ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES

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1 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES SHIDONG LI, ZHENGQING TONG AND DUNYAN YAN Abstract. For B-splineRiesz sequencesubspacesx span{β k ( n) : n Z}, thereis an exact sampling formula on integer lattices for low orders k = and whose reconstruction functions are exactly the translates of B-splines. We observe that it is possible to derive a similar sampling formula for X on integer lattices plus a small number of finitely many non-integer points for orders k 3. The sampling formula is simple with some adaptive property as well. Sampling reconstruction functions {ϕ(n, x)} are simple and exact linear combinations of the translates of B-splines. In particular, we show that for B-spline Riesz sequence subspace X of order k = 3, there is a sampling formula on the integer lattice plus any non-integer point on any finite interval I. It is also shown that any half-integer point in I, as the additional sampling point, is optimal. Such a treatment of sampling mechanisms on interval I overcomes the difficulty of using existing sampling formula on finite interval I when the reconstruction function S(x) is (typically) not compactly supported. It is therefore expected such simple treatment is to have direct applications in various computer graphics applications.. Introduction In a study of biorthogonal duals from the pseudoframe point of view[5], [7], we observed that pseudo-duals to very low orders of B-spline Riesz sequences behave peculiarly. They can have (pseudo-) duals of arbitrarily small support that goes to zero. In view of the related pseudoframe expansion, the nullifying support of the duals certainly signals a sampling formula for subspaces spanned by these low order B-spline Riesz sequences. Frames and sampling theory are closely related, e.g., [3], [4], [5], [7], [8]. Studies of sampling in relation to frames and sampling properties for wavelet subspaces and/or shift invariant subspaces have had a rather long history, e.g., [], [], [6], [9], [], [], [], [3], [6], [7], [4], [8], [9], [], [], [], [3], [4], [5]. Regular and irregular sampling mechanisms were all considered in these works. In the case of regular sampling, constructible sampling reconstruction functions S(x) are also characterized in [3], and later extended in, e.g., [3], [4], [5]. In such cases, integer Mathematics Subject Classification. Primary 4C5, 46C5, 47B. Key words and phrases. frames, B-spline Riesz sequences, Pseudoframes, pseudo-duals, frame representations, sampling formula.

2 S. LI, Z. TONG AND D. YAN sampling formulas exist, i.e., f X, f = n f(n)s( n). The characterizations of S(x) in these works typically result in fairly simple Fourier domain and/or Zak transform descriptions, though as orders k goes higher the expression of S(x) can get complicated, and is typically non-compact. Direct applications of such non-compact sampling functions in finite interval I of any practical application can be inconvenient. From the study of pseudo-dual biorthogonal sequences of B-spline Riesz sequences[5], we observe that sampling formulae for B-spline Riesz sequence subspaces X exist on almostall integers (integer plus finitely many non-integer points). The corresponding sampling reconstruction functions, with a different symbol, ϕ(n, x) are simple linear combinations of integer shifts of B-spline functions. Method of construction of such sampling formulae over any finite interval I is reported. The required additional non-integer sampling points are independent to the length of I, suggesting that such sampling formulae can be extended to infinite dimensions. The sampling reconstruction functions ϕ(n, x) are typically related to n in the form of f X, f = f(x n )ϕ(n,x), x n Λ where Λ {x n } consists of relevant integer and the finitely many non-integer sampling points. Shifting characteristics among ϕ(n, x) over integer sampling points are also observed in examples. The simplicity of such a treatment may have value in practical applications where typically finite intervals are concerned. Sampling reconstruction functions so derived are exact for the given interval I. This overcomes the difficulty of using existing sampling theorems on a finite interval I when (typically) S(x) is not compactly supported. The adaptivity and the state of stability of such sampling schemes are also quite strong. Taking any uniformly distributed random variable on ( 4, 4 ) around the almost-all integer sampling points {x n }, reconstruction functions ϕ(n,x) can be evaluated adaptively. Numerical examples are stable and will be demonstrated.. The pseudo-dual observation toward sampling properties Let ϕ = β be the nd order symmetric B-spline. Let X L (R) be defined by X = span{ϕ( n) : n Z}. (.) We have seen that there are pseudo-duals ϕ α such that supp ϕ α [ α,α] [,], and ϕ α is continuous, differentiable and symmetric for every < α. For any f X, we have f = n Z f, ϕ α ( n) ϕ( n).

