Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling

arXiv:1804.06638v1 [math.FA] 18 Apr 2018

QUATERNIONIC FUNDAMENTAL CARDINAL SPLINES: INTERPOLATION AND SAMPLING JEFFREY A. HOGAN1,2,3 AND PETER R. MASSOPUST3

Abstract. B-splines Bq , Sc q > 1, of quaternionic order q, for short quaternionic B-splines, are quaternion-valued piecewise M¨ untz polynomials whose scalar parts interpolate the classical Schoenberg splines Bn , n ∈ N, with respect to degree and smoothness. As the Schoenberg splines of order ≥ 3, they in general do not satisfy the interpolation property Bq (n − k) = δn,k , n, k ∈ Z. P b Bq (ξ + 2πk)—if However, the application of the interpolation filter 1/ k∈Z

well-defined—in the frequency domain yields a cardinal fundamental spline of quaternionic order that does satisfy the interpolation property. We handle the ambiguity of the quaternion-valued exponential function appearing in the denominator of the interpolation filter and relate the filter to interesting properties of a quaternionic Hurwitz zeta function and the existence of complex quaternionic inverses. Finally, we show that the cardinal fundamental splines of quaternionic order fit into the setting of Kramer’s Lemma and allow for a family of sampling, respectively, interpolation series. Keywords and Phrases: Quaternions, cardinal B-splines, Clifford algebra, fundamental cardinal spline, Hilbert module, interpolation, sampling, Kramer’s lemma AMS Subject Classification (2010): 15A66, 30G35, 41A05, 42C40, 65D07, 94A20

1. Introduction In [7], a new class of B-splines with quaternionic order, quaternionic B-splines for short, were introduced with the intention to obtain interpolants and approximants for data that require a multi-channel description. For example, seismic data has four channels, each associated with a different kind of seismic wave: the so-called P (Compression), S (Shear), L (Love) and R (Rayleigh) waves. Similarly, the color value of a pixel in a colour image has three components – the red, green and blue channels. In order to perform the tasks of processing multi-channel signals and data an appropriate set of analyzing basis functions is required. These basis functions should have the same analytic properties as those enjoyed by classical B-splines but should in addition be able to describe multi-channel structures. In [11, 12], a set of analyzing functions based on wavelets and Clifford-analytic methodologies were introduced in an effort to process four channel seismic data. A multiresolution structure for the construction of wavelets on the plane for the analysis of fourchannel signals was outlined in [7]. 1 Corresponding

author. partially supported by Australian Research Council grant DP160101537. 3 Research partially supported by Bavarian Research Alliance Grant BaylntAn UPA 2017 76. 2 Research

1

JEFFREY A. HOGAN1,2,3 AND PETER R. MASSOPUST3

2

The Schoenberg cardinal spline interpolation problem on the real line R consists of finding a spline L with the property that L(k) = δk,0 , for all P k ∈ Z, and which is a linear combination of shifts of a cardinal B-spline B: L = ck B( · − k). It k∈Z

is therefore natural to ask whether the newly introduced quaternionic B-splines Bq also solve a cardinal interpolation problem. In the classical case, only fundamental cardinal splines of even order exist and one should expect that some restrictions on q in the quaternionic setting will occur as well. The solution of the cardinal spline interpolation problem in the quaternionic setting is based on finding zero-free regions of a certain linear combination of quaternion-valued Hurwitz zeta functions. These zero-free regions are found by considering zero-free regions for associated complex-valued Hurwitz zeta functions. In the complex-valued case, such zero-free regions were derived in [5] and they will determine nonempty regions in the real quaternionic algebra HR for which the quaternionic cardinal spline interpolation problem has a unique solution. Associated with the cardinal spline interpolation problem is Kramer’s sampling theorem which gives conditions under which a function can be represented as an infinite series involving sampled values and a cardinal spline-type sampling function. We will extend Kramer’s lemma to the complex quaternionic setting by first deriving a Riesz representation theorem for left Hilbert modules. 2. Notation and Preliminaries The real, associative algebra of real quaternions HR is given by ( ) 3 X HR = a + vi ei : a, v1 , v2 , v3 ∈ R , i=1

where the imaginary units e1 , e2 , e3 satisfy e21 = e22 = e23 = −1, e1 e2 = e3 , e2 e3 = e1 and e3 e1 = e2 . Because of these relations, HR is a non-commutative algebra. 3 P Each quaternion q = a + vi ei may be decomposed as q = Sc q + Vec q where i=1

Sc q = a is the scalar part of q and Vec q = v =

3 P

vi ei is the vector part of q. The

i=1

conjugate q of the real quaternion q = a + v is the quaternion q = a − v. Note 3 3 P P that qq = qq = |q|2 = a2 + |v|2 = a2 + vi2 . Note also that if v = vj ej and i=1

w=

3 P

j=1

wj ej are quaternionic vectors, then

j=1

vw = −hv, wi + v ∧ w, where hv, wi =

3 P

(2.1)

vj wj is the scalar product of v and w and

j=1

v ∧ w = (v2 w3 − v3 w2 )e1 + (v3 w1 − v1 w3 )e2 + (v1 w2 − v2 w1 )e3 is the vector (cross) product of v and w. 3 3 P P If q = q0 + qi ei ∈ HC = a + vi ei : a, v1 , v2 , v3 ∈ C is a complex quateri=1

i=1

nion, we define the conjugate q of q by q = q0 −

3 P i=1

qi ei , where qi is the complex

QUATERNIONIC FUNDAMENTAL SPLINES: INTERPOLATION AND SAMPLING

conjugate of the complex number qi . If p = p0 +

3 P

3

pi ei ∈ HC , we define the inner

i=1 3 P

product hp, qi to be the complex number hp, qi =

p1 q i = Sc pq. We also define

i=0

eq by the usual series: eq =

∞ qj P . We require the following bounds on |eq |, which j=0 j!

we state without proof. Lemma 1. Let q = a + v ∈ HR . Then we have 1. |eq | = ea ≤ e|q| . √ 0 0 2. If z ∈ C and q 0 = zq ∈ HC , then |eq | ≤ e 2|q | . For q = a + v = a +

