A Bernstein-like operator for a mixed algebraic-trigonometric space

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1 XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, septiembre 2009 (pp. 1 7) A Bernstein-like operator for a mixed algebraic-trigonometric space J. M. Carnicer 1, E. Mainar 2, J. M. Peña 1 1 Dpto. de Matemática Aplicada/IUMA, Universidad de Zaragoza, Zaragoza s: carnicer@unizar.es, jmpena@unizar.es. 2 Dpto. de Matemáticas, Estadística y Computación, Universidad de Cantabria. mainare@unican.es. Palabras clave: Bernstein operator, shape preserving representations, normalized B-basis Abstract Bernstein-like operators are useful to measure the approximation properties of a vector space of functions. Especially important for measuring the degree of approximation and describing these approximations are their spectral properties. A property usually required for these approximation spaces is that there exists a normalized totally positive basis because it allows us to provide shape preserving representations. The normalized B-basis of a space has optimal shape preserving properties. We present the normalized B-basis of the space T 1/2 generated by 1, t, cos t, sin t, cos(t/2), sin(t/2), computed using the techniques in the paper [2]. The spectral properties of the Bernstein-like operator associated to this space are discussed and indicate how close is the curve to its control polygon. The third greatest eigenvalue gives a rough idea of the approximation power of the space. For the classical Bernstein operator, the third eigenvalue is λ 2 = 0.8 and for the space T 1/2 we find λ In contrast to the classical Bernstein operator, whose corresponding Greville abscissae are strictly increasing, we find coincident abscissae in this case, leading to an example of a Bernstein-like operator which is not a bijection from the space to itself. 1 Introduction The Bernstein basis is optimal among all other shape preserving bases of the space of polynomials of degree not greater than n on a given compact interval (see [5]). Roughly speaking, this means that the curve represented by this basis is closer to its control polygon than with other kinds of representations for polynomial curves. In [6] it was proved that each space of functions admitting shape preserving representations (in the sense of [7]) 1

2 J. M. Carnicer, E. Mainar, J. M. Peña always has an optimal shape preserving basis called the normalized B-basis. For polynomial curves, the shape preserving representations exist on intervals of any length. However, for more general spaces interesting in curve design, it is not clear whether there exist shape preserving representations on a given interval (see [8]). In the last years, a growing interest in the design of curves in spaces mixing algebraic, trigonometric and hyperbolic functions has arisen. In [3] we have found all six dimensional spaces invariant under translations and reflections containing the first degree polynomials and the trigonometric functions cos t, sin t and admitting shape preserving representations on the interval [0, 2π] (this implies that a complete circular arc can be represented using a single control polygon). Among these spaces, we find spaces mixing trigonometric functions with two angular frequencies T w := 1, t, cos t, sin t, cos(wt), sin(wt), 0 < w < 1. In [4], we have chosen the space T 1/2 to show a procedure to obtain the optimal bases of T 1/2. We have also obtained exact representations of remarkable curves in design and engineering. The space T w is the set of solutions of the linear differential equation with constant coefficients L[u] = 0, where L is the linear differential operator L = D 6 +(w 2 +1)D 4 +w 2 D 2. The problem of analyzing Bernstein-like operators B[f] = f(t i )p i for the set of solutions of linear differential equations with constant coefficients has been studied by Aldaz, Kounchev and Render (cf. [1]). The functions p 0,..., p n form a basis that generalizes the Bernstein basis. In most cases it corresponds to the normalized B-basis of the space. If the operator fixes linear functions, then the points t 0,..., t n are called the Greville abscissae. An important question analyzed in this context is to find conditions that ensure that t 0 < < t n. On the other hand, examples where the Greville absissae are nondecreasing and some of them are coincident are interesting cases of limit spaces in this theory. We show that the space T 1/2 is such an example. In contrast to the classical Bernstein operator, whose corresponding Greville abscissae are strictly increasing, the fact that the Greville abscissae are coincident, provides an example of a degenerate operator in the sense that it is not a bijection from the space to itself. We also remark that this space arises in practical curve design problems as mentioned above. On the other hand, the spectral properties of the Bernstein-like operator associated to T 1/2 gives a rough idea of the approximation power of the space. The distance of the third greatest eigenvalue to one gives a measure of how close is the curve to its control polygon. For the classical Bernstein operator, the third eigenvalue is λ 2 = 0.8 and for the space T 1/2 we show λ Section 2 is devoted to introduce the normalized B-basis of the space T 1/2 obtained in [4]. We include a table of coefficients of remarkable functions in the space with respect to the normalized B-basis. Section 3 shows the spectral properties of the associated Bernsteinlike operator. 2

