Discrete weierstrass transform in discrete hermitian clifford analysis

Discrete Weierstrass transform in discrete Hermitian Clifford analysis

A. Mass´e^a,∗, F. Sommen^a, H. De Ridder^a, T. Raeymaekers^a

aClifford Research Group, Department of Electronics and Information Systems, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium

Abstract

The classical Weierstrass transform is an isometric operator mapping elements of the weightedL2−spaceL2(R,exp(−x²/2)) to the Fock space. It has numere- ous applications in physics and applied mathematics. In this paper, we define an analogue version of this transform in discrete Hermitian Clifford analysis, where functions are defined on a grid rather than the continuous space. This new transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Her- mite polynomials. Furthermore, a discrete Weierstrass space with appropriate inner product is constructed, for which the discrete Hermite polynomials form a basis. In this setting, we also investigate the behaviour of the discrete delta functions and check if they are elements of this newly defined Weierstrass space.

Keywords: Discrete Clifford analysis, Weierstrass transform, Hermite polynomials, Weierstrass space, Delta functions

1. Introduction

The Weierstrass transform (or Gauss transform, Gauss-Weierstrass transform, [1]), named after Karl Weierstrass is one of the many convolution transforms in functional analysis. It averages the values of a functionf :R→R, by making the convolution with a Gaussian function to obtain a ’smoothed’ version of f. Evaluating the smoothed function at the point x, points in the original function close toxare given a higher weight, while points that are further away from xget lower weights. It can be applied to solve the heat equation in one dimension: the Weierstrass transformW[f] gives the temperature distribution

∗Corresponding author

Email addresses: [email protected](A. Mass´e),[email protected](F.

Sommen),[email protected](H. De Ridder),[email protected](T.

Raeymaekers)

of an infinitely long rod, one time unit later than the initial temperature given by the function f. In image processing, the transform is also known as the Gaussian blur, which helps to reduce the level of noise in the image. In signal processing, it is a simple kind of low-pass filter, whereas in statistics, it is used as a kernel density estimation. The Weierstrass transform is clearly related to various Fourier type transforms and it is a major part of (discrete) transform analysis in general. Applications are thus widely around.

This article is to be situated in the context of discrete Hermitian Clifford analy- sis, where a discrete version of the Weierstrass transform is constructed. Clifford analysis is a generalisation of classical analysis, in which null solutions of the Dirac operator play the key role. On the one hand, it can be viewed as a higher-dimensional theory of holomorphic functions in the complex plane. On the other hand, as the Dirac operator factorises the Laplacian, it is also a refine- ment of classical harmonic analysis. For further information about continuous Clifford analysis, we refer to [2]. Discrete Clifford analysis is a recent branch in functional analysis, which originated as an appropriate framework for the development of discrete counterparts of the function theory in classical Clifford analysis. Both numerical problems and theoretical results have been studied (see for example [3, 4, 5, 6, 7, 8]). We will continue working in the discrete Clifford algebra model described in for example [9, 10, 11], where both forward and backward differences are introduced and combined with forward and back- ward basis elements. It should be noted that other discrete counterparts for the Weierstrass transform have been studied, see e.g. [12] and [13], which is used to approximate discrete data sets. Whereas these transforms are scalar valued, we use Clifford valued functions and the Clifford framework, where we can use the tools in this theory such as the factorisation of the Laplace operator. While, in this paper, we will restrict ourselves to the one dimensional case, discrete Clifford analysis is a theory that is well understood in higher dimensions. This will allow us to develop a higher dimensional Weierstrass transform in future work.

Together with a discrete Weierstrass transform, a discrete Weierstrass space will be constructed. The main idea is to use the discrete Gauss distribution Gas a weight function and the discrete Hermite polynomials as basis elements of the space. We ask ourselves if the delta-functions, the building blocks of discrete function theory, are elements of this space. For the discrete Weierstrass transform of a discrete functionf ∈ W, we use the discrete Gauss distribution Gtogether with the convolution kernel exp −z²/2 +ξz

, resulting in a direct analogue to the continuous Weierstrass transform.

This paper is organised as follows. In Section 2, we set up the preliminaries and describe the discrete Clifford setting in more detail. In Section 3, we define the discrete Weierstrass transform and continue to the definition of the discrete Weierstrass space in Section 4. In Section 5, we investigate the building blocks of discrete function theory, theδ−functions, and ask whether they are elements of the discrete Weierstrass space. In this paper, we limit ourselves to the one-

dimensional case.

2. Preliminaries

2.1. Clifford analysis

Clifford analysis - basis elements. Consider them-dimensional real vector space R^mspanned by the orthonormal basis{e1, . . . , em}, endowed with a non-degenerate quadratic form with signature (p, q), p+q = m. The algebra generated by ej, j= 1, . . . , m, satisfies the relations

e²_j = 1, j = 1, . . . p, e²_j =−1, j =p+ 1, . . . , m, e_je_k+e_ke_j= 0, j 6=k, j, k= 1, . . . , m.

(2.1)

The Clifford algebra thus constructed is denoted by Rp,q. Denote for the set A = {j1, . . . , jr} ⊆ {1, . . . , m} the element eA = ej₁ej₂, . . . ej_r, for 1 ≤ j1 <

j2< . . . < jr≤m, withe_∅= 1. The complexification of this real algebra is the compositionCp,q :=C⊗Rp,q which has dimension 2^m over C. In this article, we will consider the complex algebraC0,m: every basis element ej squares up to−1.