3 P j=1

vj ej ∈ HR and z ∈ C, the quaternionic power z q ∈ HC

is defined by v sin(|v| log z)]. (2.2) |v| This definition of z q allows for the usual differentiation and integration rules: Z d q z q+1 z = qz q−1 and z q dz = + const., q 6= −1. (2.3) dz q+1 z q := z a [cos(|v| log z) +

In general, however, the semigroup property z q1 z q2 = z q1 +q2 fails to hold. (See, for instance, [7].) If F ∈ {R, C, HR , HC } and 1 ≤ p < ∞, then Lp (R, F) denotes the Banach space of R∞ measurable functions f : R → F for which |f (x)|p dx < ∞, where the meaning −∞

of |f (x)| is dependent on F. The Banach space L∞ (R, F) is defined similarly. R∞ On L2 (R, C), we define an inner product by hf, gi = f (x)g(x) dx, and on −∞

L2 (R, HC ) by Z

∞

hf, gi = Sc

Z f (x)g(x) dx =

−∞

∞

hf (x), g(x)i dx.

(2.4)

−∞

The Fourier-Plancherel transform F is defined on L1 (R, F) by Z (F f )(ξ) := fb(ξ) := f (x)e−iξx dx R 2

and may be extended to L (R, F), on which it becomes a multiple of a unitary mapping: hF f, F gi = 2πhf, gi. 3. Quaternionic B-splines In this section, we state the definition of B-splines of quaternionic order and briefly review some of their properties. More details and proofs can be found in [7]. The complex B-spline Bz (z ∈ C) is defined in the Fourier domain to be the cz : R → C given by function B z 1 − e−iξ c , Re z > 1, Bz (ξ) = iξ


4

and in the time domain by Bz (t) =

∞ 1 X z z−1 (t − k)+ , Γ(z) k

t ∈ R, Re z > 1.

k=0

(See [3], [7].) A B-spline Bq of quaternionic order q (for short quaternionic B-spline) is defined cq : R → HC given by in the Fourier domain to be the function B −iξ q cq (ξ) := 1 − e B , Sc q > 1. (3.1) iξ 1 − e−iξ , we obtain the precise meaning of (3.1), namely, iξ cq (ξ) = Ξ(ξ)Sc q cos(|v| log Ξ(ξ)) + v sin(|v| log Ξ(ξ)) . B |v|

Setting Ξ(ξ) :=

(3.2)

The function Ξ has a removable singularity at ξ = 0 with Ξ(0) = 1. It follows from Re Ξ(ξ) = ξ −1 sin ξ and Im Ξ(ξ) = ξ −1 (1 − cos ξ), that graph Ξ ∩ (R− × {0}) = ∅. cq are well-defined when −π < arg Ξ ≤ π. Equation (3.2) Hence, Ξ and therefore B cq ∈ L2 (R, HC ) for a fixed q with Sc q > 1 and in L1 (R, HC ) for a also implies that B 2 fixed q with Sc q > 1 as the fractional B-splines [17] satisfy these conditions. Note cq ∈ L1 (R, HC ) implies that Bq is uniformly continuous on R. that B The following results are a direct consequence of the corresponding properties of fractional B-splines [17]. To this end, for a real number s ≥ 0 and 1 ≤ p ≤ ∞, the Bessel potential space H s,p (R, HC ) is given by n o H s,p (R, HC ) = f ∈ Lp (R, HC ) : F −1 [(1 + |ξ|2 )s/2 F f ] ∈ Lp (R, HC ) . Proposition 1. Let Bq be a quaternionic B-spline with Sc q > 12 . Then Bq enjoys the following properties: (i) Decay: Bq ∈ O(|ξ|− Sc q ) as |ξ| → ∞. (ii) Smoothness: Bq ∈ H s,p (R, HC ) for 1 ≤ p ≤ ∞ and 0 ≤ s < Sc q + p1 . (iii) Reproduction of Polynomials: Bq reproduces polynomials up to order dSc qe, where the ceiling function d · e : R → Z is given by r 7→ min{n ∈ Z : n ≥ r}. The time domain representation of the quaternionic B-spline Bq is given by ∞ 1 X q q−1 Bq (t) = (−1)k (t − k)+ , t ∈ R, Sc q > 1. (3.3) Γ(q) k k=0

This equality holds in the sense of distributions and in L2 (R). Here, the quaternionic binomial coefficient is defined via the quaternionic Gamma function: Z ∞ Z ∞ v Γ(q) := ta−1 cos(|v| log t)e−t dt + ta−1 sin(|v| log t)]e−t dt. (3.4) |v| 0 0 It was shown in [7] that Γ(q) can be written as Γ(σ + i|v|) + Γ(σ − i|v|) v (Γ(σ + i|v|) − Γ(σ − i|v|) + 2 |v| 2i v = Re Γ(z) + Im Γ(z), z := σ + i|v|. |v|

Γ(q) =

(3.5)


5

The next result shows that there exists also a structure formula for the quaternionic B-splines. Theorem 1. If q = a + v ∈ HR and z = a + i|v| ∈ C, then Bq (t) = Re Bz (t) +

v Im Bz (t), |v|

t ∈ R.

Proof. We recall from [7] the time domain representation (3.3) of Bq and the following decomposition of the quaternionic Pochhammer symbol (q)j : (q)j = Re(z)j +

v Im(z)j . |v|

(3.6)

By (3.6) we have v Re(z)k + |v| Im(z)k q (q)k z v z = = = Re + Im . k k! k! k |v| k Furthermore, tq−1 = ta−1 [cos(|v| log t) + = ta−1 [Re ti|v| +

v sin(|v| log t)] |v|

v v Im ti|v| ] = Re tz−1 + Im tz−1 . |v| |v|

Thus, q z v z v z−1 z−1 = Re Im((t − k) ) (t − k)q−1 + Re((t − k) ) + + + + k k |v| k |v| z z = Re Re(t − k)z−1 − Im Im(t − k)z−1 + + k k v z z z−1 z−1 + [Re Im(t − k)+ + Im Re(t − k)+ ] |v| k k z v z z−1 z−1 = Re (t − k)+ + Im (t − k)+ . (3.7) k |v| k Substituting (3.7) into (3.3) yields X X ∞ ∞ 1 z v z z−1 z−1 Bq (t) = Re (t − k)+ + Im (t − k)+ Γ(q) k |v| k k=0

k=0

1 = [Re(Γ(z)Bz (t)) + Im(Γ(z)Bz (t))]. Γ(q)