3 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES 3 In fact it suffices to point out (and easy to verify) that one pseudo-dual of {ϕ( n)} is given by 45 ϕ α (t) = α 5t4 + 5 α 4t3 3 α 3t + 5 α, t [, α], 45 α 5t4 5 α 4t3 3 α 3t + 5 (.) α, t [ α, ]. It turns out as α +, an exact sampling formula holds. Theorem. Let ϕ be the nd order symmetric B-spline function. Let X be the subspace spanned by the second order B-spline Riesz sequence as in (.). Then, for all f X, f = n Zf(n)ϕ( n). Proof. For any f X, we have f(x) = n Z f, ϕ α ( n) ϕ(x n). Let us now study the inner product f, ϕ α ( n) as α approaches +. We claim It suffices to prove that lim α + f, ϕ α( n) = f(n). (.3) f() = lim α + f, ϕ α( ). (.4) The formula (.3) would naturally hold if (.4) is true. To this end, we have that lim α + f, ϕ α α( ) f() = lim f(x) ϕ α (x)dx f() α + where the fact that α lim α + α α α α φ α (x)dx = f(x) f() ϕ α (x) dx, is used. Since function f is continuous, for any ε >, there exists δ > such that when x < δ, we have f(x) f() < ε. Consequently, for < α < δ, α α f(x) f() ϕ α (x) dx ε α α ϕ α (x) dx εc, where we have applied a fact that α ϕ α (x) dx C <. α This is equivalent to saying that (.4) holds.

4 4 S. LI, Z. TONG AND D. YAN Indeed, exact sampling formulae on integer lattices exist for B-spline Riesz sequence subspaces of orders and whose reconstruction functions are translates of the B-spline functions. Though re-observed from pseudo-dual point of view, such a sampling property of B-spline shift-invariant subspaces of orders and can also be obtained directly from so-designed algebraic requirement. For instance, suppose that ϕ is symmetric B-splines of orders or. For any f X, it follows that f = c i ϕ( i). Since {ϕ( n) : n Z} is i Z a basis, the coefficients {c i } is unique. Observe that {, t =, ϕ(t) =, t Z\{}. Hence, f(n) = c i ϕ(n i) = c n ϕ() = c n for all n Z. We confirm, however, there is no such simple sampling formula (with B-spline functions as reconstruction functions) on integer lattices only for B-spline Riesz sequence subspace X of higher order (3 and up). But there are related sampling formulae that use integer sampling points and a small number of finitely many non-integer points. They behave quite simple, and have interesting adaptive nature as well. Reported in this article is the construction methodology of such sampling formulae for X on any practically concerned finite internal I. 3. Sampling reconstruction functions can not be B-splines on integer grid for orders 3 and higher Let ϕ be the k th order symmetric B-spline (k 3). We confirm that, for k 3, it is not possible to have f = n Zcf(n)ϕ( n) for every f X = span{ϕ( n) : n Z} and some constant c. Proposition. Suppose that ϕ = β k, the k th symmetric B-spline function. Define X = span{ϕ( n) : n Z}. Then for every f X, if and only if k =,. f = n Zcf(n)ϕ( n) Proof. In view of Theorem, we merely need to show that there exists f X for which f = n Zcf(n)ϕ( n) does not hold for B-spline shift-invariant subspaces of order k 3. Assume that f = n Zcf(n)ϕ( n)

5 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES 5 always holds. Let f = ϕ, then ϕ = n Zcϕ(n)ϕ( n). Thus it implies from the linear independent property of {τ n ϕ : n Z} that cϕ() =. So we have c = /ϕ(). Set f = ϕ+τ ϕ, then we have ϕ+τ ϕ = n Zc ( ) ϕ(n)+τ ϕ(n) τ n ϕ. Thus it implies from the linear independent property of {τ n ϕ : n Z} that ( ) ϕ()+τ ϕ() = ϕ() and ( ) ϕ()+τ ϕ() = ϕ() must simultaneously hold. This is obviously impossible for B-splines of order 3 and up. We show in the following section, however, that sampling formulae using almost-all integer points exist with reconstruction functions S n (x) being linear combinations of the B-spline function and its shifts, for orders k Almost-all integer sampling for B-spline Riesz sequence subspaces of orders 3 and higher We shall denote by β k the k th order symmetric B-spline supported on [ k, k ]. Define X = span{β k ( n) : n Z}. Then for any f X and every x R, there exists {c i } l, i Z such that f(x) = c i β k (x i). i For any finite interval I, say I [n,n +M] for some integers n and M, f(x) = c i β k (x i), (4.5) [n,n +M] [n,n +M] i Λ where { } Λ = i : [n,n +M] suppβ k ( i) { } k k k = n +,n +,,n,n,n +,,n + +M. That is, there are (M + k ) relevant-shifts of β k involved on the interval [n,n +M]. Here the symbol θ stands for the least integer greater than and equal to the real number θ.