3 A Bernstein-like operator for a mixed algebraic-trigonometric space Figure 1: Normalized B-basis of T 1/2 on [0, 2π]. 2 The normalized B-basis of the space It is a well-known fact that the space 1, cos s, sin s, cos(2s), sin(2s) of trigonometric polynomials of degree 2 is an extended Chebyshev space on [0, 2π), that is, any nonzero function of the space has at most dim T 1/2 1 = 4 zeros (counting multiplicities). From this fact, it follows that there exists a normalized B-basis on the space T 1/2 on [0, 2π] as shown in [4]. The normalized B-basis (B 0,..., B 5 ) is given by and B 5 (t) := 1 (3t 8 sin(t/2) + sin t), 6π B 4 (t) := 1 4 (1 cos(t/2))2 1 (3t 8 sin(t/2) + sin t), 6π B 3 (t) := 1 2π (t sin t) 1 4 (1 cos(t/2))2, B 2 (t) := B 3 (2π t), B 1 (t) := B 4 (2π t), B 0 (t) := B 5 (2π t). Figure 1 shows the graphs of the functions of the normalized B-basis of T 1/2 on [0, 2π]. This basis can be used for the design of curves γ(t) = 5 P i B i (t), t [0, 2π], whose components are functions in T 1/2. The polygon P 0 P 1 P 5 is called the control polygon. The fact that the basis (B 0,..., B 5 ) is a normalized B-basis implies that the curves inherits common shape properties of the control polygon, analogously as in the case of Bézier curves. In [4], we have obtained exact representations of circles, complete cycloidal arcs, cardioids, deltoids, Descartes trifolium and limaçon of Pascal among other remarkable curves. 3

4 J. M. Carnicer, E. Mainar, J. M. Peña function c 0 c 1 c 2 c 3 c 4 c t 0 3π/4 3π/4 5π/4 5π/4 2π cos t sin t 0 3π/4 3π/4 3π/4 3π/4 0 cos(t/2) sin(t/2) 0 3π/8 3π/8 3π/8 3π/8 0 1 cos t t sin t π 2π 2π Table 1: Coefficients of relevant functions in T 1/2 Table 1 contains the coefficients with respect to the normalized B-basis of the functions 1 and t as well as usual trigonometric functions cos t, sin t, cos(t/2), sin(t/2), and other auxiliary functions used for the design of cycloids as the cycloidal cosine, 1 cos t, and the cycloidal sine, t sin t. 3 The Bernstein operator The spectral properties of Bernstein-like operators are crucial for analyzing the convergence properties. Moreover they provide a measure of the distance of the control polygon to the curve. The classical Bernstein operator on the space of polynomials of degree not greater than n ( ) n B[f] = f(i/n) (1 t) n i t i, t [0, 1], i can be generalized to other spaces of functions U, dim U = n + 1 with shape preserving representations: B[f] = f(t i )B i (t), t [a, b], where (B 0,..., B n ) is the normalized B-basis of U and t 0,..., t n are real values called nodes. If the space U contains the function t, then there exist unique real coefficients c 0,..., c n such that c i B i (t) = t, usually called Greville abscissae. An important property of Bernstein-like operators is the preservation of linear functions. We observe that if we choose as nodes the Greville abscissae t i = c i, then the resulting operator satisfies: B[1] = B i (t) = 1, B[t] = 4 t i B i (t) = t.

5 A Bernstein-like operator for a mixed algebraic-trigonometric space The equation n t ib i (t) = t determines the nodes, but does not imply that they form an increasing sequence. However, the fact that a t 0 t n b is desirable for certain purposes. On the one hand, this condition is related to sufficient conditions for the existence of Bernstein operators in certain spaces. On the other hand, we can give geometrical interpretations of the operator relating functions with curves in the plane. The graph of (t, B[f](t)) is a curve generated by the control polygon ( t0 f(t 0 ) ) ( t1 f(t 1 ) ) ( tn f(t n ) which is the graph of a continuous piecewise linear function if and only if the Greville absissae are strictly increasing. In addition, if the nodes are distinct and lie on the interval [a, b], a t 0 < < t n b, then the analysis of convergence and convexity-preserving properties of the operator can be performed. In Theorem 19 of [1], it is shown that there exists a Bernstein-like operator on spaces of exponential polynomials on intervals of sufficiently small length with a = t 0 < < t n = b. This result is used for proving convergence properties of the operator in Theorem 23 of [1]. Let us now restate Theorem 20 of [1]. Theorem 1 Let λ 0,..., λ n be complex numbers such that λ 0 = λ 1 is real. Suppose that the space of exponential polynomials ( d ) ( d ) E (λ0,...,λ n) := ker dt λ 0 dt λ n is closed under complex conjugation and that b a < π/m n, where M n := max{ Im λ i : i = 0,..., n}. Let (p (λ0,...,λ n),i),...,n denote a Bernstein-like basis (B-basis) of E (λ0,...,λ n). Then there exist unique nodes a = t 0 t n = b and unique positive numbers α 0,... α n such that the operator B (λ0,...,λ n)f = α i f(t i )p (λ0,...,λ n),i fixes the functions exp(λ 0 t) and exp(λ 0 t)t. From Theorem 9 of [1], it can be deduced that B i (t) := α i p (λ0,...,λ n),i(t), i = 0,..., n, is the normalized B-basis of E (λ0,...,λ n). If λ 0 = λ 1 = 0, then t 0,..., t n must coincide with the Greville abscissae. If we apply Theorem 20 of [1] to the space T 1/2 = E (0,0,0,0,i, i) on [0, l], we deduce that, if l < π, then the Greville abscissae form an increasing sequence. The space T 1/2 on [0, 2π] has Greville abscissae (see Table 1) t 0 = 0, t 1 = t 2 = 3π/4, t 3 = t 4 = 5π/4, t 5 = 1, ) 5