(3.8)

On the other hand, using (3.5), we obtain v Re Γ(z) − |v| Im Γ(z) 1 = ,. Γ(q) |Γ(z)|2

(3.9)


6

and substituting (3.9) into (3.8) gives

Re Γ(z) −

Bq (t) =

v |v|

Im Γ(z)

|Γ(z)|2

[Re(Γ(z)Bz (t)) + Im(Γ(z)Bz (t))]

1 [[Re Γ(z) Re(Γ(z)Bz (t)) + Im Γ(z) Im(Γ(z)Bz (t))] |Γ(z)|2 v + [Re Γ(z) Im(Γ(z)Bz (t)) − Im Γ(z) Re(Γ(z)Bz (t))]] |v| v 1 [Re(|Γ(z)|2 Bz (t)) + Im(|Γ(z)|2 Bz (t))] = 2 |Γ(z)| |v| v Im Bz (t). = Re Bz (t) + |v| =

4. Quaternion Inverses Let q = z +

3 P i=1

q q˜ = z 2 +

3 P i=1

wi ei ∈ HC (i.e., z, w1 , w2 , w3 ∈ C). Then with q˜ = z − w we have

wi2 , so that q is invertible if and only if z 2 +

case q −1 = q˜/(z 2 +

3 P i=1

3 P i=1

wi2 6= 0 and in this

wi2 ). When q = a + v ∈ HR , then q is invertible provided

q 6= 0 and then q −1 = q/|q|2 . Lemma 2. If λ ∈ C and q ∈ HR then eλq is invertible in HC and (eλq )−1 = e−λq . Proof. By direct calculation we find that eλq = eλa eλv = eλa (cos(λ|v|) +

v sin(λ|v|)) |v|

and therefore, v v eλq e−λq = eλa cos(λ|v|) + sin(λ|v|) e−λa cos(λ|v|) − sin(λ|v|) |v| |v| = cos2 (λ|v|) + sin2 (λ|v|) = 1.

Lemma 3. For all t ∈ R+ and q ∈ HR , we have (−t)q = eiπq tq .


7

Proof. From the definition of a quaternionic power, we have v sin(|v| log(−t))] |v| v sin(|v|(log t + iπ))] = (−t)a [cos(|v|(log t + iπ)) + |v| = (−t)a [cos(|v| log t) cosh(π|v|) − i sin(|v| log t) sinh(π|v|) v + (sin(|v| log t) cosh(π|v|) + i cos(|v| log t) sinh(π|v|))] |v| iv = (−t)a (cos(|v| log t)[cosh(π|v|) + sinh(π|v|)] |v| v cosh(π|v|)]) + sin(|v| log t)[−i sinh(π|v|) + |v| v = eiπa eiπv ta (cos(|v| log t) + sin(|v| log t)) = eiπq ta+v = eiπq tq . |v|

(−t)q = (−t)a [cos(|v| log(−t)) +

Here, we used the usual convention that ta := eiπa |t|a , for t < 0.

5. A Quaternionic Zeta Function In this section we study the quaternionic analog ζ(q, a) of the classical Hurwitz zeta function ζ(s, a), which is defined by ζ(s, a) :=

∞ X k=0

1 , (a + n)s

Re s > 1 and Re a > 0.

To do so, we first define functions χ+ and χ− by χ± (v) =

v 1 1±i , 2 |v|

(v =

3 X

v 1 e i ∈ HR )

i=1

and note that χ± (v)2 = χ± (v);

χ+ (v)χ− (v) = χ− (v)χ+ (v) = 0.

(5.1)

Lemma 4. Given q = σ + v ∈ HR , σ > 1, we define the quaternionic zeta function ζ(q, · ) : R → HR by ζ(q, a) :=

∞ X k=0

1 , (a + k)q

a > 0.

Then, with w = σ + i|v| ∈ C, we have ζ(q, a) = χ+ (v)ζ(w, a) + χ− (v)ζ(w, a), where ζ(w, a) and ζ(w, a) denote the complex-valued Hurwitz zeta functions. Further, for all positive integers n, ζ(q, a)n = χ+ (v)ζ(w, a)n + χ− (v)ζ(w, a)n .


8

Proof. Note that (2.2) and the assumption Sc q = σ > 1 imply that the infinite ∞ P 1 series (a+k)q , a > 0, converges absolutely. For each z ∈ C we have k=0

v z q = z a [cos(|v| log z) + sin(|v| log z)] |v| i|v| log z + e−i|v| log z v ei|v| log z − e−i|v| log z a e =z + 2 |v| 2i = z a [χ− (v)z i|v| + χ+ (v)z −i|v| ] = χ− (v)z w + χ+ (v)z w where w = a + i|v| ∈ C. Thus, ζ(q, a) = =

∞ X

(a + k)−q

k=0 ∞ X k=0

χ− (v) χ+ (v) + = χ− (v)ζ(w, a) + χ+ (v)ζ(w, a). (a + k)w (a + k)w

Hence, because of (5.1), we have n X n ζ(q, a)n = χ− (v)j ζ(w, a)j χ+ (v)n−j ζ(w, a)n−j j j=0 = χ− (v)ζ(w, a)n + χ+ (v)ζ(w, a)n .