6 6 S. LI, Z. TONG AND D. YAN For convenience, let m = M+ k and write Λ = {i,i,,i m }. If we choose a set of (different) sampling points {x,x,,x m } in [n,n +M], (4.5) produces the following system of linear equations at sample points {x i } i Λ, β k (x i ) β k (x i )... β k (x i m ) β k (x i ) β k (x i )... β k (x i m ) β k (x m i ) β k (x m i )... β k (x m i m ) c i c i. c im = f(x ) f(x ).. (4.6) f(x m ) If the matrix in the left side of (4.6) is invertible, then the solution to the coefficients {c i } of (4.5) uniquely exists. Consequently, there are constants {h ij } such that for all f X and t [n,n +M], f(t) = m h ij f(x j ) β k (t i) (4.7) i Λ j= m = f(x j )ϕ k (j,t), where ϕ k (j,t) j= are the sampling reconstruction functions for order k. i Λ h ij β k (t i), j =,,,m (4.8) Obviously, the choice of sampling points {x i } is not arbitrary. In fact, for inappropriate selections of {x i }, the matrix in (4.6) can be singular. For example, suppose that k = 3, M = and Λ = {n,,n +3}. In an extreme example, if we select all {x i } in the interval I = [n,n +], the last column of the matrix are zeros because supp(β 3 ) [ 3, 3 ]. Of course, {x i } is meant to be scattered (as any sampling formula should be). We shall show in detail in Section 6 that, for the case of order k = 3, {x i } can be all integer points and one arbitrary non-integer point in the interval I. But, first, let us exam some examples. Example. We demonstrate in this example that one is capable of constructing a sampling formula on any given interval. Suppose that β 3 is the 3 rd order symmetric B-spline function. Consider f [n,n +], the relevant shifting index set Λ contains four integer points Λ = {n,n,n +,n +}. 4 sampling points are therefore needed. If we choose evenly spread 4 points x = n, x = n + 4, x 3 = n +, x 4 = n + 3 4,

7 equation (4.6) becomes ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES c n c n c n + c n + = f(n ) f(n + 4 ) f(n + ). f(n ) The solution to {c i } clearly exists in terms of the sample values of {f(x i )} 4 i=. A sampling formula for all functions in X on the interval [n,n + ] (and thereby all real line) is therefore established. If n =, x [,], ϕ 3 (,x) = 3β 3 (x+) β 3 (x)+β 3 (x ) 3β 3 (x ). ϕ 3 (,x) = β 3 (x+)+4β 3 (x) 4β 3 (x )+5β 3 (x ). ϕ 3 (3,x) = 8β 3 (x+) β 3 (x)+4β 3 (x ) 7β 3 (x ). ϕ 3 (4,x) = 3β 3 (x ) Figure 4.. ϕ 3 (,x) Figure 4.. ϕ 3 (,x) Figure 4.3. ϕ 3 (3,x) Figure 4.4. ϕ 3 (4,x) Example. We demonstrate in this example that one is capable of deriving a sampling formula on a given set of (admissible) points. For instance, if one would like to derive a sampling formula from the integer points and half-points, there will be M + points in [n,n + M]. Since there are M + k equations, it follows that the inequality M + M + k must be satisfied, which in turn implies M k. Take β 3 for example. M = is sufficient, and equation (4.6)