Clifford analysis - discrete setting. We will work in the so-called split discrete setting, introduced in [4]. Split up the basis elementse_jofCm,0into forward and backward basic vectorse⁺_j ande⁻_j, such thate⁺_j +e⁻_j =ej, j= 1, . . . , m, there are no preferential Cartesian directions (rotational invariance) and moreover, the positive and negative orientations of any cartesian direction play an equivalent role (reflection invariance). We consider the algebra over the set{e⁺_j,e⁻_j |j = 1, .., m}, satisfying the following relations:

{e⁻_j,e⁻_k}:=e⁻_je⁻_k +e⁻_ke⁻_j = 0, j, k∈ {1, . . . , m}, {e⁺_j,e⁺_k}:=e⁺_je⁺_k +e⁺_ke⁺_j = 0, j, k∈ {1, . . . , m}, {e⁺_j,e⁻_k}:=e⁺_je⁻_k +e⁻_ke⁺_j =δ_jk, j, k∈ {1, . . . , m}.

(2.2)

Remark that these relations are consistent with the relations mentioned in (2.1).

The complex fieldCcan be identified with the subalgebra of scalars. The basis elements{e⁺_j,e⁻_j |j= 1, . . . , m} form a basis of the space of 1-vectors. Let the wedge product (also called exterior product) be defined as follows

e^±_j ∧e^±_k :=e^±_je^±_k −e^±_ke^±_j. (2.3) Then the elements in {e^±_j ∧e^±_k |j, k = 1, . . . m} form a basis for the space of 2-vectors or bivectors. In general,{e^±_j

1∧e^±_j

2∧. . .∧e^±_j

r |j1, . . . , jr∈ {1, . . . , m}

form the basis for ther-vectors.

Remark 2.1. We started from the classical Clifford algebraCm,0 with dimen- sion2^m and split up the basis elements in a forward and backward vector. Be- cause of that, the dimension raises to(2^m)²= 2^2m. This raising can be seen as a lift from Cm,0 toCm,m: one can compare the sum e⁺_j +e⁻_j to a continuous basis elementej with e²_j = 1, while the basis element e⁺_j −e⁻_j behaves like ej

withe²_j =−1.

Definition 2.2. A main type of involution is theHermitian conjugation^†: (e⁺_j)^†:=e⁻_j and(e⁻_j)^†:=e⁺_j.

Hermitian conjugation switches the order of multiplication:

(ab)^† :=b^†a^†,

with a, b∈ Cm,m. For any r ∈ C, (ra)^† =ra^†, where r is the usual complex conjugation.

Remark 2.3. Hermitian conjugation is the discretization of hermitean conju- gation inCm,m and a generalisation of complex conjugation inC. It holds that (e⁺_j +e⁻_j)^†=e⁺_j +e⁻_j and(e⁺_j −e⁻_j)^†=e⁻_j −e⁺_j.

We will from now on work with functions defined on the equidistant gridZ^m, with values in the Clifford AlgebraC^m,0. Classical partial derivatives are re- placed by the so-calledforward and backward differencesforj= 1, . . . , m, by analogy with the classical left and right limit of differentiation:

∆⁺_jf(x) :=f(x+ej)−f(x),

∆⁻_j f(x) :=f(x)−f(x−ej), where nowx∈Z^m

Definition 2.4. The discrete Dirac operatoris defined as

∂:=

m

X

j=1

e⁺_j∆⁺_j +e⁻_j∆⁻_j.

This Dirac operator factorises thestar Laplacian

∆^∗:=

m

X

j=1

∆⁺_j∆⁻_j =

m

X

j=1

∆⁻_j∆⁺_j =∂²,

analogeous to classical Clifford analysis where the continuous Dirac operator also factorises the Laplacian ∆x. On the coordinate level, we will denote ∂j = e⁺_j∆⁺_j +e⁻_j∆⁻_j, such that ∂=Pm

j=1∂j.

The Dirac operator and the forward and backward differences are lowering op- erators: they lower the degree of a polynomial by 1. In order to create corre- sponding raising operators, which we will denote by

ξ=

m

X

j=1

ξ_j =

m

X

j=1

e⁺_jX_j⁻+e⁻_jX_j⁺, we use theskew Weyl-relations

∆⁺_jX_j⁺−X_j⁻∆⁻_j = 1,

∆⁻_jX_j⁻−X_j⁺∆⁺_j = 1,

∂_jξ_j−ξ_j∂_j= 1.

(2.4)

Moreover, we define thediscrete Euler operatorasE=Pm

j=1ξj∂j, for which the same intertwining relations as in the continuous setting hold:

∂ξ+ξ∂= 2E+m, (2.5)

∂E=E∂+∂, (2.6)

Eξ=ξE+ξ. (2.7)

From the relations (2.4) and (2.5), we find how the raising operators X_j^± and henceξj act on polynomials. In particular,ξ^k_j[1], which are thebasic discrete polynomialsof degreek >1, are given by

ξ^2k+1_j [1] =xj k

Y

s=1

x²_j −s² e⁺_j +e⁻_j ,

ξ^2k_j [1] =

x²_j+k x_j

e⁺_je⁻_j −e⁻_je⁺_j^k−1 Y

s=1

x²_j−s² .

