Spherical Bessel functions and critical lengths

The critical length of a space of functions can be described as the supremum of the length of the intervals where Hermite interpolation problems are unisolvent for any choice of nodes. We analyze the critical length for spaces containing products of algebraic polynomials and trigonometric functions. We show the relation of these spaces with spherical Bessel functions and bound above their critical length by the first positive zero of a Bessel function of the first kind.


Introduction
Interpolation methods in spaces containing oscillating functions, such as the trigonometric functions, can fail in large intervals. An extended Chebyshev space on an interval is a finite dimensional space such that the Hermite interpolation problem has a unique solution for any choice of nodes. This property is equivalent to the fact that the number of zeros (counting multiplicities) of any nonzero function of the space is less than or equal to the dimension of the space. The critical length is the supremum of the lengths of the intervals where Hermite interpolation problems are unisolvent for any choice of nodes (see [6]). The critical length is also relevant in the construction of Bernstein-like operators [2,3] and in Computer-Aided Geometric Design [6][7][8].
Spaces containing oscillating functions can be used for design purposes since they represent exactly some curves and solutions of differential equations related with the physical nature of some problems. In particular, the spaces of solutions of differential equations with constant coefficients can be used to model versatile shapes of curves. These spaces are invariant under translations, which allows us to represent the same curve in different parameter intervals of the same length. In order to obtain shape preserving representations in spaces invariant under translations, the length of the parameter domain must be less than a given value, called the critical length for design purposes. In [6], is is shown that the critical length for design purposes is the critical length of the space of the derivatives.
The critical length of an (n + 1)-dimensional extended Chebyshev space containing the trigonometric functions cos x, sin x must be not greater than (n + 1)π. The cycloidal spaces C n , generated by algebraic polynomials of degree less than or equal to n − 2 and the trigonometric functions cos x, sin x, have been analyzed by many authors [2,3,6,10,12,14]. In [9], it was shown that the critical lengths of cycloidal spaces are related to zeros of Bessel functions of the first kind, given by (1) Some spaces invariant under translations can be described as the space generated by products of functions in other spaces. In particular, the space P n C 1 is the space generated by the functions x k cos x, x k sin x, k = 0, . . . , n. These spaces are the set of solutions of differential equations with constant coefficients. In this paper, we show that the fundamental solutions of these differential equations can be expressed in terms of Bessel functions. A description of a canonical basis in terms of Bessel functions is obtained. We prove that the critical length of the space P n C 1 is bounded above by j n+1/2,1 , the first positive zero of the Bessel function J n+1/2 . The paper is organized as follows. Section 2 describes the space P n C 1 , its fundamental solution and a canonical basis for this space. We focus on the relation of the basis with spherical Bessel functions. In Sect. 3 we discuss the critical lengths of P n C 1 and obtain an upper bound. In Sect. 4, we derive some formulae for simplifying the computation of some wronskians arising in the determination of the critical length. We use them to deduce that the critical length of the spaces P n C 1 is j n+1/2,1 for n = 0, 1. We have also evaluated several wronskians to confirm numerically that (P n C 1 ) = j n+1/2,1 for n = 2, 3. In Sect. 5, some applications to Computer Aided Design are discussed.

Fundamental solutions of the differential equation and Bessel functions
Let us recall that the wronskian matrix of a system of functions If a space of functions is the set of solutions of a linear differential equation with constant coefficients, then the wronskian matrix of any basis is nonsingular at any point. A canonical basis at the origin is any basis The last element of a canonical basis is a multiple of the fundamental solution of the linear differential equation. The fundamental solution can be defined as the unique element φ n in the space satisfying Canonical bases play a relevant role in the field of total positivity [11] and can be used to compute the critical length of an extended Chebyshev space. Let us denote by D f = f , the derivative operator. The 2(n + 1)-dimensional space of solutions of the differential equation is generated by the functions cos x, sin x, x cos x, x sin x, . . . , x n cos x, x n sin x (4) and can be described as the space P n C 1 , generated by the set of products of a function of the space P n of polynomials of degree less than or equal to n and a trigonometric function of the space C 1 = cos x, sin x . Spherical Bessel functions can be defined by the Rayleigh's formula (formula 10.1.25 of [1]) In particular, we have and the following recurrence relation holds The spherical Bessel function j n satisfies the second order self-adjoint differential equation and can be related with the Bessel function of the first kind (1) with index ν = n + 1/2 by the following formula (see 10 Let j ν,k denote the k-th positive zero of the Bessel function J ν . Then j n+1/2,1 is the first positive zero of the spherical Bessel function j n . For any nonnegative integer n, we define and we have that Now, we derive some properties of f n to show that the fundamental solution of the differential equation (3) can be expressed in terms of spherical Bessel functions. We shall use the double factorial notation where (k − 1)/2 is the greatest integer less than or equal to (k − 1)/2.

