can be written as {u −q , . . . , u −1 , u 0 , u 1 , . . . , u q }. Each non-zero integer n has a unique representation in the form

n = ak qk + · · · + a1 q + a0 , in which −q  ai  q for each index i, and with ak = 0. Let m = (t ), and set

un = uak t k + · · · + ua1 t + ua0 .

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Note that u −i = −u i . To see that {un }n0 is an E-sequence, let m and n be arbitrary distinct integers, 

and write m = ak qk +· · ·+ a1 q + a0 and n = bk qk +· · ·+ b1 q + b0 , where −q  ai  q and −q  b i  q for each index i, and at least one of ak or bk is non-zero. Then both v (un − um ) and v q (n − m) equal the smallest index i such that ai − b i is non-zero. Since this holds for any integers m and n, it follows that, for any integer k, the k-extension of {un }n0 is a V.W.D.W.O. sequence. If q is even and {un }n0 is an E-sequence of V , then −u 1 , u 0 , u 1 , u 2 , . . . is a V.W.D.W.O. sequence. If q > 4, {−u 1 , u 0 , . . . , u q−1 } must be a complete set of residues of m in V . Since u 1 − (−u 1 ) = 2u 1 ∈ m, it follows that q = 2. Also, −u 1 and u 1 must be in different residue classes of m2 in V , so 2u 1 = u 1 − (−u 1 ) ∈ / m2 implies 2 ∈ / m2 , and hence m = (2). Conversely, if q = 2 and m = (2), then N is an E-sequence for V , since mk = (2)k implies that v (m − n) = v 2 (m − n). 2 In particular, our results on E-sequences hold for the ring of p-adic integers for any rational prime p. We turn now to the ring Int( V ) of integer-valued polynomials on the discrete valuation domain V , and we assume for the remainder of this paper that V satisfies the conditions of Theorem 1, so that we can fix an E-sequence {un }n0 of V . Just as a V.W.D.W.O. sequence can be used to construct a V -basis for the algebra Int( V ) [2, Theorem II.2.7], we can use an E-sequence to construct a V -basis for the algebra Inte ( V ) of even integer-valued polynomials and a V -basis for the module Into ( V ) of odd integer-valued polynomials on V , extending the exercises of Polyá and Szegö’s [10] for Inte ( Z ) and Into ( Z ) mentioned in the introduction. Theorem 2. Given the E-sequence {un }n0 of the discrete valuation domain V , the sequence { E n }n0 of polynomials defined by

E0 = 1

and, for n  1,

En =

n −1

X 2 − uk2

k =0

un2 − uk2

forms a V -basis of the algebra Inte ( V ) of even integer-valued polynomials on V . Proof. Let s be a fixed but arbitrary integer, and set

 uns =

−u s−n if n < s, u n−s if n  s.

Since {un }n0 is an E-sequence of V , by definition {uns }n0 is a V.W.D.W.O. sequence of V . Squaring each uns gives



uns

2

 =

(u s−n )2 if n < s, (un−s )2 if n  s.

Observe that





E s uns =



0 if 1  n  2s − 1, 1 if n = 0 or 2s.

s Hence E s (u 0s ), E s (u 1s ), . . . , E s (u 2s ) ∈ V , and clearly E s is a polynomial in K [ X ] of degree 2s, so by [2, Corollary II.2.8], E s ∈ Int( V ). Also note that E s ∈ K [ X 2 ], so that E s ∈ Inte ( V ). To complete the proof, we imitate the proof of [2, Theorem II.2.7], as follows. The sequence of polynomials E n clearly forms a K -basis for K [ X 2 ], since the degree of E n is 2n for each n, so if f ∈ Inte ( V ) ⊂ K [ X 2 ] is of degree 2n, we can write f = λ0 E 0 + · · · + λn E n for some λ0 , λ1 , . . . , λn