(2.8)

The roots ofξ_j^2k+1[1] are given by−k, . . . , k, while those ofξ_j^2k[1] are given by

−(k−1), . . . , k−1.

Equations (2.8) show the action ofξj from the left on discrete polynomials. It will be necessary to translate the right action to a left action, when we will work with discrete distributions. Note that, asξj are vectorial operators and since the (discrete) Clifford algebra is non commutative, the position of the basis elementse^±_j on the left or on the right of the functionξacts upon, can play an important role. It has been proven in [14], lemma 3.5.1 that:

(ξ_j^k[1])∂_j^†=∂j(ξ_j^k[1]), (2.9) (ξ^k_j[1])ξ_j^†=ξ_j^k+1[1]. (2.10)

Definition 2.5. The discrete Euler operatoris given by

E:=

m

X

j=1

e⁺_je⁻_j X_j⁻∆⁻_j +e⁻_je⁺_jX_j⁺∆⁺_j

. (2.11)

A polynomialPk is calleddiscrete homogeneous of degreek if it is an eigen- function of the discrete Euler operator with eigenvaluek:

EPk=kPk.

The basic discrete polynomialsξ^k_j[1] are homogeneous of degree k. Moreover, they span the subvectorspace of discrete homogeneous polynomials of degree k. From (2.8), it is also clear that the discrete notion of homogenity does not coincide with the continuous definition. For example, a restriction of a continu- ous homogeneous polynomial to the grid does not give a discrete homogeneous polynomial in the discrete sense.

Every discrete function admits a unique discrete Taylor series expansion, which is a finite or infinite series of the basic homogeneous polynomials. In one dimension (m= 1), this Taylor series (see e.g. [15]) is given by

f(x) =

∞

X

k=0

1

k!ξ^k[1](x)[∂^kf(u)]u=0 (2.12) wheref is a discrete function. The Taylor series expansionf(ξ) = P

k∈Nξ^kc_k is the corresponding operator to this function. Letting this operator act on the identity function 1, one obtains a function in the discrete variable x. By identifying a discrete function with its corresponding operator, the set of discrete functions is a right Clifford module.

Similar to classical analysis, distributions are a class of linear functionals acting on a particular space of functions.

Definition 2.6. A discrete distribution F is defined as a linear functional acting on the left module of Clifford polynomials onZ^m.

hF, Vi:=D

F,[1]V(ξ^†)E .

Here,V(ξ^†) is the polynomialV where everyξis replaced by its conjugate ξ^†. The action of the operators ∆^±_j and X_j^± on the distribution F are defined in such a way that they are consistent with the skew Weyl relations:

D∆^±_jF, VE

=−D

F, V∆^∓_j E

=−D

F,∆^∓_j[V]E ,

D

X_j^±F, VE

=D

F, V X_j^∓E

=D

F, X_j^∓[V]E .

For Clifford numbera, b, we define

hF a, bVi:=bhF, Via.

This implies in particular for the Dirac operator∂ and the vector variableξ:

h∂F, Vi=−D

F, V ∂^†E ,

hξF, Vi=D

F, V ξ^†E .

(2.13)

A special and important class of distributions are the discrete delta distributions δj, which evaluate a discrete function in the pointp∈Z^m:

δp, f

=f(p).

2.2. Weierstrass transform and Hermite polynomials

Of particular interest in classical analysis areHermite polynomialsH_n and their correspondingfunctionsψ_n.

Definition 2.7. The continuous Hermite functionsψk are defined as

ψk₁,...,k_m(x) :=Hk₁,...,k_m(x)e^−|x|2/4 (2.14) where

H_k1,...,km(x)e^−|x|2/2:= (−1)^k1+...+km∂_x^k1

1. . . ∂_x^km

me^−|x|2/2.

Hk₁,...,k_m(x) are the corresponding monic Hermite polynomials of degree k in R^m.

We used the notationx= (x1, . . . , xm)∈R^m. In one dimension, this definition reduces to

ψ_k(x) = 1

2x− d dx

^k

e^−x2/4=H_k(x)e^−x2/4, (2.15) with

H_k(x) = (−1)^ke^x

2 2 d^k

dx^k

e^−x

2 2

.

Definition 2.8. The Weierstrass space W, is the weighted L₂−space with Gaussian weight functionG:

L2(R^m, G) :=

f :R^m→C| Z

R^m

f(x)

2 e^−|x|2/2dx <∞

with inner product

hf, giW := 1

√2π^m Z

R^m

f(x)g(x)e^−|x|2/2dx.

The set of polynomials is dense inWand the Hermite polynomials{Hn|n∈N} form an orthogonal basis for m = 1. It holds that hHn, HmiW = δm,nn!. If m >1, higher dimensional Hermite polynomialsHk₁,...,k_m form an orthogonal basis withkHk1,...,kmk²=

Hk1,...,km, Hk1,...,km

=k1!. . . . .km!.

Definition 2.9. The Weierstrass transformof an integrable functionf on Ris defined as the convolution off with the Gaussian kernel:

W[f](u) := 1

√2π^m Z

R^m

e^{−|u−x|2/2}f(x)dx.

The Weierstrass transform is a bounded operator on L2(R^m), with values in the Fischer space, the space of real analytic functions equipped with the inner product

hf, gi=f(∂₁, . . . , ∂_m)g(x₁, . . . , x_m)

_x=0. (2.16)

A basis of this Fischer space is given by the natural powers ofu: {u^k |k∈N}, satisfying

u^k, u^l

=δklk!, and therefore, constituting an orthonormal system.