6. Interpolation with Quaternionic B-Splines In order to solve the cardinal spline interpolation problem using the classical Curry-Schoenberg splines [1, 15], one constructs a fundamental cardinal spline function which is a linear bi-infinite combination of polynomial B-splines Bn of fixed order n ∈ N that interpolates the data set {δm,0 : m ∈ Z}. More precisely, one solves the bi-infinite system n X (n) ck Bn + m − k = δm,0 , m ∈ Z, 2 k∈Z

(n) {ck

for the sequence : k ∈ Z} and then defines the fundamental cardinal spline P (n) Ln : R → R of order n ∈ N by Ln (x) = ck Bn n2 + x − k . The Fourier k∈Z

transform of Ln is given by \ b n (ξ) = X τ−n/2 Bn (ξ) L \ τ−n/2 Bn (ξ + 2πk)

(6.1)

k∈Z

where τα f (x) = f (x − α) (x, α ∈ R). Using the Euler-Frobenius polynomials associated with the B-splines Bn , one can show that the denominator in (6.1) does not vanish on the unit circle. For details, see for instance [1, 15]. One of our goals is to construct a fundamental cardinal spline Lq : R → HR of quaternionic order q = a + v, a > 1, of the form X (q) Lq := ck Bq ( · − k) , σ > 1, (6.2) k∈Z


9

satisfying the interpolation problem

Lq (m) = δm,0 ,

m ∈ Z,

(6.3)

(q)

for an appropriate bi-infinite quaternion-valued sequence {ck : k ∈ Z} and for appropriate q belonging to some nonempty open subset of HR . The case q ∈ C reduces to the setting for complex B-splines and the existence of fundamental cardinal splines for complex B-splines was investigated and proven in [4, 5]. Taking the Fourier transform of (6.2) and applying (6.3) to eliminate the ex(q) pression containing the unknowns {ck : k ∈ Z} gives (formally) a formula for Lq similar to (6.1):

b b q (ξ) = P Bq (ξ) L , Bq (k) z k

z = eiξ , ξ ∈ R.

(6.4)

k∈Z

Equation (6.4) contains a complex quaternion in the denominator. ThePnext result gives necessary and sufficient conditions for the complex quaternion Bq (k) z k k∈Z P P to have an inverse. In the following, we write to mean . k

k∈Z

P Theorem 2. Let q = a + v ∈ HR and w = a + i|v| ∈ C. Then Bq (`)z ` is ` P P invertible in HC if and only if Bw (`)z ` Bw (k)z k 6= 0. In this case we `

k

have

X `

Bq (`)z `

−1

Ph

i v Re Bw (`) − |v| Im Bw (`) z ` . = ` P P Bw (`)z ` Bw (`)z ` `

`

(6.5)


10

Proof. First note that if q = a + v ∈ HR and (w)k is the Pochhammer symbol of the complex number w = a + i|v| then X XX (q)k Γ(q) Bq (`)z ` = (−1)k (` − k)q+ z ` k! ` ` k X (−1)k v = Re(w)k + Im(w)k (` − k)q+ z ` k! |v| `,k X (−1)k v = Re(w)k + Im(w)k (` − k)a+ cos(|v| log(` − k)+ ) k! |v| `.k v sin(|v| log(` − k)+ ) z ` + |v| X (−1)k (` − k)a+ [Re(w)k cos(|v| log(` − k)+ ) − Im(w)k sin(|v| log(` − k)+ )] = k! `,k v + [Im(wk ) cos(|v| log(` − k)+ ) + Re(w)k sin(|v| log(` − k)+ )] z ` |v| X (−1)k v a i|v| log(`−k)+ i|v| log(`−k)+ (` − k)+ Re((w)k e Im((w)k e ) z` = )+ k! |v| `,k " ! X X (−1)k i|v| = Re (` − k)a+ (w)k (` − k)+ k! ` k !# X (−1)k v i|v| a + z` Im (` − k)+ (w)k (` − k)+ |v| k! k X v = Re(Γ(w)Bw (`)) + Im(Γ(w)Bw (`)) z ` |v| ` X v v Im(Γ(w)) Re(Bw (`)) + Im(Bw (`)) z ` = Re(Γ(w) + |v| |v| ` X v = Γ(q) Im(Bw (`)) z ` , Re(Bw (`)) + |v| `

where we used (3.5). Let i 1 X 1 Xh Z= Re(Bw (`))z ` = Bw (`)z ` + Bw (`)z ` = (A + B) 2 2 `

`

and, for i ∈ {1, 2, 3}, " # X 1 vi X 1 vi vi X ` ` ` Im((Bw (`))z = Bw (`)z − Bw (`)z = (A − B) Vi = |v| 2i |v| 2i |v| `

so that

P `

3 X

Bq (`)z ` = Z +

`

3 P

`

Vi ei . Then

i=1

1 1 Z2 + Vi2 = (A + B)2 − (A − B)2 = AB = 4 4 i=1

! X `

Bw (`)z `

! X `

Bw (`)z `

.


11

However for w ∈ C with Pochhammer symbol (w)j we have (w)j = (w)j and if t is a real number, tw = tw . Therefore, Bw (t) =

1 X (−1)k 1 X (−1)k = (w)j (t − k)w (w)j (t − k)w + = Bw (t), + Γ(w) k! Γ(w) k! k

k

so that 2

Z +

3 X

! Vi2

=

i=1

Thus,

P

X

Bw (`)z

! X

`

`

Bw (`)z

`

.

`

Bq (`)z ` is invertible if and only if

P

`

Bw (`)z l

P

`

Bw (`)z `

6= 0. In

`

this case the inverse is given by

X `

Bq (`)z `

−1

Bq (`)z ` ˜ ` = P P ` ` Bw (`)z Bw (`)z P

`

`

Ph

i v Re Bw (`) − |v| Im Bw (`) z ` . = ` P P ` ` Bw (`)z Bw (`)z `

By the Poisson summation formula, we have

`

P

Bw (`)z ` =

`

Pb Bw (z + 2π`). The `

latter term was shown in [5] to be equal to ζ(w, α) + e−iπw ζ(w, 1 − α),

(6.6)

where we take the principal value of the multi-valued function e−iπ( · ) and where ξ ζ(w, α) denotes the Hurwitz zeta function with parameter 2π =: α ∈ (0, 1). In [5], regions R for w = a + i|v|, a > 2, were constructed for which (6.6) is nonzero. In particular, R = R and R is either a rectangular region centered at points (2n, 0) ∈ C, n ∈ N, of width less than one and strictly positive height, or an annular-shaped region centered at points (2n, 0) ∈ C, n ∈ N, of width less than one and angular extend 0 6= ϑ ∈ (−π, π). Let QR := {q = a + v ∈ HR : a + i|v| ∈ R}. Then, since R = R, the denominator of (6.5) is nonzero provided q ∈ QR . Applying the Poisson summation formula to the denominator of (6.4) yields b b q (ξ) = P Bq (ξ) L , bq (ξ + 2πk) B