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in K . Note that λ0 = f (u 0 ) ∈ V . Suppose, by induction on k  n, that λi ∈ V for i < k. Then gk = λk E k + · · · + λn E n = f − (λ0 E 0 + · · · + λk−1 E k−1 ) ∈ Inte ( V ). Since gk is an integer-valued polynomial, λk = gk (uk ) ∈ V . 2 Similarly, we can get a basis for the module of odd integer-valued polynomials on V . We omit the proof, which is a minor variation of the proof of Theorem 2. Theorem 3. Given the E-sequence {un }n0 of V , the sequence { O n }n1 of polynomials defined by

On =

n −1 X  X 2 − uk2

un

un2 − uk2

k =1

forms a V -basis of the module Into ( V ) of odd integer-valued polynomials on V . Next we prove the irreducibility of certain basis polynomials, using an analog of the factorial function. Definition 3. If {un }n0 is an E-sequence for the discrete valuation domain V , we set un ! E =

n−1

2 k=0 (un

− uk2 ) for each n  0.

We isolate out the following technical computation for use in our irreducibility result. Proposition 2. Given the E-sequence {un }n0 of V , if r and s are positive integers such that r + s = qm for some positive integer m, then

u r +s ! E u r ! E ·u s ! E

is a non-unit element of V .

Proof. Recall that v q (a) denotes the largest power of q that divides the integer a. Using the kextension {unk }n0 of {un }n0 , we compute that

v (u k ! E ) =

k −1

v (u k − u i ) +

i =0

=

=

k −1



v uk2k − ukk+i +

v q (k − i ) +

i =0

=

k i =1

=

2k

v (u k + u i )

i =0

k −1  i =0

k −1

k −1

k −1  i =0

v uk2k − ukk−i



v q (k + i )

i =0

v q (i ) +

2k −1

v q (i )

i =k

v q (i ) − v q (2).

i =1

Without loss of generality we may assume that 2r < k. Using also the corresponding formulas for v (u r ! E ) and v (u s ! E ), and the fact that v q (i ) = v q (2qm − i ), we obtain

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v

uk ! E ur ! E · u s ! E

 = v (u k ! E ) − v (u r ! E ) − v (u s ! E ) =

=

2k

v q (i ) −

2r

i =1

i =1

2k

2qm −1

v q (i ) −

i =1

2s

v q (i ) −

v q (i ) + v q (2)

i =1



v q (i ) −

i =2s

2s

v q (i ) + v q (2)

i =1

  = v q 2qm − v q (2s) + v q (2) > 0.

2

Note that irreducibility of a polynomial in Int( V ) need not imply its irreducibility in the subring Inte ( V ) of even integer-valued polynomials on the discrete valuation domain V . For example, the polynomial X 2 factors as X · X in Int( V ) but certainly cannot be factored as a product of two non-units in Inte ( V ). We use E-sequences to give a criterion that will ensure irreducibility of certain polynomials in Inte ( V ) and then use this theorem to show that certain of the basis elements E n constructed in Theorem 2 are irreducible in the ring Inte ( V ) (even though they are reducible in Int( V )). Theorem 4. Let V be a discrete valuation domain with residue field of cardinality q, and suppose that {un }n0 is an E-sequence of V . Let { E n }n0 be the associated V -basis of polynomials for the ring Inte ( V ) constructed n in Theorem 2, and let f = k=0 ai E i be a polynomial in Inte ( V ). If n = ql for some positive integer l, if an is not divisible in V by

un ! E u r ! E ·u s ! E

for all positive integers r and s such that r + s = n, and if there exists some index

k for which ak is a unit, then f is irreducible in Inte ( V ). Proof. If we factor f = ag for some constant a ∈ V , then since ak is a unit for some k, it follows that a must instead that we can factor f = gh for some non-constant polynomials, ralso be a unit. Suppose s g = i =0 b i E i and h = i =0 c i E i ; hence 2n = 2r + 2s. Equating the leading coefficient of f with the product of the leading coefficients of g and h yields

an un ! E

contradicting the assumption that an is not divisible in V by

=

br ur ! E

un ! E u r ! E ·u s ! E

·

.

cs us !E

2

, so that an = br c s u !un·!uE ! , r E s E

Using Proposition 2, the special case of Theorem 4 in which an = 1 and ak = 0 for k < n yields the following immediate consequence. Corollary 1. Given the E-sequence {un }n0 of V , for each positive integer k, the associated basis element E qk of Inte ( V ) is irreducible in the ring Inte ( V ). Corollary 2. The ring Inte ( V ) is not factorial. Moreover, if the cardinality q of the residue field of V is odd, then q−1 the elasticity of Inte ( V ) is at least 2 , while if q = 2, then the elasticity is at least 2. Proof. For positive integer n, we note that q n −1

u qn ! E · E qn =





X 2 − uk2 .