This definition can be naturally extended with z ∈ C so that the result is a holomorphic function, i.e. an element of the Fock space. Whether the result of the Weierstrass transform is a real or complex function (in the Fischer space, resp. Fock space), does of course not influence its isometric property.

As an informative example, let us calculate the Weierstrass transform of the Hermite polynomials in one dimension:

W[H_n](u) = 1

√2π Z

R

e^{−(u−x)2/2}(−1)ⁿe^x2/2 dⁿ

dxⁿ(e^−x2/2)dx

= (−1)ⁿ 1

√ 2π

Z

R

e^−u2/2+xu dⁿ dxⁿ

e^−x2/2 dx

= 1

√2π Z

R

dⁿ dxⁿ

e^−u2/2+xu

e^−x2/2dx

= uⁿ

√2π Z

R

e^{−(u−x)2/2}dx=uⁿ.

(2.17)

This easy calculation shows that the Weierstrass transform of Hn equals uⁿ. This proves that we deal with an isometry between the Weierstrass space and the so-called Fischer space: the space of real analytic functions equipped with the inner product (2.16).

2.3. Discrete Gauss distribution and Hermite functions

From now on, we will restrict our work to the one-dimensional case, for ease of calculation and notation. Yet our methods may be generalized to higher

dimensional cases which would be a topic for future research. Higher dimensions involve more complicated calculations, mainly because of the non-commutative character of elements in the Clifford algebra.

The continuous Gauss distribution with its density function f(x) = e^−x2/2 is unavoidable in classical analysis. As it will be one of the main ingredients to define the discrete Weierstrass space, we give the definitions of the discrete analogue.

Thediscrete Gauss distribution Gis defined through its action on the dis- crete homogeneous polynomials, similar to the classical Gaussian distribution acting on the classical homogeneous polynomialsxⁿ (see [6]):

G, ξⁿ[1]

:=











√2π n= 0,

0 nodd,

(n−1)!!√

2π neven,

(2.18)

with (2l−1)!! = ^(2l)!_2ll! the double factorial. The dual Taylor series representation ofGis given by

G:=√

2πexp ∂² 2

! δ₀ and its density function is

X

p∈Z

√

2π e⁻¹BesselI(p,1)δ_p,

where BesselI(p,1) is the modified Bessel function of the first kind andδp are the delta functions, which evaluate to 1 inp∈ Zand 0 everywhere else. This can be obtained by expanding∂^2lδ0into discrete delta distributions.

LetP(ξ) be a discrete polynomial in the discrete vector variableξ. The distribu- tionP(ξ)Gis well defined and can be calculated using the skew Weyl relations for distributions. Alternatively, the useful calculation rule

∂G=−ξG (2.19)

enables us to writeP(ξ)G as a sum of derivatives of the δ0 distribution. For a given polynomialP, defined on the grid, there exists a unique operator P(ξ) such thatP(ξ)[1](n) =P(n),∀n∈Z, hence there exists a unique distribution of the formP(ξ)G.

Discrete Hermite polynomials were introduced in [16] through the action of Hermite operatorH_n(ξ) on the constant function 1, i.e. H_n(ξ)[1]. H_n(ξ) is, as an operator, a polynomial of degreeninξ with real coefficients.

The Hermite operators are given by H2l(ξ) =

l

X

j=0

a^2l_2jξ^2j, H2l+1(ξ) =

l

X

j=0

a^2l+1_2j+1ξ^2j+1, (2.20)

where

a^2l_2j = (−1)^j2^l−j l

j

Γ(l+¹₂)

Γ(j+¹₂), a^2l+1_2j+1= (−1)^j2^l−j l

j

Γ(l+¹₂+ 1) Γ(j+¹₂+ 1).

(2.21) The absolute values of the coefficientsa^2l_2j and a^2l+1_2j+1 are equal to those of the coefficients of the continuous Hermite polynomials.

Even Hermite polynomials and operators have a scalar and a bivectorial part, while odd Hermite functions only have a vectorial part, as they only contain even, respectively odd powers ofξ.

A useful tool for calculations isRodrigues’ Formula ([14], p. 9-4):

H2k,m(ξ)G= (−1)^k∂^2kG, H2k+1,m(ξ)G= (−1)^k+1∂^2k+1G, (2.22) which is an immediate consequence of equation (2.19)

Hermite functions are analogously defined as the product of a Hermite polyno- mial withe^−ξ2/4:

ψn(ξ) :=Hn(ξ)e^−ξ2/4[1].

Note that this is the action of the polynomial operatorHn(ξ) one^−ξ2/4[1] and not the multiplication of the functionHn(ξ)[1] withe^−ξ2/4[1].

3. Discrete Weierstrass transform

3.1. First considerations

We want a discrete version of the Weierstrass transform, sending a discrete function (i.e. defined on the grid) to an analytic function in a Clifford valued Fock space. Therefore, we try to mimic the ideas of the classical definition.

A basis for the continuous Weierstrass space, is given by the set of Hermite polynomials. In the calculations of W[Hn](z) =zⁿ (see (2.17)), the Gaussian function appears. Both the Hermite polynomials and Gaussian function are introduced in the discrete setting in the previous section. These elements will allow us to translate the Weierstrass transform to the split discrete setting.