8 8 S. LI, Z. TONG AND D. YAN becomes If n =, c n c n c n + c n + c n +3 = f(n ) f(n + ) f(n +). f(n + 3 ) f(n +) ϕ 3 (,x) = 8β 3 (x+). ϕ 3 (,x) = 9 β 3(x+)+ 5 β 3(x) β 3(x )+ β 3(x ) 5 β 3(x 3). ϕ 3 (3,x) = β 3 (x+) β 3 (x)+β 3 (x ) β 3 (x )+β 3 (x 3). ϕ 3 (4,x) = 5 β 3(x+)+ β 3(x) β 3(x )+ 5 β 3(x ) 9 β 3(x 3). ϕ 3 (5,x) = 8β 3 (x 3) Figure 4.5. ϕ 3 (,x) Figure 4.6. ϕ 3 (,x) Figure 4.7. ϕ 3 (3,x) Figure 4.8. ϕ 3 (4,x) Figure 4.9. ϕ 3 (5,x) Remark Example demonstrates how a sampling formula can be established from a given interval. The minimum number of sampling points is given by M + k. Example demonstrates how a sampling formula can be established if a given set of (admissible) sampling points is given. In this example, samples at integers and half points are considered. In such cases, the minimum length of the interval M is shown to be M k. It is worth of mentioning that there are quite some adaptive flavor to the sampling formulae so derived.

9 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES 9 In Section 6, we will apply this adaptiveness to derive a sampling formula using mostly integer sampling points, plus (k ) non-integer sampling points (independent of the length M and the starting point n ). But first, let us point out a slight variation of the sampling mechanism aforementioned using non-symmetric B-splines (supported on [,k]) over integer intervals [n,n + M]. We will demonstrate that the required sampling points for interval [n,n +M] is exactly M +k, which is reduced by for splines of odd orders. 5. The case of non-symmetric B-spline From this point and on, we shall not assume the symmetry of the B-splines. With an abuse of notation, we shall also denote by β k the k th order B-spline, but the support is now on [,k]. With everything similar to that of Section 4, let us observe that for all f X, f(x) = c i β k (x i), (5.9) [n,n +M] [n,n +M] i Λ where { } Λ = i : [n,n +M] suppβ k (x i) { } = n k +,n k+,,n,n,n +,,n +M. Λ differs Λ in that supp(β k (x i)) = [i,i + k] and the interval [n,n + M] are both integer intervals. Consequently, for odd order splines, the starting index of Λ is n k+, resulting a total of (M +k ) relevant-shifts of β k over the interval [n,n +M]. This, for odd order k, translates into less in the cardinality of Λ than that of Λ. Similarly, let m = M +k and write Λ = {i,i,,i m }. Then (5.9) will give rise to exactly the same system of linear equations (4.6) at sample points {x i } i [n,n +M], with B-splines β k supported on [,k]. The method would apply as well to this set-up if we replace Λ with Λ. The Examples and in Section 4 for order k = 3 would now have slightly different format. Example 3. Suppose that β 3 is the 3 rd order B-spline function supported on [,3]. Considering f [n,n +], the set Λ contains 3 numbers Λ = {n,n,n }. If we choose x = n, x = n + 3, x 3 = n + 3,

10 S. LI, Z. TONG AND D. YAN then equation (5.9) gives rise to If n =, c n c n c n = f(n ) f(n + 3 ). f(n + 3 ) ϕ 3 (,x) = 3 4 β 3(x+) 5 4 β 3(x+)+ 3 4 β 3(x). ϕ 3 (,x) = 3β 3 (x+)+3β 3 (x+) 9β 3 (x). ϕ 3 (3,x) = 3 4 β 3(x+) 3 4 β 3(x+)+ 7 4 β 3(x) Figure 5.. ϕ 3 (,x) Figure 5.. ϕ 3 (,x) Figure 5.. ϕ 3 (3,x) Example 4. Under the same set-up of Example 3, suppose that we also want to derive a sampling formula with integer and half points. The M + k equations with M + integer and half points on [n,n +M] yields M = k =. Consequently, we are looking at M +k = +3 = 3 equations. Take, Then equation (5.9) gives rise to x = n, x = n +/, x 3 = n +. If n =, c n c n c n = f(n ) f(n + ). f(n +) ϕ 3 (,x) = 5 β 3(x+) β 3(x+)+ β 3(x). ϕ 3 (,x) = β 3 (x+)+β 3 (x+) β 3 (x). ϕ 3 (3,x) = β 3(x+) β 3(x+)+ 5 β 3(x).