(1)

k =0

Certainly each factor X 2 − uk2 is irreducible in the ring Inte ( V ), as is the polynomial E qn (Corollary 1). Since u qn ! E is a constant, it follows immediately that Inte ( V ) is not factorial. Moreover, in Proposition 2 we computed that v (u qn ! E ) =

2qn

v (k) for odd q, while for q = 2, k=1 q 2n+1 v (u 2n ! E ) = ( k=1 v 2 (k)) − 1. Since u qn ! E factors as the product of v (u qn ! E ) irreducible constants, 2qn qn −1 the left-hand side of Eq. (1) factors as the product of ( k=1 v q (k)) + 1 = 2 q−1 + 1 irreducible

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factors (when q is odd), while the right-hand side consists of qn irreducible factors. The ratio qn −1 q−1 qn /(2 q−1 + 1) → 2 , as n → ∞, a lower bound on the elasticity of Inte ( V ) if q is odd. If q = 2,

the reciprocal ratio becomes (2n+1 − 1)/2n → 2, as n → ∞, a lower bound on the elasticity of Inte ( V ) if q is even. 2 3. Stone–Weierstrass Theorem

In this section we use the basis elements for the even and odd integer-valued polynomials from the last section to give analogs of the Stone–Weierstrass Theorem for even and odd continuous functions on a discrete valuation domain. The representation we will give is similar to that of a continuous periodic function as the sum of sine and cosine series. The proofs are inspired by the case for the ring of integer-valued polynomials on a discrete valuation ring, which can be found in Chapter III of [2], where it was shown that, if the characteristic functions of cosets of ideals can be approximated, then all continuous functions can be approximated. Since these characteristic functions need not be even or odd, we use the even and odd parts of the characteristic function, coupled with the existence of an E-sequence and the basis elements constructed in the previous section, to get our results. We remark that the results in [2] apply to all discrete valuation domains, while our results apply only to discrete valuation domains with an E-sequence (cf. Theorem 1). As in the previous section, throughout this section we let V be a discrete valuation domain with field of fractions K , valuation v, maximal ideal m, and finite residue field V /m of order q. We also assume that V has an E-sequence {un }n0 with corresponding V -basis { E n }n0 for Inte ( V ) (Theorem 2).

Writing Vˆ for the m-adic completion of V , and C e ( Vˆ , Vˆ ) for the ring of even functions continuous on Vˆ , we show that, for each positive integer n, each function in C e ( Vˆ , Vˆ ) can be approximated ˆ n by polynomial in Inte ( V ). modulo m We begin by setting up the notation for “even” characteristic functions on cosets. Definition 4. Let m be the maximal ideal of the discrete valuation domain V . For any element a ∈ V and any integer h, let



h a (x)

χ

h = 1 if x ∈ a + mh , 0 if x ∈ /a+m

be the characteristic function of the coset a + mh , and



ah =

χah if a + mh = −a + mh , χah + χ−h a otherwise

be the even part of the characteristic function of the coset a + mh . We first show that, in order to be able to approximate all functions in C e ( Vˆ , Vˆ ), it suffices to be able to approximate the even part of the characteristic functions of all cosets in V . Lemma 1. For all positive integers k, the following assertions are equivalent.