Drawing inspiration from Rodriguez’ formula (2.22), we define the discrete Weierstrass transformation of the discrete Hermite polynomials as follows:

Definition 3.1. The discrete Weierstrass transformation of the discrete Her- mite polynomials are defined as

W[H_n](z) := 1

√2πhH_n(ξ)G, e^{−z2/2+ξ z}[1]i. (3.1)

Proposition 3.2. W[Hn](z) = (−1)bⁿ₂czⁿ,∀n∈N Proof. This is a straight forward calculation.

hHn(ξ)G, e^{−z2/2+ξ z}[1]i=h(−1)dⁿ2e∂ⁿG, e^{−z2/2+ξ z}[1]i

= (−1)dⁿ2ee^−z2/2h∂ⁿG,

∞

X

i=0

ξⁱzⁱ[1]

i! i

= (−1)bⁿ2ce^−z2/2

∞

X

i=0

zⁱ

i!hG, ξⁱ[1](∂^†)ⁿi

= (−1)bⁿ₂ce^−z2/2

∞

X

i=0

zⁱ

i!hG, ∂ⁿξⁱ[1]i

= (−1)bⁿ₂ce^−z2/2

∞

X

i=n

zⁱ i!

i!

(i−n)!hG, ξⁱ⁻ⁿ[1]i

= (−1)bⁿ₂ce^−z2/2

∞

X

j=0

z^j+n

j! hG, ξ^j[1]i

= (−1)bⁿ2ce^−z2/2zⁿ

∞

X

j=0

z^2j (2j)!

√ 2π(2j)!

2^jj!

= (−1)bⁿ2c√ 2π zⁿ. This proves the proposition.

These transformations correspond to the continuous case, taking into account that the discrete Hermite polynomials are formally equal to their continuous counterparts, up to signs.

This definition can be naturally extended to any finite linear combination of Hermite polynomials:

f =X

n∈N

H_nc_n ⇒ W[f](z) =X

n∈N

W[H_n](z)c_n, c_n∈Cm,m.

In the next section, we want to extend this definition for general discrete func- tions and find a condition which has to be fulfilled in order that the function has a Weierstrass transform.

3.2. Discrete Weierstrass space

The classical Weierstrass transform acts on the L₂(R^m, G)-space with results in the Fock space. In the previous section, we established a discrete version

of this transform, only for any finite linear combination of discrete Hermite polynomials. Can we find an appropriate space of functions, similar to the classical weighted L2−space, such that the discrete Weierstrass transform is extendable to this space?

The elementary principles of the continuous setting will lead to the definition of a discrete Weierstrass spaceW. The set of discrete polynomials should be a dense subset ofW and a basis will be formed by the Hermite polynomials. The inner product will be defined using the discrete Gauss distribution.

The set of discrete functions forms a right Clifford module in which the right submodule of polynomials is dense, by writing a discrete functionf in its Taylor seriesf(ξ)[1], withf(ξ) =P

k∈Nξ^kck the corresponding operator.

Remark that the set of discrete Hermite polynomials in ξ of degree less or equal to n, as its classical counterpart, is a set of n+ 1 linear independent polynomials. Hence, the space of discrete polynomials of degree≤nis spanned by these Hermite polynomials, i.e. every discrete polynomial inξcan be written as a finite linear combination of Hermite polynomials.

Define a sesquilinear form as follows:

(f, g) = f(ξ)[1], g(ξ)[1]

:=D

f(ξ)G,[1](g(ξ))^†E

(3.2) We will prove that this form defines an inner product on the space spanned by the discrete Hermite polynomials, in which the latter will form an orthogonal basis.

Remark that

f(ξ)[1]†

= [1]X

k∈N

c^†_k(ξ^†)^k=X

k∈N

c^†_kξ^k[1].

In particular, iff is a real function (i.e. everyc_k ∈R) then [1] f(ξ)†

=f(ξ)[1].

Lemma 3.3. The discrete Hermite polynomials are orthogonal with respect to the inner product (3.2)

Proof. We calculate (Hn, Hm). Ifn6=m, it follows from [16] that (Hn, Hm) = 0.

Supposem=n= 2kis even.

(H2k, H2k) =

* _k X

j=0

a^2k_2jξ^2jG,[1](

k

X

i=0

a^2k_2iξ²ⁱ)^† +

=

* _k X

j=0

a^2k_2jξ^2jG,

k

X

i=0

a^2k_2i[1](ξ²ⁱ)^† +

=

k

X

i,j=0

a^2k_2ja^2k_2i D

G, ξ^2i+2j[1]E

=

k

X

i,j=0

a^2k_2ja^2k_2i√

2π (2i+ 2j)!

2^i+j(i+j)!

=√

2π(2k)! =:η2k.

While form=n= 2k+ 1, we have similarly that (H_2k+1, H_2k+1) = X

i,j∈N

a^2k+1_2j+1a^2k+1_2i+1√

2π (2i+ 2j+ 2)!

2^i+j+1(i+j+ 1)!

=√

2π(2k+ 1)! =:η2k+1.

The last step was calculated by filling in the explicit values of the coefficients a^k_j.

In conclusion, forn, m∈N: (Hn, Hm) =√

2π n!δn,m=:ηn.

Remark 3.4. The results are the same as the inner products of Hermite poly- nomials in the continuous setting.

Lemma 3.5. The bilinear form (3.4)is conjugate symmetric with respect to†.

