Tight Frame with Hahn and Krawtchouk Polynomials of Several Variables

Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With ${\mathsf r}(d,n):= \binom{n+d-1}{n}$, the first method generates, for each $m \ge n$, two families of tight frames in ${\mathbb R}^{{\mathsf r}(d,n)}$ with ${\mathsf r}(d+1,m)$ elements. The second method generates a tight frame in ${\mathbb R}^{{\mathsf r}(d,N)}$ with $1 + N \times{\mathsf r}(d+1, N)$ vectors. All frame elements are given in explicit formulas.


Introduction
For a real finite dimension space V with inner product (·, ·), a set of vectors {v k : k = 1, . . . , M } in V is called a (Parseval) tight frame with M elements if Tight frames are connected to a wide variety of problems in pure and applied mathematics and they have been studied intensively in recent years (see, for examples, [1,2,3,4,9] and references therein). We construct tight frames with the Hahn and Krawtchouk polynomials of several variables, and use them to construct several families of tight frames for the Euclidean spaces that are given in explicit formulas. Let Π d n denote the space of polynomials of total degree at most n in d-variables. Let ·, · denote an inner product defined on the space of polynomials or on Π d N for a fixed integer N . A polynomial p ∈ Π d n is called an orthogonal polynomial of degree n with respect to the inner product if p, q = 0 for all q ∈ Π d n−1 .
Let V d n denote the space of orthogonal polynomials of degree n. Then dim V d n = n + d − 1 n , n = 0, 1, 2, . . . .
In the present paper, we study monic Hahn and Krawtchouk polynomials instead. These are orthogonal projections of (−1) |α| (−x) α on either V d n (H κ,N ) or V d n (K ρ,N ). For α ∈ Z d+1 m and m ≥ n, let U α,n (x) = (−1) |α| (−x) α + · · · denote the monic orthogonal polynomial of degree n. Then Ξ n,m := {U α,n : α ∈ Z d+1 m } is a subset of V d n and it contains an orthogonal basis of V d n . In the special case of m = n and α = (α , n − |α |) with α ∈ N d 0 and |α | ≤ n, the monic Hahn and Krawtchouk polynomials were studied in [13,14]. We shall describe generating functions, connection to the mutually orthogonal polynomials, and explicit expressions for these polynomials. Furthermore, the set Ξ n,m contains far more elements than the dimension of V d n , we shall prove that the set Ξ n,m is a tight frame for the space V d n and these frames are given in explicit formulas. The connection between tight frames and orthogonal polynomials of several variables has been explored for the Jacobi polynomials on the simplex [16] (see also [17,19]). The discrete orthogonal polynomials have not been used in this connection until now.
A finite tight frame is equivalent to a tight frame on an Euclidean space. From our Hahn and Krawtchouk polynomial frames, we are able to derive a large family of tight frames for Euclidean spaces that are given by explicit formulas. There are many ways of constructing tight frames for the Euclidean spaces. One can, for example, use the classical discrete Fourier transform to construct tight frame for C n or R n that contains m elements for all m > n. There are also algorithms that can be used to construct tight frames with certain additional features [3]. Our frames are all given by explicit formulas. In fact, we shall give two methods for constructing such frames. For n, d ∈ N, let For a positive integer m ≥ n, our first method uses either the Hahn polynomial frames or the Krawtchouk polynomial frames to construct explicitly tight frames in R n(d,n) that contain n(d, m) elements. Our second method uses the Hahn polynomial frames with the parameter κ = 0 to construct tight frames in R n(d,N ) with N × n(d, N ) + 1 elements. All frames are given by explicit formulas and those derived from the second method have the additional feature that all but one frame elements are vectors whose entries consist of only rational numbers. The paper is organized as follows. In the next section, we recall necessary background on orthogonal polynomials. Monic Hahn polynomials are introduced and developed in Section 3, from which monic Krawtchouk polynomials are deduced by taking an appropriate limit in Section 4. Tight frames for the Hahn and the Krawtchouk polynomials, as well as for the Euclidean spaces are discussed in Section 5, where several examples are explicitly given.