ˆ k by a polynomial in Inte ( V ). (1) Every function in C e ( Vˆ , Vˆ ) can be approximated modulo m (2) For all positive integers h and all elements a ∈ V , the function ah can be approximated modulo mk by a polynomial in Inte ( V ). Proof. (1) ⇒ (2) For any positive integer h, the extension ˆah of ah to Vˆ is either the characteristic ˆ h (if a + m ˆ h = −a + m ˆ h ), or the sum of the characteristic functions of a + m ˆ h and function of a + m ˆk ˆ h in Vˆ , and hence is an even continuous function. If f ∈ Inte ( V ) approximates ˆah modulo m −a + m h k h k ˆ for all r ∈ Vˆ , so that f (r ) − a (r ) ∈ m for all r ∈ V . on Vˆ , then f (r ) − ˆa (r ) ∈ m

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(2) ⇒ (1) If φ ∈ C e ( Vˆ , Vˆ ), then since Vˆ is compact, the function φ is uniformly continuous. Thus, ˆ h implies φ(a) − for each positive integer k, there exists a positive integer h such that a − b ∈ m k k h ˆ on a + m ˆ ; that is, φ is constant modulo m ˆ . Since φ is also even, it follows that φ ≡ φ(b) ∈ m h k h a φ(a) · a (mod m ) on V , where a ranges over coset representatives for m in V such that, if h h a + m = −a + m , then we include only one of a or −a. By hypothesis, for each such representative a, there exists a polynomial f a ∈ Inte ( V ) such that f a ≡ ah mod mk , hence φ ≡ a φ(a) · f a (mod mk ) on V . Since V is dense in Vˆ , (1) follows.

2

Using Lemma 1, we can now prove a Stone–Weierstrass Theorem for even continuous functions on the completion of a discrete valuation domain with an E-sequence. Theorem 5. Let V be a discrete valuation domain with maximal ideal m and m-adic completion Vˆ , and suppose that V has an E-sequence. Then for all positive integers k, each even continuous function on Vˆ can be approximated modulo mk by an even integer-valued polynomial on V . That is, for all positive integers k and all ˆ k for all a ∈ Vˆ . φ ∈ C e ( Vˆ , Vˆ ), there exists f ∈ Inte ( V ) such that φ(a) ≡ f (a) mod m Proof. Let q be the cardinality of the residue field V /m, and let {un }n0 be an E-sequence for V , with corresponding V -basis { E n }n0 for Inte ( V ) (Theorem 2). We first prove the theorem for k = 1. For given positive integer h, by [2, Proposition III.3.1], for each integer-valued polynomial f with deg( f ) < qh , we have that a ≡ b mod mh implies that f (a) ≡ f (b) mod m; that is, f is constant modulo m on each coset a + mh . Suppose first that q is odd, and let q = that 0 < n  q , using the q -extension of {un }n0 , we have 

En ≡

q



E n (u i ) · χ

i =−q

h ui

≡

q

E n (u i ) ·



i =1

h ui

h



χ + χ− u i ≡

qh −1 . 2

Then for integer n such



q

E n (u i ) · uhi

(mod m)

i =1

(since E n is an even function and polynomial of degree 2n  2q < qh , and E n (u 0 ) = 0 for n > 0). If instead q = 2 (which implies that m = 2V ), let q = 2h−1 , and note that −a + mh = a + mh holds if and only if 2a ∈ mh = 2h V , that is, if and only if a ∈ 2h−1 V = mh−1 , so that there are only two such “even” cosets 0 + mh and 2h−1 + mh = q + mh . With this definition, using the (q − 1)-extension of {un }n0 , the above congruence becomes 

En ≡

q i =−q +1

q  −1

E n (u i ) · χ

h ui

≡ E n (u q  ) · χ

h u q



+

i =1

E n (u i ) ·



h ui

h



χ + χ− u i ≡



q

E n (u i ) · uhi

(mod m)

i =1

for q = 2 and 0 < n  q as well. Indeed, the above congruence also holds for n = 0 (since E 0 = 1 is constant), and thus for n = 0, . . . , q , we have q + 1 equations 

En =

q

E n (u i ) · uhi + δn

(2)

i =1

where δn ∈ m · Inte ( V ). Eq. (2) may be represented in matrix form, E = MΥ + , where E, Υ and  are (q + 1) × 1 column matrices whose coefficients are, respectively, the functions E n , uhi , and δn , and M is the (q + 1) × (q + 1) matrix ( E i (u j )). Note that M is upper triangular with coefficients on the diagonal equal to 1; hence M is invertible in V , so that Υ = M−1 E − M−1 . Therefore, the functions uhi can be approximated by even integer-valued polynomials modulo m. By Lemma 1, the theorem follows for k = 1.