6 J. M. Carnicer, E. Mainar, J. M. Peña which is an increasing sequence notwithstanding that the length of the parameter domain is greater than π. On the other hand, this is an example where the Greville abscissae are not strictly increasing. The Bernstein operator B = B (0,0,0,0,i, i) for the space T 1/2 takes the form B[f] = f(0)b 0 + f(3π/4)(b 1 + B 2 ) + f(5π/4)(b 3 + B 4 ) + f(1)b 5, where B 0,..., B 5 is the normalized B-basis obtained in Section 2. Observe that B is an endomorphism of the vector subspace of T 1/2 generated by the functions 1, t, sin(t/2), sin t, whose normalized B-basis is given by (B 0, B 1 + B 2, B 3 + B 4, B 5 ). The linear functions are preserved by B, which means that 1, t are eigenfunctions corresponding to the eigenvalue λ 0 = λ 1 = 1. The function sin(t/2) = 3π 8 (B 1(t) + B 2 (t) + B 3 (t) + B 4 (t)) satisfies B[sin(t/2)] = sin(3π/8)(b 1 (t) + B 2 (t)) + sin(5π/8)(b 3 (t) + B 4 (t)) = sin(3π/8) sin(t/2), 3π/8 which means that sin(t/2) is an eigenfunction corresponding to the third eigenvalue λ 2 = sin(3π/8) Analogously, the function 8 3π sin t = 3π 4 (B 1(t) + B 2 (t) B 3 (t) B 4 (t)) satisfies B[sin t] = sin(3π/4)(b 1 (t) + B 2 (t)) + sin(5π/4)(b 3 (t) + B 4 (t)) = sin(3π/4) 3π/4 sin t, which means that sin t is an eigenfunction corresponding to the fourth eigenvalue λ 3 = 4 3π sin(3π/4) This completes the spectral analysis of this Bernstein-like operator. The space T 1/2 is not so simple as the space of polynomials and the corresponding Bernstein operator is singular. However the third eigenvalue is similar to the third eigenvalue of the classical Bernstein operator. So, this space can be used as a subtitute of the space of polynomials of degree less than or equal to 5 for simple approximation problems in which trigonometric functions have to be included. The spectral analysis reveals approximation properties similar to the space of polynomials of fifth degree and we may expect a good approximation of a curve through its control polygon. Acknowledgements Research partially supported by the Spanish Research Grant BFM , Gobierno de Aragón and Fondo Social Europeo. 6

7 A Bernstein-like operator for a mixed algebraic-trigonometric space References [1] J. M. Aldaz, O. Kounchev and H. Render, Bernstein operators for exponential polynomials, Constr. Approx. 29 (2009), [2] J. M. Carnicer, E. Mainar and J. M. Peña, Critical length for design purposes and extended Chebyshev spaces. Constr. Approx. 20 (2004), [3] J. M. Carnicer, E. Mainar and J. M. Peña, Shape preservation regions for six-dimensional spaces. Adv. Comput. Math. 26 (2007), [4] J. M. Carnicer, E. Mainar and J. M. Peña, Optimal bases of spaces with trigonometric functions. In: Tenth International Conference Zaragoza-Pau on Applied Mathematics and Statistics, Monografías del Seminario Matemático García de Galdeano (2009). [5] J. M. Carnicer and J. M. Peña, Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math. 1 (1993), [6] J. M. Carnicer and J. M. Peña, Totally positive bases for shape preserving curve design and optimality of B-splines. Comput. Aided Geom. Design 11 (1994), [7] T. N. T. Goodman, Shape preserving representations. In: Lyche T., Schumaker L. L., editors. Mathematical methods in CAGD, Boston: Academic Press; 1989, [8] J. M. Peña, Shape preserving representations for trigonometric polynomial curves. Comput. Aided Geom. Design 14 (1997),