Proof. Let f, g be two discrete Clifford-valued polynomials and write them in their Hermite polynomial expansion, i.e. f =P

k∈NH_ka_k andg =P

l∈NH_lb_l. We than have

(f, g) =D

f(ξ)G,[1](g(ξ))^†E

= X

k,l∈N

Hk(ξ)akG,[1]hHl(ξ)bl†

i

= X

k,l∈N

b^†_lhH_k(ξ)G,[1]H_l(ξ^†)ia_k

=X

n∈N

b^†_nη_na_n,

SinceGis scalar, the second step a_kG=Ga_k is allowed. Also, asη_n∈R, (g, f)^†=X

n∈N

a^†_nηnbn

†

=X

n∈N

b^†_nηnan.

Using the orthogonality of the Hermite polynomials H_k = H_k(ξ)[1], we can proof that the bilinear form in definition 3.2 is not indefinite. As the form is Clifford-valued, positive definiteness does not apply.

Remark 3.6. It is possible to take the scalar part of the right hand side of definion (3.2) in order to obtain a positive definite inner product. In that case however, one obtains a complex Hilbert space, in which the discrete Hermite polynomials are no longer basis elements.

Lemma 3.7. For a discrete functionf, it holds that if(f, f), thenf = 0.

Proof. Let us first calculate r^†r, where r is an arbitrary clifford element r = a+be⁺+ce⁻+d(e⁺∧e⁻), wherea, b, c, dare real or complex numbers.

(a+be⁺+ce⁻+d(e⁺∧e⁻))^†(a+be⁺+ce⁻+d(e⁺∧e⁻))

= (a+be⁻+ce⁺+d(e⁺∧e⁻))(a+be⁺+ce⁻+d(e⁺∧e⁻))

=a²+abe⁺+ace⁻+ad(e⁺∧e⁻) +abe⁻+b²e⁻e⁺+bde⁻(e⁺∧e⁻)+

ace⁺+c²e⁺e⁻+cde⁺(e⁺∧e⁻) +ad(e⁺∧e⁻) +bd(e⁺∧e⁻)e⁺ +cd(e⁺∧e⁻)e⁻+d²(e⁺∧e⁻)²

=a²+d²+b²+c²

2 + (ab+ac−cd+bd)e⁺+ (ab+ac−cd+bd)e⁻ +c²−b²+ 4ad

2 (e⁺∧e⁻) We invoked the relations (2.2), (2.3) and

e^±(e⁺∧e⁻) =∓e^±, (e⁺∧e⁻)e^±=±e^±, (e⁺∧e⁻)²= 1,

(e⁺∧e⁻)^†=e⁺∧e⁻.

Letf be a Clifford-valued polynomial, written as a finite linear combination of Hermite polynomials: f =P

k∈NHkck. Then (f, f) =X

k,l

c^†_l(Hk, Hl)ck =X

k∈N

ηkc^†_kck. (3.3) It follows that if (f, f) = 0 then f = 0.

Proven all of the defining properties of an inner product, we have come to the next definition:

Definition 3.8(Inner product). Theinner product of two discrete functions is given by

(f, g) = f(ξ)[1], g(ξ)[1]

:=D

f(ξ)G,[1](g(ξ))^†E

, (3.4)

withf(ξ)andg(ξ)the Taylor series expansions off andg.

The scalar part of a Clifford number is used to define the norm:

Definition 3.9. The normof a Clifford numbera is defined as:

kak:= [a]₀.

This naturally leads to the definition of norm of a function:

Definition 3.10. The scalar part of(f, f)is defined as thenormof the discrete functionf:

kfk:=

(f, f)

0.

By Lemma 3.7 and equation (3.3), this definition is well-defined. Remark that this definition can be naturally applied to the norm of a general Clifford-element.

Based on the findings in this section, we define the discrete Weierstrass space as follows:

Definition 3.11. The discrete Weierstrass space W is the completion of the right Clifford module of Hermite polynomials in ξ in the norm (3.4): f ∈ W ⇔f =P∞

n∈NH_na_n with(f, f)<∞.

Remark that this condition can be rewritten as_[a

n]₀

√η_n

n

∈`<sub>2</sub>, where`₂is defined in the usual way as the space of sequences (x_n)_n satisfying

X

n∈N

|x_n|²<∞.

With this newly introduced Weierstrass space, we can expand the definition of the discrete Weierstrass transform to all functions inW, as they are convergent (for theW-inner product) series of Hermite polynomials.

Definition 3.12(Discrete Weierstrass transform). For a discrete functionf ∈ W,f =P∞

k∈NHkak, its Weierstrass transform is defined as W[f](z) = 1

√2πhf(ξ)G, e^{−z2/2+ξ z}[1]i=X

n∈N

W[H_n]a_n=X

n∈N

(−1)^bnc2 a_nzⁿ

The distributionf(ξ)Gis calculated by writingf in its Taylor series expansion, hence a (possible infinite) series of polynomials in the operatorξ.

Iff =P∞

n∈NHnan∈ W, thenW[f] =P

n∈Nzⁿan, and W[f],W[f]

= 1

√2π X

n∈N

*

zⁿan,X

m∈N

z^mam

+

= 1

√ 2π

X

m,n∈N

amanhzⁿ, z^mi

= 1

√2π X

n∈N

|an|²n!.

As

an

√η_n

n ∈`2, this series converges in the Fock space.

Of course any polynomial is an element of W. What other elements can we distinguish? Let us have a look at some examples.