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ˆ ). Suppose, by induction, that Let φ ∈ C e ( Vˆ , Vˆ ), and let t be a generator of m (and hence also of m we can express φ = f 0 + t f 1 + · · · t k−1 f k−1 + t k φk for some f 0 , f 1 , . . . , f k−1 ∈ Inte ( V ) and some φk ∈ C e ( Vˆ , Vˆ ). (The previous paragraphs prove this for k = 1.) By the case k = 1 above, φk = f k + t φk+1 for some f k ∈ Inte ( V ) and some φk+1 ∈ C e ( Vˆ , Vˆ ). Therefore, φ = f 0 + t f 1 +· · ·+ t k−1 f k−1 + t k f k + t k+1 φk+1 , where f 0 , f 1 , . . . , f k−1 , f k ∈ Inte ( V ) and φk+1 ∈ C e ( Vˆ , Vˆ ), completing the induction and hence the proof of the theorem. 2 For discrete valuation domain V having an E-sequence, every even continuous function on the completion Vˆ may be expressed as a series, using the corresponding V -basis of even integer-valued polynomials. Theorem 6. Let V be a discrete valuation domain with valuation v and V -basis of polynomials { E n }n0

for the algebra Inte ( V ). Then every continuous function φ ∈ C e ( Vˆ , Vˆ ) can be expanded as a sum of series ∞ φ = i =0 ai E i for coefficients ai ∈ Vˆ such that v (ai ) → ∞. The coefficients ai are uniquely determined by the recursive formula

φ(un ) = an +

n −1

ai E i (un ).

(3)

i =0

Proof. Let t be a generator of the maximal ideal m of V . As shown in the proof of Theorem 5, given φ ∈ C e ( Vˆ , Vˆ ), we may write φ = f 0 + t f 1 + · · · + t k f k + t k+1 φk+1 for some polynomials f 0 , f 1 , . . . , f k ∈ ∞ Inte ( V ) and function φk+1 ∈ C e ( Vˆ , Vˆ ). Hence, we can express φ = n=0 t n f n for a sequence of even integer-valued polynomials { f n }n0 .

k

n a E , Using the V -basis { E n }n0 for Inte ( V ), for each natural number n we can write f n = i = 0 i ,n i for some positive integer kn and coefficients ai ,n ∈ V . Set ai ,n = 0 for each i > kn , and let ai = ∞ ∞ l ˆ l=0 t ai ,l , an element of V . Let k(r ) = sup{k0 , . . . , kr }, so that, if i > k(r ), then ai = l=r +1 ai ,l , and ∞ hence v (ai ) > r. It follows that v (ai ) → ∞, and therefore i =0 ai E i is convergent. Also, the difference

∞

k(r )

∞

∞

n f n − i =0 ai E i is divisible by t r +1 , so that i =0 ai E i converges to n=0 t f n = φ . Finally, since E n (un ) = 1 and E i (un ) = 0 for all integers i > n (Theorem 2), the recursive equation (3) follows. 2 n=0 t

n

As mentioned in the introduction, the expansion of an even continuous function on Vˆ as a series of even integer-valued polynomials on the discrete valuation domain V does not automatically follow from the corresponding result for continuous functions and integer-valued polynomials in general, as the following example shows. Example 1. The ring Z5 , the localization of the ring of rational integers at the prime 5, is a discrete valuation domain with maximal ideal 5Z5 . The set of natural numbers  X N forms n−a1V.W.D.W.O. sequence of Z5 , and the sequence { B n }n0 of polynomials defined by B n = n = n1! k=0 ( X − k) forms a Z5 basis of the algebra Int(Z5 ) [2, Theorem II.2.7]. ˆ 5 by Define a function A on Z

 A (x) =

ˆ 5 ) ∪ (4 + 5Zˆ 5 ), 1 if x ∈ (1 + 5Z 0 otherwise.