11 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES Figure 5.3. ϕ 3 (,x) Figure 5.4. ϕ 3 (,x) Figure 5.5. ϕ 3 (3,x) 6. Analysis of the sampling formula for order 3 In this section, we exam the adaptivity of such sampling formulas over a finite interval that requires one arbitrary non-integer point besides integer lattices for B-splines of order 3. Lemma 3. Suppose that β 3 is the 3 rd order B-spline function. If y (,), then β 3 (+ y) β 3 (+y) β 3 (y) >. Proof. By the formula of the 3 rd order B-spline function, we obtain that β 3 (+y) = ( (+y) 3(+y) +3y ), β 3 (+y) = ( (+y) 3y ), β 3 (y) = y. It follows that β 3 (+y) β 3 (+y) β 3 (y) = y( y). Since y (,), we have that β 3 (+y) β 3 (+y) β 3 (y) >. Theorem 4. Let X = span{β 3 ( n)} n. For any integer interval [n,n + M], M, a function f X restricted on [n,n + M] can be reconstructed by the M + integer sampling values at n = n,,n +M and at one arbitrary additional non-integer point in (n,n +M). Proof. With the derivation in Section 5, we see that on the closed integer interval [n,n + M], therearem+integerpoints. SincetherequirednumberofsamplingpointsisM+k, an additional k = 3 = non-integer point x is needed in (n,n +M). We note that the number of additional non-integer sampling points is independent to the length M. We are to establish that the corresponding matrix of (5.9) is non-singular for any non-integer choice of this additional point x. Take x = n, x = n +,, x M+ = n +M, x M+ = x, where x = n +y = n + y +y is any non-integer point in (n,n +M), M. Here y stands for the greatest integer less than and equal to y, and y (,) is such that

12 S. LI, Z. TONG AND D. YAN y = y +y. With Λ = {n,n,,n +M }, equation (5.9) gives rise to the system of equations where Ac = f A = , p q r and c = c n. c n +M c n +M, f = f(x ). f(x M+ ). f(x M+ ) p = β 3 (+y), q = β 3 (+y), r = β 3 (y), y (,). (6.) For convenience, denote by A the entire, but the last row, of the matrix A. Write the last row of A as A t = pe y + +qe y + +re y +3. Define also B t = (q p r)e y +. Here θ t stands for the matrix transpose of θ. Rewrite the matrix A and define a new matrix B by, respectively, A = ( ) A, and B = A ( ) A, B where e i is the M + dimensional column vector with the i th element and others zero. By elementary row operation, adding both ( p) times ( y +) th row and ( r) times ( y +) th row to the bottom row of A, A is reduced to B. It follows that there exists unique elementary matrix P such that PA = B and Pe i = e i. It is easy to calculate detb = q p r M+. By Lemma 3, we know that both A and B are invertible.

13 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES 3 Example 5. Take M =, there are M+k = +3 = 4 required sampling points. On the closed interval [n,n +], besides 3 integer points n, n +,n +, one additional nonintegerpoint,say, n +/in[n,n +]canbeselected. WithΛ = {n,n,n,n +}, equation (5.9) gives rise to the system of equations If n =, c n c n c n = c n + f(n ) f(n +) f(n +). f(n + ) ϕ 3 (,x) = 5 β 3(x+) β 3(x+)+ β 3(x) β 3(x ). ϕ 3 (,x) = β 3(x+) β 3(x+)+ 5 β 3(x) 5 β 3(x ). ϕ 3 (3,x) = β 3 (x ). ϕ 3 (,x) = β 3 (x+) β 3 (x+)+β 3 (x) β 3 (x ) Figure 6.6. ϕ 3 (,x) Figure 6.7. ϕ 3 (,x) Figure 6.8. ϕ 3 (3,x) Figure 6.9. ϕ 3 (,x) Applying Theorem 4 to any finite interval Ω, we have the following sampling formula. Corollary 5. Let X be the 3 rd order B-spline shift invariant subspace. Let Ω be any finite interval. There exists a sampling formula using integers and one arbitrary non-integer point