Proposition 1
The following properties hold for the functions f n defined in (7): where c n is an odd polynomial in P n and s n is an even polynomial in P n . Hence f n is an odd function in P n C 1 .
(e) f n is a solution of the second order differential equation (g) f n has a zero of multiplicity 2n + 1 at the origin. Moreover, and so, the fundamental solution of the equation Proof (a) From (6), it follows that (b) It follows by induction. The result is clear for n = 0, since f 0 (x) = sin x. Let us assume that (b) holds for n. Then we have by (a) that Since c n , s n ∈ P n are respectively odd and even polynomials, we deduce that are odd and even polynomials in P n+1 , respectively. (c) It follows by induction. Clearly f 1 (x) = x sin x = x f 0 (x). Assuming that (c) holds for n ≥ 1, we have from (a) that Differentiating and applying the induction hypothesis, we use (a) to deduce that and, since f n (0) = 0, we also deduce that f n (x) = x 0 t f n−1 (t)dt. (d) follows directly from (a) and (c). (e) Differentiating (a) and using (c), we derive So, (e) follows. (f) The first statement readily follows from (e) and (c). The second statement follows by induction on n. .
Assuming that we deduce that A canonical basis of P n C 1 at the origin can be obtained from the fundamental solution n , . . . , φ n , φ n ). From Proposition 1 (g), the next result follows.
Proposition 2 A canonical basis of P n C 1 at the origin is given by the functions f 0 , f 0 , f 1 , f 1 , . . . , f n , f n .

An upper bound for critical lengths of P n C 1
Let us recall that an (n + 1)-dimensional space of C n (I ) functions defined on the interval I is an extended Chebyshev space if the number of zeros, counting multiplicities, of any nonzero function of the space is less than or equal to n.
The space P n C 1 is invariant under translations because it is the set of solutions of a differential equation of order 2n + 2 with constant coefficients. For this kind of spaces a critical length can be defined as follows. If the space is the set of solutions of a differential equation of order 2n + 2 whose characteristic polynomial is an even polynomial, then it is invariant under reflections in the sense that if u ∈ U , then x → u(ξ − x) also belongs to U (see Sect. 3 of [6]). Let us observe that P n C 1 is the space of solutions of the differential Eq. (3), whose characteristic polynomial is (λ 2 + 1) n+1 . So, the space P n C 1 is invariant under reflections.
The following result can be used to compute critical lengths of finite dimensional spaces of differentiable functions invariant under reflections.
By Proposition 3.2 (ii) of [6], U is an extended Chebyshev space on each interval of length less than α, that is, (U ) ≥ α. Since w j,n (α) = 0 for some j > n/2, we deduce from  i (0). Then, we can apply the result to the basis (s 0 u 0 , . . . , s n u n ). Since w j,n (x) = det W (u j , u j+1 , . . . , u n )(x) = s j s j+1 · · · s n det W (s j u j , s j+1 u j+1 , . . . , s n u n )(x), we conclude that the wronskian associated to both bases (u 0 , . . . , u n ) and (s 0 u 0 , . . . , s n u n ) coincide up to a sign and both have the same set of zeros. Then the result follows.
In the following result, we show that the critical length of P n C 1 is not greater than the first positive zero of the spherical Bessel function j n .

Theorem 1 The critical length of the space P n C 1 is the first positive zero of the functions
Moreover, Proof Since P n C 1 is invariant under translations and reflections, the first part of the statement follows from Proposition 3. The wronskian function w 2n+1,2n+1 (x) = f n (x) = x n+1 j n (x) coincides with the last function of the canonical basis of P n C 1 . Therefore, the critical length must be less than or equal to its first positive zero.
Observe that C 2n+2 and P n C 1 are both 2(n + 1)-dimensional spaces containing trigonometric functions and This implies that Hermite interpolation problems on cycloidal spaces can be posed on longer intervals than the same kind of problems on spaces P n C 1 of the same dimension.

Critical lengths of the spaces P n C 1 , for n ≤ 1
Let us first show that the function has no positive zeros.