ξ ∈ R.

k∈Z

b q (0) = 1 and L b q (π) = 0. As the expression in the denominator is We note that L 2π-periodic, it suffices to assume that 0 M

1 |ξ + 2πk|a

∞ 2|1 − e | X −a 2 k ≤ a a (2π) π (a − 1)M a−1 −iξ a

≤

k=M

and we conclude that 1 1 1−a − ). Fq (ξ) F M (ξ) = O(M q

(6.13)

The coefficients cqk , obtained via (6.12) are to be estimated from the discrete Fourier transform N −1 2π X e−2πijk/N cqN,M,k = N j=0 FqM (2πj/N ) for N sufficiently large. We then have Z 2π −ikξ N −1 2π X e−2πijk/N e q q |ck − cN,M,k | = dξ − Fq (ξ) N j=0 FqM (2πj/N ) 0 Z ≤

0

2π

N −1 e−ikξ 2π X e−2πijk/N dξ − Fq (ξ) N j=0 Fq (2πj/N )

−1 −2πijk/N N −1 2π NX e 2π X e−2πijk/N =: A + B. − + N j=0 Fq (2πj/N ) N j=0 FqM (2πj/N )

(6.14)

Quantity B on the right hand side of (6.14) is easily handled with an application of (6.13), which gives B = O(M 1−a ). To estimate A, we employ the following simplification of a result due to Epstein [2]: Theorem 4. Suppose f is a 2π-periodic function on the real line with ` continuous 2π NP −1 R derivatives (` ≥ 1), fb(k) = f (ξ)e−ikξ dξ and fbN,k = 2π f (2πj/N )e−2πijk/N N 0

j=0

is the N -point Riemann sum approximation of the integral defining fb(k). Then for |k| ≤ N/2, ` 12 2 b b |f (k) − fN,k | ≤ (2π) kf (`) k∞ . N It is easily checked that for quaternionic functions on the line of the form f (ξ) = v fv (ξ) with v 6= 0 a fixed quaternionic vector and fs , fv complex-valued, fs (ξ) + |v|

16


d f 0 (ξ) at all ξ for which f (ξ) 6= 0. Applying Theorem 4 to (f (ξ)−1 ) = − dξ f (ξ)2 the estimate of A in (6.14) gives 12 k(Fq0 )s k∞ + k(Fq0 )v k∞ A ≤ (2π)2 = O(N −1 ). (6.15) N inf ξ |Fq (ξ)|2 we have

Returning now to (6.14), we have |cqk − cqN,M,k | = O(M 1−a + N −1 ).

(6.16)

Of course, Fq is indefinitely differentiable and the N −1 on the right hand side of (6.16) may be replaced by higher powers N −` at the expense of possibly larger multiplicative constants. It remains then to compute (numerically) estimates for the multiplicative conk(Fq0 )s k∞ + k(Fq0 )v k∞ . By direct calculation, we have stant Sq = inf ξ |Fq (ξ)|2  −iq/2, if ξ ∈ Z; Fq0 (ξ) = iq  [Fq+1 (ξ) − ie−iξ Fq (ξ)], if ξ ∈ / Z, 1 − e−iξ 1 1 e − 1 e + 1 e and q2 = 2.5+ 4√ e +1e − and this is used (with q1 = 6.2+ 2√ 2 1 4 2 4 3 2 1 8 2 in Figures 4 and 5 below.

√

13 8 e3 )

Figure 4. Absolute values of the scalar parts (dashed), vector parts (dotted) of the functions Fq1 (left) and Fq2 (right). The solid lines are graphs of |Fq1 | (left) and |Fq2 | (right). These calculations provide numerical estimates of the constants of (6.15). We find min |Fq1 | ≈ 0.1568, min |Fq2 | ≈ 0.7799, k(Fq1 )0s k∞ + k(Fq1 )0v k∞ ≈ 3.7889, k(Fq2 )0s k∞ + k(Fq2 )0v k∞ ≈ 2.1753. In these calculations, we have applied Theorem 4 with ` = 1, but of course the use of higher derivatives (to obtain faster decay of errors as N increases) is possible, at the expense of larger multiplicative constants in the estimates. This will not be pursued here, but we note that given the calculations of Fq and Fq0 already obtained,


17

Figure 5. Absolute values of the scalar parts (solid) and vector parts (dashed) of Fq01 (left) and Fq02 (right). the second derivative may be obtained from  q(q + 1)   , − 4 −iξ Fq00 (ξ) = 0 (1 − e )(Fq+1 (ξ) − ie−iξ Fq0 (ξ)) + e−iξ (iFq+1 (ξ) − Fq (ξ))   , iq (1 − e−iξ )2

if ξ = 0; if ξ 6= 0.

7. A Sampling Theorem In this section, we derive a sampling theorem for the fundamental cardinal spline Lq , where q ∈ QR . For this purpose, we employ and generalize the following version of Kramer’s lemma [9] which appears in [6] to the complex-quaternionic setting. Theorem 5 ([6], p. 501). Let ∅ = 6 I ⊆ R and let {φk : k ∈ Z} be an orthonormal basis of L2 (I). Suppose that {Sk : k ∈ Z} is a sequence of functions Sk : Ω → C on a domain Ω ⊂ R and t := {tk : k ∈ Z} a numerical sequence in Ω satisfying the conditions C1. S k (tl ) = ak δkl , (k, l) ∈ Z × Z, where ak 6= 0; X C2. |Sk (t)|2 < ∞, for each t ∈ Ω. k∈Z

Define a function K : I × Ω → C by K(x, t) :=

X

Sk (t)φk (x),

k∈Z

and a linear integral transform K on L2 (I) by Z (Kf )(t) := f (x)K(x, t) dx. I

Then K is well-defined and injective. Furthermore, if the range of K is denoted by H := g : R → C : g = Kf, f ∈ L2 (I) , then


18

(i) (H, h · , · iH ) is a Hilbert space isometrically isomorphic to L2 (I) (H ∼ = L2 (I)) when endowed with the inner product hF, GiH := hf, giL2 (I) , where F := Kf and G = Kg. (ii) {Sk : k ∈ Z} is an orthonormal basis for H. (iii) Each function f ∈ H can be recovered from its samples on the sequence t via the formula X Sk (t) , f (t) = f (tk ) ak k∈Z

the series converging absolutely and uniformly on subsets of R where kK( · , t)kL2 (I) is bounded. Proof. For the proof and further details, we refer the reader to [6].