Clearly A ∈ C e (Zˆ5 , Zˆ5 ), and one easily computes that the Mahler expansion begins A = B 1 − 2B 2 + 3B 3 − 3B 4 + · · · , which is not an expansion by even polynomials. On the other hand, the set of natural numbers also forms an E-sequence of Z5 , and the sequence { E n }n0 of polynomials defined in Theorem 2 forms a Z5 -basis of the algebra Inte (Z5 ). Again one easily computes that the series expansion begins A = E 1 − 4E 2 + · · · , which is an expansion by even polynomials.

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We conclude this section by proving analogs of the above results for odd continuous functions, showing that every odd continuous function on Vˆ can be approximated by an odd integer-valued polynomial on V . We denote the set of odd continuous functions on Vˆ by C o ( Vˆ , Vˆ ). We begin by setting up the notation for “odd” characteristic functions on cosets. Definition 5. Let V be a discrete valuation domain. For element a ∈ V and integer h, let

h a

γ =

χah if a + mh = −a + mh , h h χa − χ−a otherwise

be the odd part of the characteristic function of the coset a + mh . As in the even case, in order to be able to approximate all functions in C o ( Vˆ , Vˆ ), it suffices to be able to approximate the odd part of the characteristic functions of all cosets in V . Lemma 2. For all positive integers k, the following assertions are equivalent.

ˆ k by a polynomial in Into ( V ). (1) Every function in C o ( Vˆ , Vˆ ) can be approximated modulo m (2) For all positive integers h and all elements a ∈ V , the function γah can be approximated modulo mk by a polynomial in Into ( V ). Proof. (1) ⇒ (2) For any positive integer h, the extension γˆah of γah to Vˆ is the difference of the ˆ h and −a + m ˆ h in Vˆ , and hence is an odd continuous function. If characteristic functions of a + m ˆ k on Vˆ , then f (r ) − γˆah (r ) ∈ m ˆ k for all r ∈ Vˆ , so that f (r ) − f ∈ Into ( V ) approximates γˆah modulo m h k γa (r ) ∈ m for all r ∈ V . (2) ⇒ (1) If φ ∈ C o ( Vˆ , Vˆ ), then as in the proof of Lemma 1, compactness of Vˆ implies that, for ˆ k on a + m ˆ h. each positive integer k, there exists a positive integer h such that φ is constant modulo m Since φ is also odd, it follows that φ ≡ a φ(a) · γah (mod mk ) on V , where a ranges over coset representatives for mh in V such that we include only one of a + mh or −a + mh . By hypothesis, for each such representative a, there exists a polynomial f a ∈ Into ( V ) such that f a ≡ γah mod mk , hence φ ≡ a φ(a) · f a (mod mk ) on V . Density of V in Vˆ then implies that (1) holds. 2 As in the case of even functions, we use Lemma 2 to prove a Stone–Weierstrass Theorem for odd continuous functions on the completion of a discrete valuation domain. Theorem 7. Let V be a discrete valuation domain with E-sequence. For all positive integers k, each odd continuous function φ ∈ C o ( Vˆ , Vˆ ) can be approximated modulo mk by an odd integer-valued polynomial f ∈ Into ( V ). Proof. Let q be the cardinality of the residue field V /m, and let {un }n0 be an E-sequence for V , with corresponding V -basis { O n }n1 for Into ( V ) (Theorem 3). We first prove the theorem for k = 1. For given positive integer h, by [2, Proposition III.3.1], for each integer-valued polynomial f such that deg( f ) < qh , the function f is constant modulo m on each coset a + mh . If q is odd, let q = that, for each integer n such that 1  n  q , using the q -extension we have 

On ≡

q i =−q



O n (u i ) · χ

h ui

≡

q i =1

O n (u i ) ·



h ui

h



χ − χ− u i ≡

qh −1 , 2

so



q

O n (u i ) · γuhi

(mod m)

i =1

(again since O n is an odd function and polynomial of degree 2n − 1 < 2q < qh , and O n (u 0 ) = 0 for all n).