Proposition 4
For any x > 0, the following inequalities hold Proof (a) Let us show that Differentiating, we have that Using Proposition 1 (e) and Proposition 1 (c) we deduce that Dividing by x 2n−1 we have that Hence x −2(n−1) w(x) is a nondecreasing function. Now, we use Proposition 1 (c) and (g) to deduce that Differentiating and using Proposition 1 (f), we get So v(x) is a strictly increasing function on (0, +∞). By Proposition 1 (g), we have and then v(x) is positive in (0, +∞).
Let us observe that the proof of Proposition 4(c) is based on Proposition 4(b), which corresponds to the following Turán type inequality for Bessel functions [4,13]). We have included a proof based on the properties shown in Proposition 1 for the sake of completeness.
By Proposition 3, it follows that (P 0 C 1 ) is the first positive zero of the function that is, If n = 1, we can use Proposition 3 to show that (P 0 C 1 ) is the first positive zero of the functions By Proposition 4 (c), det W ( f n , f n )(x) does not vanish on (0, +∞). So The difficulty of the analysis of the sign of w j,2n+1 increases when j < 2n. However, we conjecture that the least positive zero of the functions w j,2n+1 is attained for the function w 2n+1,2n+1 = f n , or equivalently, that (P n C 1 ) = j n+1/2,1 for n > 1.
We have computed several wronskians for low degree n and they confirm our conjecture at least for n = 2, 3, as shown in Figure 1. We have depicted the graphs of the wronskians, conveniently normalized and divided by a factor of the form x k . The graphs show that the first positive zero of the wronskians is attained for f n = w 2n+1,2n+1 , giving rise to (P 2 C 1 ) = j 3/2,1 ≈ 5.763459196842433, (P 3 C 1 ) = j 3/2,1 ≈ 6.987924414992913.

Applications to computer-aided design
Most design tools describe curves in the parametric form where u i : [a, b] → R, i = 0, . . . , n, are nonnegative functions such that n i=0 u i (t) = 1 for all t ∈ [a, b]. In order to avoid redundancy, u 0 , . . . , u n are required to be linearly independent and they form a basis of a space U = u 0 , . . . , u n . The polygon P 0 · · · P n is called the control polygon of γ . Usually, such parametric representation of curves is shape preserving, in the sense that the shape of the curve imitates the shape of its control polygon. Shape preserving representations of curves are associated to the fact that the system of functions (u 0 , . . . , u n ) is normalized and totally positive [5].
A totally positive matrix is any matrix such that all its minors are nonnegative. A system (u 0 , . . . , u n ) of functions defined on an interval I , u i : I → R, i = 0, . . . , n, is totally positive if any collocation matrix (u j (t i )) i, j=0,...,n is a totally positive matrix for any t 0 < · · · < t n in I . The system is normalized if all functions add up to one, that is n i=0 u i (t) = 1, for all t ∈ I .
The spaces U where curves are designed might be invariant under translations. If they contain trigonometric functions, they do not possess normalized totally positive bases on intervals of arbitrary length (see Section 6 of [6]). This motivates the following definition. In Corollary 4.1 of [6] it was shown that the critical length for design purposes can be expressed in terms of the critical length. For a given a space U ⊂ C[a, b], let D −1 U := {u ∈ C 1 [a, b]|u ∈ U }. Then the results in the previous sections can be interpreted in terms of the existence of shape preserving representations of curves in the space D −1 (P n C 1 ).

Proposition 6
A canonical basis at the origin of D −1 (P n C 1 ) = P 0 ⊕ (P n C 1 ) is given by The critical length for design purposes satisfies (D −1 (P n C 1 )) ≤ j n+1/2,1 .
Proof Since D(P 0 ⊕ (P n C 1 )) = D(P n C 1 ) = P n C 1 and dim(P 0 ⊕ (P n C 1 )) = 2n + 3 = dim(P n C 1 ) + 1, we deduce that D −1 (P n C 1 ) = P 0 ⊕ (P n C 1 ). By Proposition 1 (g), we deduce that x 0 f n (t)dt has a zero of multiplicity 2n + 2 at x = 0 and deduce that the given basis is canonical at the origin. Finally, we conclude from Theorem 1 and Proposition 5 that the critical length for design purposes of D −1 (P n C 1 ) satisfies (D −1 (P n C 1 )) = (P n C 1 ) ≤ j n+1/2,1 .