To carry out the extension to complex quaternions, let Cn be the complexified Clifford algebra Cn = {zA eA : zA ∈ C, A ⊆ {1, 2, . . . , n}}. Here, if A = {i1 , i2 , . . . , ik } with 1 ≤ i1 < i2 < · · · < ik then eA := ei1 ei2 · · · eik where e2j = −1 and ej ek = −ek ej for j 6= k. We also insist that e∅ := 1. We define an involution on Cn as the C-conjugate linear mapping ∗ : Cn → Cn by X ∗ X zA eA , (7.1) zA eA := A

A

where zA is the complex conjugate of the complex number zA and eA is the Clifford conjugate of the Clifford algebra basis element eA , determined by ej = −ej (1 ≤ P j ≤ n), e0 = e0 , and eA eB = eB eA . The expression denotes the sum over all A

subsets A ⊆ {1, 2, . . . , n}. Note that if z, w ∈ Cn then (zw)∗ = w∗ z ∗ . P P For z = zA eA ∈ Cn , we define |z|2 = |zA |2 . Note that |z|2 = [zz ∗ ]0 = [z ∗ z]0 = A

A

|z ∗ |2 for all z ∈ Cn . Let H be a set on which there are two operations defined: • addition: + : H × H → H; and • scalar multiplication: · : Cn × H → H. The triple (H, +, ·) is called a left module over Cn if (H, +) is an abelian group and, for all λ, µ ∈ Cn and x, y ∈ H, the following conditions are satisfied: 1. λ · (x + y) = λ · x + λ · y; 2. (λ + µ) · x = λ · x + µ · x; 3. (λµ) · x = λ · (µ · x); 4. 1 · x = x. In case (H, +, ·) is a left module over Cn and h · , · i : H × H → Cn a mapping, we say (H, +, ·, h · , · i) is a left Hilbert module over Cn if, for all x, y, z ∈ H and λ, µ ∈ Cn , the following requirements hold: 5. hx + y, zi = hx, yi + hy, zi; 6. hx, y + zi = hx, yi + hx, zi; 7. hλx, µyi = λhx, yiµ∗ ;


19

8. hy, xi = hy, xi∗ ; 9. hf, f i = 0 ⇐⇒ f = 0; 10. kf k := [hf, f i]0 defines a pseudo-norm on H in the sense that (a) kf k ≥ 0 and (kf k = 0 ⇐⇒ f = 0); (b) kf + gk ≤ kf k + kgk; (c) kλf k ≤ C |λ| kf k, for some constant C > 0. Now suppose that X is a measure space with positive measure dx and ( ) X X L2 (X, Cn ) := f = fA eA : fA : X → C measurable and kf k22 = kfA k22 < ∞ . A

A

2

2

A pairing h · , · i : L (X, Cn ) × L (X, Cn ) → Cn is given by Z hf, gi := f (x)g(x)∗ dx. X 2

With this definition, L (X, Cn ) becomes a left Hilbert module over Cn . Note that if f ∈ L2 (X, Cn ), then Z Z |f (x)|2 dx. f (x)f (x)∗ dx = 0

X

Let H be a left Hilbert module over Cn . A mapping T : H → Cn is called a bounded left linear functional if, for all f, g ∈ H and λ ∈ Cn , (i) T (f + g) = T f + T g; (ii) T (λf ) = λ(T f ); (iii) there exists a constant M > 0 such that |T f | ≤ M kf k. We call kT k := inf{M > 0 : property (iii) is satisfied} the norm of the functional T. The next theorem is a generalization of the Riesz Representation Theorem to our particular complex-quaternionic setting. Theorem 6 (Riesz Representation Theorem). Let T be a bounded left linear functional on a left Hilbert module H over Cn for which there exists an orthonormal ∞ basis {ϕn }∞ n=1 , i.e., {ϕn }n=1 ⊂ H and for all f ∈ H, we have (i) hϕ n ∈ N; Pn , ϕm i = δ2nm , ∀m, (ii) |hf, ϕn i| = kf k2 ; n∈N

(iii) f =

∞ P

hf, ϕn iϕn .

n=1

Then there exists a unique g ∈ H such that T f = hf, gi, for all f ∈ H. Proof. Let {ϕn }∞ n=1 be an orthonormal basis for H. Let aj := T ϕj ∈ Cn , and given n P f ∈ H, let cj := hf, ϕj i ∈ Cn and fn := cj ϕj . Then kf − fn k2 → 0 as n → ∞ j=1

and since T is left linear, T fn =

n P j=1

cj T ϕj =

n P

cj aj . Also, again by the linearity

j=1

and boundedness of T , |T f − T fn | = |T (f − fn )| ≤ kT kkf − fn k → 0, as n → ∞.


20

∞ P

Consequently, we have that T f =

cj aj . Furthermore,

j=1

  n X  cj aj  j=1

0

 1/2 n n X X ≤ cj aj = |T fn | ≤ kT kkfn k2 = kT k  |cj |2  . j=1 j=1

(7.2)

Putting cj = a∗j in (7.2) yields n X

 1/2 n X |aj |2 ≤ kT k  |aj |2  ,

j=1

j=1

from which we obtain the uniform bound

n P

|aj |2

1/2 ≤ kT k.

j=1

Now let g :=

∞ P j=1

hf, gi =

a∗j ϕj ∈ H. Then

X ∞ j=1

cj ϕj ,

∞ X

a∗` ϕ`

=

∞ X ∞ X

cj hϕj , ϕ` ia` =

j=1 `=1

`=1

∞ X

cj aj = T f,

j=1

as required. To establish uniqueness, suppose that h is another such element of H for which T f = hf, hi, for all f ∈ H. Then we have hg − h, g − hi = hg − h, gi − hg − h, hi = T (g − h) − T (g − h) = 0, so that kg − hk2 = [hg − h, g − hi]0 = 0, i.e., g = h.