Background and preliminary
Throughout this paper we reserve the Greek letter κ and ρ for the parameters in the weight function H κ,N and K ρ . We shall use other Greek letters for the multiindexes. Orthogonal polynomials in V d n are indexed by ν or µ in N d 0 , and we use α, β, γ for indexes in N d+1 0 . For a, b > −1, the classical Hahn polynomial Q n (x; a, b, N ) is a 3 F 2 hypergeometric function given by These are discrete orthogonal polynomials defined on the set {0, 1, . . . , N }. For the Hahn polynomials of d-variables, a mutually orthogonal basis for V d n (H κ,N ) can be given in terms of Q n (x; a, b, N ). We need the following definition.
For ν ∈ N d 0 and x ∈ R d , define Then the polynomials in {H ν (x; κ, N ) : |ν| = n} form a mutually orthogonal basis of V d n (H κ,N ) and B ν := H ν (·; κ, N ), H ν (·; κ, N ) H κ,N is given by, setting λ κ := |κ| + d + 1, This basis was defined in [11] and given in the present form in [10,14,18]. Another basis for V d n (H κ,N ) that is of interests for our work is the monic orthogonal basis studied in [13], which will come out as a special case of the discussion in the next section. It is worth mentioning that, for ν ∈ N d 0 , m, N = 0, 1, . . . , depends only on |ν|, not the elements of ν. The expansion of H ν (x; κ, N ) in the shifted monic monomials (−x) α is given in [19].
There is a close relation between the Hahn polynomials and the orthogonal polynomials with respect to the weight function .
A mutually orthogonal basis for V d n (W κ ) can be given in terms of the classical Jacboi polynomials where a j is defined by (2.3). Then the polynomials in {P κ ν : |ν| = n} form a mutually orthogonal basis of V d n (W κ ) with P κ ν , P κ ν Wκ given by This basis is well studied, but our normalization is different from the usual definition (see [6, p. 47 (1) in the denominator. As defined in [11] and later recognized in [18], the polynomials P κ ν serve as generating function for the Hahn polynomials.
The Hahn polynomials also appear as connecting coefficients between P κ ν and monic orthogonal polynomials R κ α defined by, for α ∈ N d+1 The explicit formula of R α (x) was derived in [18].
. The Hahn polynomials serve as the connecting coefficients of P κ ν and R k α as shown recently in [19].
and, conversely, We now turn to basic results on the Krawtchouk polynomials. For 0 < p < 1, the classical Krawtchouk polynomial K n (x; p, N ) of one variable is defined by For ν ∈ N d 0 , a family of the Krawtchouk polynomials of d variables can be given in terms of Krawtchouk polynomials of one variable [10,14]. For ρ ∈ N d 0 , we use the notation (2.2), which implies that |ρ j | = ρ 1 + · · · + ρ j for j = 1, 2, . . . , d.

Monic Hahn polynomials of several variables
The monic polynomial R κ α has a single monomial x α as its highest term. For the Hahn polynomials, the role of x α is played by m α defined as follows.
It follows that m α (x) is a monic polynomial of degree |α|. For each α ∈ Z d+1 N , we define the monic Hahn polynomial as the orthogonal polynomial that has m α as its leading term.
, so that the set contains redundancy. In parallel to the relations between the monic orthogonal polynomials R κ α and the Jacobi polynomials P κ ν , in Theorem 2.6, we can derive relations between Q β (·; κ, N ) and the Hahn polynomials H ν (·; κ, N ).
Throughout the rest of this section, we often use the abbreviation Recall that the Jacobi polynomials P κ ν are generating functions of the Hahn polynomials H ν (·; κ, N ). It turns out that R κ α are generating functions of the monic orthogonal polynomials Q ν (·; κ, N ).
The generating function relation (3.3) can be used to derive an explicit expansion of Q β (·; κ, N ). The following simple lemma is useful.
Proof . Using the multinomial identity we obtain that which proves the stated identity.
Proof . Using the expansion of Y α in terms of R κ β and the generating function relation (3.3), we obtain, for x ∈ R d and X = (x, 1 − |x|), that Setting X = y/|y| with y ∈ R d+1 , we obtain an expansion of |y| N −|α| y α in terms of y γ , which implies, when comparing with (3.4), the identity (3.6).

Y. Xu
In particular, (3.10) shows that the set {Q α,n (·; κ, N ) : |α| = m}, which is a subset of the space V d n (H κ,N ), spans the space. Let P n (H κ,N ; ·, ·) denote the reproducing kernel of V d n (H κ,N ), which is characterized by the requirement that it is an element of V d n (H κ,N ) in either its variable and P n (H κ,N ; x, ·), P H κ,N = P(x), ∀ P ∈ V d n (H κ,N ).
In terms of the mutually orthogonal basis {H ν (·; κ, N ) : |ν| = n}, the reproducing kernel can be written as Our next result shows that this kernel can be expanded in Q α,n (·; κ, N ).
While (3.11) follows directly from the definition, (3.12) is by no means trivial since the set {Q α,n (·; κ, N ) : α ∈ Z d+1 N } is linearly dependent and is heavily redundant. In fact, the identity (3.12) is the key result for our discussion on the tight frames in Section 5.

Monic Krawtchouk polynomials of several variables
For each α ∈ Z d+1 N , we define the monic Krawtchouk polynomial as the orthogonal polynomial that has m α as its leading term.
The properties of such polynomials can be derived from the monic Hahn polynomials Q ν (·; κ, N ). Indeed, it is easy to see that N ).
From the first identity follows readily that the inner product f, g H κ,N becomes f, γ K ρ,N if we set κ = tρ and let t → ∞. Consequently, we conclude that Together with the limit relation (2.12), we deduce following relations from Proposition 3.3.
and, conversely, Furthermore, from Theorem 3.6 and Proposition 3.7, we deduce the expansion of monic Krawtchouk polynomials and its converse. Conversely, We now define analogues of Q α,n (x; κ, N ) polynomials.
Consequently, from Proposition 3.9, we deduce the following relation.
It can also be expressed in L α,n (·; ρ, N ) as the following analogue of Theorem 3.13 shows.