14 4 S. LI, Z. TONG AND D. YAN in Ω. In other words, let n and M be integers such that Ω [n,n +M]. Then for an arbitrary non-integer y such that n +y Ω, we have f X, f(t) = M+ j= f(j +n )ϕ 3 (j,t)+f(n +y )ϕ 3 (,t), t Ω. where ϕ 3 (j,t) = i Λ c ij β 3 (t i), j =,,, M +, ϕ 3 (,t) = i Λ d i β 3 (t i) for some finite sequences {c ij } and {d i } that exist with i Λ = [n,,n +M ]. Proof. By Theorem 4, there exist coefficients {c ij } and {d i } such that, for all f X, t Ω [n,n +M], and any non-integer point n +y Ω, f(t) = M+ c ij f(j +n )+d i f(n +y ) β 3 (t i) (6.) = = i Λ M+ j= M+ j= where we have assumed and j= f(j +n ) c ij β 3 (t i) +f(n +y ) i β 3 (t i) i Λ i Λd f(j +n )ϕ 3 (j,t)+f ( n +y ) ϕ 3 (,t), (6.) ϕ 3 (,t) = i Λ d i β 3 (t i), ϕ 3 (j,t) = i Λ c ij β 3 (t i), j =,,,M +. Remark. In practical applications, there maybe no need to find out the sampling reconstruction function {ϕ 3 (j,t)} M+ j=. Keeping the coefficients {c ij} and {d i } and using the reconstruction formula (6.) maybe equally or more convenient. (6.) may also be viewed as an average sampling formula whose reconstruction functions are simply the B-spline itself, while the coefficients are weighted averages of sampling values. For finitelengthsignals contained intheinterval [n,n +M], thesamplingreconstruction functions {ϕ 3 (j, )} M+ j= can be evaluated a priori. This is exceedingly direct forward. We

15 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES 5 show here an example with M = 8, n =. Note that there is a shifting nature in ϕ 3 (j, ), j 3. ϕ 3 (,x) = 5 β 3(x+) β 3(x+)+ β 3(x) β 3(x )+ β 3(x ) β 3(x 3) + β 3(x 4) β 3(x 5)+ β 3(x 6) β 3(x 7). ϕ 3 (,x) = β 3(x+) β 3(x+)+ 5 β 3(x) 5 β 3(x )+ 5 β 3(x ) 5 β 3(x 3) + 5 β 3(x 4) 5 β 3(x 5)+ 5 β 3(x 6) 5 β 3(x 7). ϕ 3 (3,x) = β 3 (x ) β 3 (x )+β 3 (x 3) β 3 (x 4)+β 3 (x 5) β 3 (x 6) +β 3 (x 7). ϕ 3 (4,x) = β 3 (x ) β 3 (x 3)+β 3 (x 4) β 3 (x 5)+β 3 (x 6) β 3 (x 7). ϕ 3 (5,x) = β 3 (x 3) β 3 (x 4)+β 3 (x 5) β 3 (x 6)+β 3 (x 7). ϕ 3 (6,x) = β 3 (x 4) β 3 (x 5)+β 3 (x 6) β 3 (x 7). ϕ 3 (7,x) = β 3 (x 5) β 3 (x 6)+β 3 (x 7). ϕ 3 (8,x) = β 3 (x 6) β 3 (x 7). ϕ 3 (9,x) = β 3 (x 7). ϕ 3 (,x) = β 3 (x+)+β 3 (x+) β 3 (x)+β 3 (x ) β 3 (x )+β 3 (x 3) β 3 (x 4)+β 3 (x 5) β 3 (x 6)+β 3 (x 7) Figure 6.. ϕ 3 (,x) Figure 6.. ϕ 3 (,x) 6.. Optimal choice of the non-integer point. We also should study the influence of the choice of the additional non-integer point, and see what value (half point, or some value close to an integer value) would it give rise to the least significant (small) reconstruction function ϕ 3 (, ). Theoretically, thesmallerthesignificant valueofϕ 3 (, ) is, thebetterandmorestablethe sampling formula will be. Because the sampling value at the non-integer point will become less significant if the corresponding reconstruction function ϕ 3 (, ) is the least significant (among all non-integer choices).

16 6 S. LI, Z. TONG AND D. YAN Figure 6.. ϕ 3 (3,x) Figure 6.3. ϕ 3 (4,x) Figure 6.4. ϕ 3 (5,x) Figure 6.5. ϕ 3 (6,x) Figure 6.6. ϕ 3 (7,x) Figure 6.7. ϕ 3 (8,x) Figure 6.8. ϕ 3 (9,x) Figure 6.9. ϕ 3 (,x) Theorem 6. Let X be the 3 rd order B-spline shift-invariant subspace, and let x be any noninteger point in an integer interval [n,n +M], M. Define C i (x) d i (a constant depending on x) for all i Λ, where {d i } is the set of coefficients determining ϕ as seen in the proof of Proposition 5. (I) Then C i (x) = C(x) is independent of i, and C(x). (II) The equality C(x) = holds if and only if x is a (and any) half-point in [n,n +M], M.