The theorem below is the extension of Theorem 5 to the current setting. Theorem 7 (Sampling Theorem for Cn -valued Functions). Let T be a measure space with positive measue dt, Sn : T → Cn (n ≥ 1) a sequence of Cn -valued functions, and {tk }∞ k=1 ⊂ T such that (1) Sn (tk ) = δnk , (n, k ≥ 1); ∞ P (2) |Sn (t)|2 ≤ M < ∞, for all t ∈ T . n=1 ∞ ∞ P P 2 Let H := clos`2 span {Sn } = f = an Sn : an ∈ Cn and |an | < ∞ . Then n=1

n=1

for all f ∈ H, f (t) =

∞ X

f (tn )Sn (t).

n=1

Proof. Let (X, dx) be a measure space and let {φn }∞ n=1 be an orthonormal basis for L2 (X, Cn ) in the sense that R (1) φn (x)φm (x)∗ dx = δnm , for all n, m ∈ N; X 2 ∞ R R P ∗ 2 2 (2) F (x)φn (x) dx = |F (x)| dx, for all F ∈ L (X, Cn ); n=1 X X ∞ R P (3) F (x)φn (x)∗ dx φn = F , for all F ∈ L2 (X, Cn ). n=1

X


Let K : X × T → Cn be given by K(x, t) :=

P

21

φn (x)∗ Sn (t) and define an integral

n∈N

operator K by Z KF (t) :=

F ∈ L2 (X, Cn ).

F (x)K(x, t) dx, X

Note that Z

|K(x, t)|2 dx =

X

2 Z X ∗ dx φ (x) S (t) n n X

=

n∈N

Z X X

=

∗ X φn (x)∗ Sn (t) φm (x)Sm (t) dx

n∈N

X

Sn∗ (t)

=

φ∗n (x)φm (x) dx Sm (t)

X

n,m∈N

X

0

m∈N

Z

∗

[Sn (t) Sn (t)]0 =

n∈N

0

X

2

|Sn (t)| = M < ∞.

m∈N

If f = KF , for some F ∈ L2 (X, Cn ), then Z F (x)K(x, t) dx |f (t)| = ZX ≤C |F (x)||K(x, t)| dx X Z Z √ 2 2 ≤C |F (x)| dx |K(x, t)| dx ≤ C M kF kL2 (X) . X

(7.3)

X

Hence, the mapping K : F → f is well-defined. K is also one-to-one since if KF (t) = 0 for all t ∈ T , then Z Z KF (tk ) = F (x)K(x, tk ) dx = F (x)φk (x)∗ dx = 0, for all k ∈ N, X

X

which implies that F ≡ 0. Now, let H := Ran(K) and define a pairing h · , · iH on H by Z hf, giH := F (x)G(x)∗ dx ∈ Cn , X

where F, G ∈ L2 (X, Cn ) are the unique elements for which KF = f and KG = g. We claim that with this pairing H becomes a left Hilbert module over Cn . To prove this, we need to verify that axioms 1–9 are satisfied. First, notice that the addition and scalar multiplication in H are defined in the usual way: linearity of the operator K yields that if KF = f and KG = g, then λ · f = K(λ · F ) and K(F + G) = f + g. Thus, Z Z (λ · f )(t) = (λ · F (x))K(x, t) dx = λ F (x)K(x, t) dx = λ(f (t)) X

X


22

and Z

Z (F + G)(x)K(x, t) dx = [F (x) + G(x)]K(x, t) dx X ZX Z = F (x)K(x, t) dx + G(x)K(x, t) dx = f (t) + g(t).

(f + g)(t) =

X

X

Axioms 1–4 follow immediately. Note that if f, g are as above and h = KH then Z Z Z ∗ ∗ hf + g, hiH = (F + G)(x)H(x) dx = F (x)H(x) dx + G(x)H(x)∗ dx X

X

X

= hf, giH + hg, hiH , and hence axiom 5 is satisfied. Axiom 6 follows similarly. Next, observe that if f, g are as above and λ, µ ∈ Cn , then Z Z ∗ hλf, µgiH = (λF )(x)(µG)(x) dx = λ F (x)G(x)∗ dx µ∗ = λhf, giH µ∗ , X

X

thus verifying axiom 7. For axiom 8, note that Z ∗ Z hg, f iH = G(x)F (x)∗ dx = F (x)G(x)∗ dx = hf, gi∗H . X

X

Suppose now that hf, f iH = 0. Then Z Z 2 ∗ |F (x)| dx = F (x)F (x) dx = [hf, f iH ]0 = 0, X

X

0

so that F ≡ 0. Hence f = KF ≡ 0 and axiom 9 is verified. Axiom 10(a) now follows immediately. For axiom 10(b), note that 1/2

kf + gkH = [hf + g, f + giH ]0 Z 1/2 = (F + G)(x)(F + G)(x)∗ dx X

Z

0

|F (x) + G(x)|2 dx

=

1/2

X

= kF + Gk2 ≤ kF k2 + kGk2 = kf kH + kgkH . Finally, we have kλf k2H = [hλf, λf iH ]0 =

Z

|λF (x)|2 dx ≤ C 2 |λ|2

Z

X

|F (x)|2 dx = C 2 |λ|2 kf k2H ,

X

so that axiom 10(c) is verified. We need to show that H := Ran(K) = clos`2 span {Sn }. Since {φn }∞ n=1 is an ∞ P orthonormal basis for L2 (X, Cn ), for each F ∈ L2 (X, Cn ), we have F = an φn 2 for some sequence {an }∞ n=1 ∈ ` (N). Then KF =

∞ P

n=1

an Sn ∈ clos`2 span {Sn }. This

n=1

gives the equivalence of the definitions of H. We remark that {Sn }∞ basis for H. To see this, note that since n=1 is an orthonormal R Sn = Kφn , we have hSn , Sm iH = φn (x)φm (x)∗ dx = δnm . Also, if hf, Sn iH = 0, X


for all n ∈ N, then Z F (x)φn (x)∗ dx = hf, Sn iH = 0, for all n

F ≡0

=⇒

=⇒

23

f ≡ 0.