3.3. Examples

Example 1. For any linear combination of Hermite polynomials, one clearly

has ∞

X

k,l=0

(Hkck, Hlbl) =

∞

X

k=0

b^†_k(Hk, Hk)ck=

∞

X

k=0

b^†_kηkck.

Example 2. The inner product of two basic discrete polynomials follows immediately from the definition of the GaussianG.

ξ^k[1], ξ^l[1]

=D

ξ^kG,[1](ξ^†)^lE

=D

G,[1](ξ^†)^l(ξ^†)^kE

=D

G, ξ^k+l[1]E

=











√2π k+l= 0,

√2π(k+l−1)!! k+leven,

0 k+lodd.

Example 3. We calculate the e^λξ, e^λξ , λ∈C.

e^λξ, e^λξ

=he^λξG,[1]e^λξ†i=hG, e^λξ[1]e^λξ†[1]i=hG, e^2λξ[1]i

=

∞

X

s=0

(2λ)^s

s! hG, ξ^s[1]i=

∞

X

`=0

(2λ)^2`

(2`)! hG, ξ<sup>2`</sup>[1]i=

∞

X

`=0

(2λ)^2`

(2`)!

√2π(2`)!

2<sup>`</sup>`!

=

∞

X

`=0

√

2πλ^{2`</sup>2<sup>`}

`! =√ 2πe^2λ2. Hence for everyλ∈C,e^λξ is an element ofW.

Example 4. Not every discrete exponential function is an element ofW:

e^λξ2[1], e^λξ2[1]

=hG, e^2λξ2[1]i=

∞

X

s=0

(2λ)^s

s! hG, ξ^2s[1]i

=

∞

X

s=0

(2λ)^s s!

√

2π(2s)!

2^ss! =

∞

X

s=0

√ 2π λ^s

2s s

.

The convergence of this series depends on the parameterλ. Denoting the general term byas, this series is convergent by d’Alemberts criterium, if

s→∞lim

a_s+1 as

<1⇔ |λ|< 1 4.

The cases forλ=±¹₄ are the borderline: after some tedious calculations, the series withλ=−¹₄ is also convergent with value√

π, while the positive value ¹₄ is divergent.

Other examples are

e^−ξ2/8, e^−ξ2/8

=2√ 3 3

√π,

e^−ξ2/6, e^−ξ2/6

=

√30 5

√π,

e^ξ2/8, e^ξ2/8

= 2√

π,

e^ξ2/6, e^ξ2/6

=√ 6π.

The inner product

e^±ξ2/2, e^±ξ2/2

is divergent. This example shows that it is indeed useful to put a condition on the set of functions inW. Remark that in the classical setting however, the continuous Gaussian exp(−x²/2) is an element of the Weierstrass space. The difference is found in the fact that our weight function is the Gaussian distribution, defined with ∂ instead of ξ, while the discrete functione^−ξ2/2[1] will tend to∞forx→ ∞.

Example 5. The inner product of two basic vectors is:

e⁺,e⁺

=

e⁺G,[1]e⁻

=

G,[1]e⁻e⁺

=

G,e⁻e⁺[1]

=√

2πe⁻e⁺=

√2π 2 , e⁻,e⁻

=

√2π 2 , e⁺,e⁻

=

e⁺G,[1]e⁺

=

G,e⁺e⁺[1]

= 0

We can interpret these results as orthogonality relations of the basic elements e⁺ ande⁻. As a result, e⁺±e⁻,e⁺±e⁻

= +√ 2π

Let us now calculate the Weierstrass transform of the previous examples.

1. We start with the basic discrete homogeneous polynomials of even degree Wh

ξ^2k[1]i

(z) = 1

√2πhξ^2kG, e^−z2/2+ξz[1]i

= 1

√2πe^−z2/2hξ^2kG,

∞

X

l=0

ξ^lz^l l! [1]i

= 1

√2πe^−z2/2

∞

X

l=0

z^l

l!hG, ξ^l+2k[1]i

=e^−z2/2

∞

X

l=0

z^2l

(2l)!hG, ξ^2l+2k[1]i

=e^−z2/2

∞

X

l=0

z^2l (2l)!

(2l+ 2k)!

2^l+k(l+k)!

=e^−z2/2(2k−1)!!1F1 k+1 2;1

2;z² 2

! .

For odd powers ofξ, we find Wh

ξ^2k+1[1]i

(z) =e^−z2/2

∞

X

l=0

z^2l+1

(2l+ 1)!hG, ξ^2l+2k+2[1]i

=e^−z2/2

∞

X

l=0

z^2l+1 (2l+ 1)!

(2l+ 2k+ 2)!

2^l+k+1(l+k+ 1)!

=e^−z2/2z(2k+ 1)!!1F1 k+3 2;3

2;z² 2

! .