17 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES 7 Proof. We will first consider the proof of (I). Keep all the denotations the same as in the proof of Lemma. Then Direct calculations implies that A e M+ = (B P)e M+ = B (Pe M+ ) = B e M+. B e M+ = Since Ac = f, c = A f, it follows that d. d M+ M+ ( ) y + i e i. q p r i= = A e M+ = B e M+ = M+ q p r i+ ( ) y + i e i. Hence C i (x) d i = q p r C(x) is a constant (depending x) for all i Λ. (II). For any non-integer point x [n,n +M], by Lemma, we know q p r >, and d i = q p r = y( y) =. ( ) Clearly, C(x)achieves theminimumvalueifandonlyify = y =. Thatismin x d i = when x is a half-point. Remark. Similar analysis is possible for each given k, though the general scenario for all k remains an open topic. It is one of our subsequent work. A Few More Examples: In the next few figures, we demonstrate the numerical examples forlonger length inthecaseof orderk = 3andforhigherordersk > 3-justtoillustrate how easy such sampling formula can be established, and how sampling reconstruction functions may behave. To save on space, the figures will have the set of all sampling reconstruction functions {φ k (j, )} j plotted in one axis in the order of j index. They are meant to show the behavior of these sampling functions, while individual φ k (j, ) (for each j) may not always be clearly readable.

18 8 S. LI, Z. TONG AND D. YAN In Figure 6.3, the case of order k = 3 with longer dimension of Ω = 3 is demonstrated. Besides the integer point on [, 3], one additional point y =.5 is used. One of the purposes is to show the translation relationship among sampling reconstruction functions Figure 6.3. {ϕ3 (j, x)}j on [, 3], y =.5. In Figures 6.3, 6.3, 6.33, and 6.34, the set of sampling reconstruction functions {φk (j, )}j for the orders of k = 4, k = 5, k = 6 and k = 7 are also plotted, again in the orders of the index j Figure 6.3. {ϕ4 (j, x)}j on [, 5], y =.5, Figure 6.3. {ϕ5 (j, x)}j on [, 5], y =.5,.5, 4.5. Adaptivity and Stability. As we mentioned in the introduction, the sampling formula can be built adaptively with all (admissible) sampling points, even with randomly altered

19 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES Figure {ϕ 6 (j,x)} j on [, 5], y =.5,.5, 3.5, Figure {ϕ 7 (j,x)} j on [,], y =.3,.7, 5.5, 9.3, 9.7. sampling points. We performed an exhaust number of numerical tests with sampling points {x n } being altered by uniformly distributed random variables on ( 4, 4 ). Sampling reconstructions functions all behave stable. Showing in Figure 6.35 is one typical example for k = 4 over the interval [,5]. The sampling points are altered version of the almost-all integer sampling points (integers on [,5] and k = additional points at.5 and 4.5). They are x n =.84,.87,.8, 3.78, 4.594, 5.4,.3379, Figure {ϕ 4 (j,x)} j on [,5]. Sampling points are altered to become x n =.84,.87,.8, 3.78, 4.594, 5.4,.3379, 4.43.