Finite Tight frames
Then for all Furthermore, {Q α,n (·; κ, N ) : |α| = m} is a tight frame of V d n (H n,N ), Proof . If f ∈ V d n (H n,N ), then f (x) = f, P n (H κ,N ; ·, x) H κ,N , so that the first identity follows immediately from (3.12). Taking the inner product of the first identity with f gives the second identity.
A finite tight frame is equivalent to a tight frame in an Euclidean space. From Theorem 5.1 and connecting coefficients of our Hahn polynomials, we can derive Euclidean tight frames that can be given explicitly. Recall that r(d, n) = n+d−1 n . Fixing a linear order in the set {ν ∈ N d 0 : |ν| = n}, say the lexicographical order, we denote x ∈ R r(d,n) as x = (x ν : |ν| = n, ν ∈ N d 0 ). The usual inner product in R r(d,n) then takes the form (x, y) := |ν|=n x ν y ν , x, y ∈ R r(d,n) .
Then the set H(d, n, m, κ) := {h α,n : |α| = m} is a tight frame in R r(d,n) , that is, and the frame has #H(d, n, m, κ) = m+d d elements.
Proof . For y ∈ R r(d,n) and x ∈ Z d+1 N , define By the orthogonality of H ν (x; κ, N ), we immediately have f y , f y H κ,N = (y, y). On the other hand, by (3.9), we obtain where C n (m, N ) is the constant in front of the summation in (3.9). Using (2.6), it follows readily that Substituting into (5.1) and simplifying the constant, we see that the right hand side of (5.1) becomes exactly |α|=m (y, h α,n ) 2 .

Y. Xu
Recall that P n (H κ,m ; ·, ·) denotes the reproducing kernel of V d n (H κ,m ).

(5.2)
In particular, in terms of the unit vectors h α,n = h α,n / h α,n , Proof . The identity ( is a tight frame in R 3 with 10 elements, and is a tight frame in R 4 with 10 elements. For fixed d, n, m, we obtain a family of tight frames in R r(d,n) with d + 1 parameters κ = (κ 1 , . . . , κ d+1 ). Although the proof is established for κ i > −1, the analytic continuation shows that we obtain a tight frame for κ ∈ R d+1 as long as h α,n is well defined. A frame is called normalized tight frame if all elements of the frame have the same norm [2]. Our tight frame is not normalized in general. In the case of d = 2 and m = n = 2, we can choose κ to obtain a normalized tight frame.
form a normalized tight frame in R 3 with 6 vectors.
For the Krawtchouk polynomials, the analogue of Theorem 5.1 is as follows.
Theorem 5.6. Let m ≥ n. Then for all Furthermore, {Q α,n (·; κ, N ) : |α| = m} is a tight frame of V d n (H n,N ), is a tight frame in R 4 with 10 elements.
Using the tight frames of V d n (H κ,N ) as building blocks, we can also build tight frame for the space Π d N of polynomials of degree at most N in d-variables under the inner product ·, · H κ,N .
and, in particular, Proof . The orthogonal expansion of f ∈ Π d N in the Hahn polynomials is given by f, P n (H κ,N ; x, ·) H κ,N .
There is also a straightforward analogue for the Krawtchouk polynomials, which we shall not state.
Since {H ν (·; κ, N ) : |ν| ≤ N } is a mutually orthogonal basis, by writing f as and using the connection (3.9), each tight frame in (5.3) corresponds to a tight frame for R n(d,N ) , where and the frame elements can be computed directly in terms of H ν (x; κ, m n ). We shall not written down these tight Euclidean frames but turn our attention to another way of constructing tight Euclidean frames using the Hahn polynomials, this time using Q α,n (·; κ, N ) instead of H ν (·; κ, N ). This construction is based on the observation that the weight function H κ,N of the Hahn polynomials becomes 1 when κ = 0, so that the inner product ·, · H 0,N is a constant multiple of the inner product (·, ·) of the Euclidean space R n(d,N ) . We use Theorem 5.9 with κ = 0 and m n = N for n = 1, 2, . . . , N . Proof . For κ = 0, the inner product becomes f, g H 0,N = N ! (d + 1) N |α|=N f (α)g(α).
Since f ∈ Π d N is uniquely determined by its values on the points {α : α ∈ Z d+1 N }, we see that N ) . For m = N and κ = 0, D n (N, N ) = (d + 1) N /N ! is independent of n. Consequently, we can write (5.3) as (5.5).
Evaluating these two functions at elements in Z 3 2 , we obtain q α,1 and q α,2 . The set Ξ(2, 2) consists of columns of the matrix  For d = 2 and each N = 1, 2, . . ., the identity (5.5) gives a tight frame of R n(2,N ) with M := N × n(2, N ) + 1 vectors. The next one is a tight frame of R 10 with 31 vectors. Except q 0 , all other frame elements in a tight frame of this family have rational entries.