X

Moreover, X n∈N

2 Z X Z ∗ |F (x)|2 dx = kf k2H . |hf, Sn iH | = F (x)φn (x) dx = 2

X

n∈N

X

Equation (7.3) may be interpreted as giving the boundedness of the evaluation functional Et : H → Cn defined by Et (f ) := f (t). By the Riesz Representation Theorem we conclude that there exists a kt ∈ H such that f (t) = Et (f ) = hf, kt iH . Let k(t, s) = hkt , ks iH = ks (t). Then hf, k( · , s)iH = hf, ks iH = f (s). Hence, k(t, s) is a reproducing kernel for H. Suppose k 0 is another such kernel. With ks0 (t) = k 0 (s, t), we have that ks0 (t) = hks0 , kt iH = hkt , ks0 i∗H = kt (s)∗ = hkt , ks i∗ = hks , kt iH = ks (t), i.e, k 0 (s, t) = k(s, t) for all s, t ∈ T . As {Sn }∞ n=1 is an orthonormal basis for H, we obtain X X X kt = hkt , Sn iH Sn = hSn , kt i∗H Sn = Sn (t)∗ Sn , n

n

n

so that k(t, s) = hks , kt iH =

X

X

Sn (s)∗ Sn ,

=

m∈N

H

Sn (s) hSn , Sm iH Sm (t) =

X

n∈N

X

Sm (t)∗ Sm

∗

n,m∈N

Note also that Z Z K(x, s)∗ K(x, t) dx = X

n∈N

!∗ X

X

=

X X

∗

φn (x) Sn (s)

n∈N

n,m∈N

=

Sn (s)∗ Sn (t).

Sn (s)∗

! X

∗

φm (x) Sm (t)

dx

m∈N

Z

φn (x)φm (x)∗ dx Sm (t)

X

Sn (s)∗ Sn (t) = k(t, s).

n∈N ∗

Finally, as K(x, tn ) = φn (x) , we obtain Z Z hf, Sn iH = F (x)φn (x)∗ dx = F (x)K(x, tn ) dx = f (tn ), X

(7.4)

X

and therefore, since {Sn }∞ n=1 is an orthonormal basis for H, (7.4) yields X X f= hf, Sn iH Sn = f (tn )Sn . n∈N

n∈N

Remark 1. The above results continue to hold when the index set N is replaced by any countably infinite set.

24


For our purposes, we choose T := R, {tk }k∈Z := Z, and for the interpolating function Sk := Lq ( · − k), q ∈ QR . Then Theorem 7 implies the next result. Theorem 8. Let H := clos`2 span {Lq ( · − k)} where q ∈ QR . Then, for all f ∈ H, X f (t) = f (k)Lq (t − k). k∈Z

Proof. Conditions (1) and (2) for Sk = Lq ( · − k), k ∈ Z, in Theorem 7 hold by assumption on Lq and the estimate (6.11). References [1] C. K. Chui, An Introduction to Wavelets, Academic Press, 1992. [2] C.L. Epstein, How well does the finite Fourier transform approximate the Fourier transform?, Comm. Pure. Appl. Math. 58 (2005), 1421–1435. [3] B. Forster, Th. Blu and M. Unser, Complex B-Splines, Appl. Comput. Harm. Anal. 20 (2006), 261–282. [4] B. Forster and P. Massopust, Interpolation with fundamental splines of fractional order, Proceedings of SampTA 2011, 1–4. [5] B. Forster, R. Garunkstis, P. Massopust, and J. Steuding, Complex B-Splines and Hurwitz Zeta Functions, London Math. Soc. Journal of Computation and Mathematics 16 (2013), 61–77. [6] A. G. Garcia, Orthogonal sampling formulas: A unified approach, SIAM Review 42(3), 2000, 499–512. [7] J. A. Hogan and P. R. Massopust, Quaternionic B-Splines, to appear in J. Approx. Th. [8] J A. Hogan and A. J. Morris, Quaternionic wavelets, Numer. Funct. Anal. Optim., 33(7-9), 2012, 1095–1111. [9] H. P. Kramer, A generalized sampling theorem. J. Math. Phys. 63 1957, 68–72. [10] J. P. Morais, S. Georgiev and W. Spr¨ oßig, Real Quaternionic Calculus Handbook, Birkh¨ auser, 2014. [11] S. Olhede and G. Metikas, The Hyperanalytic Wavelet Transform, Statistics Section Technical Report TR-06-02, Imperial College London, UK (2006), 1–49. [12] S. Olhede and G. Metikas, The Monogenic Wavelet Transform, IEEE Trans. Signal Proc. 57(9) (2009), 3426–3441. [13] I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Th. 2 1969, 167–206. [14] I. J. Schoenberg, Cardinal interpolation and spline functions: II Interpolation of Data of Power Growth, J. Approx. Th. 6 1972, 404–420. [15] I. J. Schoenberg, Cardinal Spline Interpolation, CBMS-NSF, Vol. 12, SIAM, 1973. [16] R. Spira, Zeros of Hurwitz zeta functions, Mathematics of Computation 30(136) (1976), 863–866. [17] M. Unser and Th. Blu, Fractional Splines and Wavelets, SIAM Review 42(1), 2000, 43–67. School of Mathematical and Physical Sciences, Mathematics Bldg V123, University of Newcastle, University Drive, Callaghan NSW 2308, Australia E-mail address: [email protected] Centre of Mathematics, Research Unit M15, Technical University of Munich, Boltzmannstr. 3, 85748 Garching b. Munich, Germany E-mail address: [email protected]