1F1(a;b;z) is the generalized hypergeometric function, converging for all finite values ofz. Substituting different values for k, we see the following concrete results:

k W[ξ^k](z)

1 z

2 z²+ 1 3 z³+ 3z 4 z⁴+ 6z²+ 3 5 z⁵+ 10z³+ 15z 6 z⁶+ 15z⁴+ 45z²+ 15 7 z⁷+ 21z⁵+ 105z³+ 105z 8 z⁸+ 28z⁶+ 210z⁴+ 420z²+ 105 9 z⁹+ 36z⁷+ 378z⁵+ 1260z³+ 945z

10 z¹⁰+ 45z⁸+ 630z⁶+ 3150z⁴+ 4725z²+ 945 11 z¹¹+ 55z⁹+ 990z⁷+ 6930z⁵+ 17325z³+ 10395z

12 z¹²+ 66z¹⁰+ 1485z⁸+ 13860z⁶+ 51975z⁴+ 62370z²+ 10395 In these polynomials, we recognize the same coefficients as in the (contin- uous) Hermite polynomials, up to sign. More specific, we see that

W[ξ^2k](z) = (−1)^kH_2k(iz), (3.5) W[ξ^2k+1](z) = (−1)^k+1iH2k+1(iz). (3.6) A natural question is to compare these results with the classical (contin- uous) case. Indeed, the continuous Weierstrass transform of the (continu- ous) polynomialsx^k are the same polynomials as in the table above.

2. Exponential functions:

W[e^λξ](z) =he^λξG, e^−z2/2+ξz[1]i

=

∞

X

l=0

1

l!λ^lhξ^lG, e^−z2/2+ξz[1]i

=

∞

X

l=0

λ^l

l!W[ξ^l](z)

=

∞

X

l=0

λ^2l

(2l)!(−1)^lH2l(iz) + λ^2l+1

(2l+ 1)!(−1)^l+1iH2l+1(iz).

To calculate this sum of Hermite polynomials, we use the formulas ([17])

∞

X

k=0

2^kH2k(√ 2x)ω^k

(2n)! =e^−ωcos(2x√

−ω),

∞

X

k=0

2^k+12H_2k+1(√ 2x)ω^k

(2n+ 1)! = e^−ω

√−ω sin(2x√

−ω).

(3.7)

Making the right substitutions and lettingω= ^−λ₂², we arrive at W[e^λξ](z) =e^−ωcos√

2iz√

−ω + λi

√−ωe^−ωsin√ 2iz√

−ω

=e^λ2/2cos √ 2iz

rλ² 2

!

− λi qλ²

2

e^λ2/2

√2 sin √ 2iz

rλ² 2

!

=e^λ2/2

cos(iλz) +isin(iλz)

=e^λ2/2+iλz. 3. Let|λ|< ¹₄, such thate^λξ2 ∈ W.

W[e^λξ2](z) =he^λξ2G, e^−z2/2+ξz[1]i

=

∞

X

l=0

1

l!λ^lhξ^2lG, e^−z2/2+ξz[1]i

=

∞

X

l=0

λ^l

l!W[ξ^2l](z)

=

∞

X

l=0

λ^l

l!(−1)^lH_2l(iz).

Adding both formulas in (3.7) and keeping in mind that H_n(−z) = (−1)ⁿH_n(z), we find

∞

X

n=0

Hn(z)ωⁿ

bⁿ₂c! = (1 + 2zω+ 4ω²) (1 + 4ω²)^−3/2e^4ω

2z2

1+4ω2. (3.8)

Substituting this result in the calculation ofW[e^λξ2](z), withω= ^−λ₂², W[e^λξ2](z) =e^4λz

2

1−4λ(1−4λ)⁻¹².

4. Discreteδ−functions

As they are the building blocks of discrete function theory, we investigate the discreteδ−functions and check if they are elements ofW. Letδ0=δ0(ξ)[1] be theδ0-function which takes values 0 everywhere except in the origin where it is 1. Letδ0(ξ) =P

l∈Zξ<sup>`</sup>c` the corresponding Taylor series operator. We search for an expression δ0 = δ0(ξ)[1] = P

n∈NHn(ξ)dn[1]. It holds that δ0(ξ)[1] = δ0 =δ^†₀ = (δ0(ξ)[1])^†, becauseδ0 is real. If the coefficientsdn exist, they must satisfy the relations

(H_n, δ₀) =

*

H_n(ξ)G,[1]X

`∈N

d^†_{`</sub>H<sub>`}(ξ)^† +

=

*

H_n(ξ)G,X

`∈N

d^†_{`</sub>H<sub>`}(ξ)[1]

+

=d^†_n

Hn(ξ)G, Hn

=d^†_nηn. On the one hand, we have

(Hn, δ0) =

Hn(ξ)G, δ0

=X

s∈Z

Hn(ξ)G

(s)δ0(s),

because the distribution Hk(ξ)G corresponds to a function on Z, where the action is pointwise. In this way, we can useδ0 as a density function onZ. So (Hn, δ0) = (Hn(ξ)G)(0). On the other hand, one can also consider the Taylor seriesδ0(ξ) =P

l∈Nξ<sup>`</sup>c`and substitute it into the inner product (Hn, δ0):

(H_n, δ₀) =

*

H_n(ξ)G,[1]X

`∈N

c^†<sub>`</sub>ξ<sup>`</sup> +

=X

l∈N

c^†_`D

H_nG,[1]ξ^`E ,

which should equald^†_nηn. Remark that this gives a finite sum, as forn > `, the result

HnG, ξ^`

is zero. This gives us two ways to calculate the coefficientsdn. d^†_n =(Hn, δ0)

ηn

.

4.1. Taylor series method

Expandingδ0by its Taylor series, we have (see [15]) δ0(x) =

∞

X

l=0

(−1)^`

l!l! ξ^2l[1](x) +

∞

X

l=0

(−1)^l+1

(l+ 1)!l!ξ^2l+1[1](x)(e⁺−e⁻).