20 S. LI, Z. TONG AND D. YAN 7. Conclusion A simple, adaptive and exact sampling formula is derived for B-spline Riesz sequence (shift invariant) subspaces X over any given interval I. The observation is that for order k B-spline subspace X, any interval [n,n +M] has M +k shifts of the k th -order splines involved. Consequently, M + k sampling points are required in the construction of a sampling formula. Since the closed interval [n,n +M] has M+ integer points, one needs (k+m ) (M+) = k additional samplingpointsatnon-integer valuesin(n,n +M). Note that the k non-integer sampling points needed is independent of the length M of the interval I. The choice of these points can be adaptive and/or adjustable. The case of order k = 3 is shown analytically that the integer-plus-any-non-integer sampling formula exists. We also show, in the case of order 3, that an optimal selection of the k = 3 = additional point is always a mid-point. The establishment of the sampling formulas amounts to the solution of (5.9). Since the choice of the k non-integer points does not depend on the length of interval I, we conjecture that a constructible similar sampling formula exists for the infinite dimensional B-spline Riesz sequence subspace X defined on R. This is one of our subsequent work. The analytical establishment of sufficient sampling conditions on integers and k non-integer points for general order k is also a subsequent work. References [] A. Aldroubi and H. Feichtinger. Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The l p -theory. Proc. Amer. Math. Soc., 6(9):677686, 998. [] A. Aldroubi and K. Gröchenig. Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces. J. Fourier Anal. Appl., 6, no. : pp 9,. [3] J. J. Benedetto. Irregular sampling and frames. Wavelets: A Tutorial in Theory and Applications, C.K. Chui, editor, Academic Press, Boston, pp445-57, 99. [4] J. J. Benedetto. Frame decompositions, sampling, and uncertainty principle inequalities. Wavelets: Mathematics and Applications, J. J. Benedetto and M. W. Frazier, editors, CRC Press Inc., Boca Raton, FL, Chapter 7, 994. [5] J. J. Benedetto and W. Heller. Irregular sampling and the theory of frames. Mat. Note, (suppl. ): pp. 3 5, 99. [6] O. Christensen. An Introduction to Frames and Riesz Bases. Birkhäuser, Boston, Basel, Berlin, 3. [7] C. K. Chui. An Introduction to Wavelets. Academic Press, Boston, 99. [8] I. Daubechies. Ten lectures on wavelets. 99. [9] C. de Boor, R. DeVore, and A. Ron. Approximation from shift-invariant subspaces of L (R d ). Trans. Amer. Math. Sco., 34: pp787 86, 994. [] C. de Boor, R. DeVore, and A. Ron. The structure of finitely generated shift-invariant subspaces in l. J. Func. Anal., no. 9:: 37 78, 994. [] H. G. Feichtinger and K. Gröchenig. Theory and practice of irregular sampling. Wavelets: Mathematics and Applications, J. J. Benedetto and M. W. Frazier, editors, CRC Press Inc., Boca Raton, FL, Chapter 8, 994.

21 ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES [] H. G. Feichtinger and K. Gröchenig. Theory and practice of irregular sampling. Wavelets: Mathematics and Applications, J. J. Benedetto and M. W. Frazier, editors, CRC Press Inc., Boca Raton, FL, Chapter 8, 994. [3] A. J. E. M. Janssen. The zak transform and sampling theorems for wavelet subspaces. IEEE Trans. on Signal Processing, vol. 4, n. :: , 999. [4] D. Li and M. Xue. Basis and Frames in Banach Spaces (in Chinese). The Science Press, Beijing, China, 995. [5] S. Li. Biorthogonal duals of b-spline reisz sequences via pseudoframes for subspace. preprint. [6] S. Li. Irregular sampling in a generic subspace. Samp. Theory Signal Image Process., 3, No. : pp 7, 4. [7] S. Li and H. Ogawa. Pseudoframes for subspaces with applications. J. Fourier Anal. Appl., June,, no. 4: pp 49 43, 4. [8] Y. Liu. Irregular sampling for spline wavelet subspaces. IEEE Trans. Infor. Theory, 4: pp63 67, 996. [9] R. Long. Multidimensional Wavelet Analysis. World Publishing Corporation, Beijing, China, 995. [] H. Ogawa. A unified approach to generalized sampling theorem. Proc. of IEEE-IECEJ-ASJ Int. Conf. on Acoustics, Speech, and Signal Processing, pages pp , April, 986. [] H. Ogawa. A generalized sampling theorem. Electronics and Communications in Japan, Vol. 7, No. 3: pp.97 5, March, 989. [] A. Ron and Z. Shen. Frames and stable bases for shift-invariant subspaces of L (R d ). Can. J. Math., 47(5): pp. 5 94, 995. [3] G. G. Walter. A sampling theorem for wavelet subspaces. IEEE Trans. Inform. Theory, vol. 38, no. : , 99. [4] X. Zhou and W. Sun. Frames and sampling theorem. Science in China, series A, vol. 4, no. 6:: 66 6, 998. [5] X. Zhou and W. Sun. On the sampling theorem for wavelet subspaces. J. Fourier Anal. Appl., vol. 5, no. 4:: , 999. Department of Mathematics, San Francisco State University, San Francisco, CA 943, USA address: shidong@sfsu.edu Graduate University of Chinese Academy of Sciences, Beijing 9, PRC address: tong @63.com Graduate University of Chinese Academy of Sciences, Beijing 9, PRC address: ydunyan@gucas.ac.cn