L. Klingler, M. Marshall / Journal of Number Theory 133 (2013) 1525–1536

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If instead q = 2, let q = 2h−1 , and again note that there are only two even cosets 0 + mh and 2h−1 + mh = q + mh , so that, using the (q − 1)-extension, the above congruence becomes 

On ≡

q i =−q +1

q  −1

O n (u i ) · χuhi ≡ O n (u q ) · χuh  +



q

O n (u i ) ·

i =1





χuhi − χ−h u i ≡



q

E n (u i ) · γuhi

(mod m)

i =1

for q = 2 and 1  n  q as well. Thus for n = 1, . . . , q , we have q equations 

On =

q

O n (u i ) · γuhi + δn

(4)

i =1

where δn ∈ m · Into ( V ). Eq. (4) may be represented in matrix form, O = MΓ + , where O, Υ and  are q × 1 column matrices whose coefficients are, respectively, the functions O n , γuhi , and δn , and M is the q × q matrix ( O i (u j )). Again the matrix M is invertible, being upper triangular with coefficients on the diagonal equal to 1, so that Γ = M−1 O − M−1 . Therefore, the functions γuhi can be approximated by odd integer-valued polynomials modulo m, so by Lemma 2, the theorem follows for k = 1. Exactly as in the proof of Theorem 5, by induction on k, each φ ∈ C o ( Vˆ , Vˆ ) can be expressed in the form φ = f 0 + t f 1 + · · · + t k−1 f k−1 + t k φk for some f 0 , f 1 , . . . , f k−1 ∈ Into ( V ) and some φk ∈ C o ( Vˆ , Vˆ ), where t is a generator of m, which completes the proof of the theorem. 2 As in the even case, for discrete valuation domain V having an E-sequence, we can express each odd continuous function on the completion Vˆ as a series, using the V -basis of polynomials { O n }n1 for Into ( V ). (We omit the proof, which consists of substituting Theorem 7 for Theorem 5 in the proof of Theorem 6.) Theorem 8. Let V be a discrete valuation domain with V -basis of polynomials { O n }n1 for the module Into ( V ). Every continuous function φ ∈ C o ( Vˆ , Vˆ ) can be expanded as a sum of series φ =

∞

i =1 ai O i

for

ˆ coefficients ai ∈ V such that v (ai ) → ∞. The coefficients ai are uniquely determined by the recursive formula φ(un ) = an + ni =−11 ai O i (un ). We end with a corollary that gives an analog of Fourier series expansion. Corollary 3. If V is a discrete valuation domain with residue field of odd cardinality, then every continuous function φ ∈ C ( Vˆ , Vˆ ) can be expanded as the sum of two series

φ=

∞ i =0

ai E i +

∞

bi O i

i =1

for some ai , b i ∈ Vˆ such that v (ai ), v (b i ) → ∞, where { E n }n0 and { O n }n1 are sequences of even and odd integer-valued polynomials, respectively, on V . Proof. Write 2φ = φe + φo , where φe ( X ) = φ( X ) + φ(− X ) is an even continuous function on Vˆ and φo ( X ) = φ( X ) − φ(− X ) is an odd continuous function on Vˆ . Since the residue field of V is assumed to have odd cardinality, by Theorem 1 V has an E-sequence, so by Theorems 6 and 8, φe and φo can be expanded as sums of the relevant series. Since 2 is a unit in V , dividing both sides by 2 yields the desired result. 2

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We remark that Corollary 3 holds in a slightly modified form in case V is a discrete valuation domain with an E-sequence but residue field of even cardinality. In that case, by Theorem 1, V has maximal ideal 2V and hence characteristic 0. Writing 2φ = φe + φo as in the proof of Corollary 3, both the even continuous function φe and the odd continuous function φo can be expanded as sums of the relevant series, so that

1

1

2

2

φ = φe + φo =

where

1 φ 2 e

and

1 φ 2 o

∞ ai i =0

2

Ei +

∞ bi i =1

2

Oi

are even and odd continuous functions, respectively, from Vˆ to

1 ˆ V. 2

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