# fachbereich mathematik funktionalanalysis und

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fachbereich mathematik funktionalanalysis und

SCHRIFTEN ZUR FUNKTIONALANALYSIS UND GEOMATHEMATIK M. J. Fengler, W. Freeden, M. Gutting The Spherical Bernstein Wavelet Bericht 20 – Mai 2005 FACHBEREICH MATHEMATIK THE SPHERICAL BERNSTEIN WAVELET M. J. FENGLER∗ , W. FREEDEN∗ , M. GUTTING∗ Abstract. In this work we introduce a new bandlimited spherical wavelet: The Bernstein wavelet. It possesses a couple of interesting properties. To be specific, we are able to construct bandlimited wavelets free of oscillations. The scaling function of this wavelet is investigated with regard to the spherical uncertainty principle, i.e., its localization in the space domain as well as in the momentum domain is calculated and compared to the well-known Shannon scaling function. Surprisingly, they possess the same localization in space although one is highly oscillating whereas the other one shows no oscillatory behavior. Moreover, the Bernstein scaling function turns out to be the first bandlimited scaling function known to the literature whose uncertainty product tends to the minimal value 1. Key words. Spherical Wavelets, Bernstein Polynomials, Constructive Approximation on the Sphere, Uncertainty Principle AMS subject classifications. 33C55, 42C10, 42C15, 42C40, 65T99 1. Motivation. Over the last decade wavelets have found important applications in numerous areas of mathematics, physics, engineering and computer science. Wavelets form versatile tools for representing general functions or data sets. They especially become more and more important in Earth sciences since most recent satellite missions deliver millions of data scattered around the globe. Meanwhile spherical wavelets introduced by [5, 7, 8, 9] play the fundamental role in the analysis of regional, high-frequent phenomena observed in geophysical, geodetic, meteorological and seismological applications (see, e.g., [2, 4, 5, 6] and the many references therein). The spherical wavelets discussed here are based on expansions of Legendre polynomials. Hence, they form radial basis functions on the sphere whose argument depends only on the spherical distance between the localization of the wavelet and its evaluation point. Moreover, they are usually designed to fulfill the partial differential equations corresponding to the considered geophysical context (see [9]). In view of constructive approximation on the sphere the wavelet techniques have shown their strength in detecting local and regional (also time-dependent) effects (see [5, 6]) for more than one decade. For example, studies of local changes of mass in the gravitational field observed from satellite data or crustal field modeling from Earth’s magnetic data represent recent examples of application. Also vectorial data are analyzed by an intrinsic extension of scalar wavelets to vectorial wavelets, e.g., for modeling atmospherical flows and oceanographic streams. Along this experience the wavelet techniques have been extended to (non-linear) Galerkin methods for solving spherical partial differential equations, such as the Navier-Stokes equation (see [2, 3]). In the literature bandlimited and non-bandlimited wavelets are distinguished. Bandlimited wavelets are generated by a finite series of Legendre polynomials whereas for non-bandlimited wavelets this Legendre expansion is infinite. Examples for both types can be found in [5, 6, 9]. However, due to the construction as series of finitely many Legendre polynomials in frequency domain, it has been common belief that it is not possible to build spherical bandlimited wavelets that show no oscillations (except for one that is required to obtain a vanishing zero moment). Just at this point this work comes into play. We introduce a smooth, bandlimited, spherical wavelet: The spheri∗ University of Kaiserslautern, Department of ({fengler/freeden/gutting}@mathematik.uni-kl.de). 1 Mathematics, Geomathematics Group 2 M. J. FENGLER, W. FREEDEN, M. GUTTING cal Bernstein wavelet. Its great advantage is that it possesses the appealing property of fast and stable numerical evaluation because it can be formulated in closed form by an elementary function. In fact, it has been constructed via its explicit formula, i.e., in the space domain whereas usually spherical wavelets are generated in the momentum domain via its Legendre coefficients. This different type of construction is also reflected by the proofs in this article – many properties of the Bernstein wavelet (especially its localization properties) are shown differently as for other well-known wavelets. As the Bernstein wavelet shows no oscillatory behavior in space domain it is also much easier to derive error estimates for spatial truncation as required in many numerical procedures, e.g., spherical panel clustering [10]. The paper is organized as follows: First, we briefly introduce the mathematical tools and notations that are needed later on. Second, we define the Bernstein wavelet in the space domain in closed form and prove that all essential properties of spherical wavelets are fulfilled. Surprisingly, when comparing the Bernstein scaling function with kernels known to the literature, we realize that its space uncertainty is exactly the same as in the Shannon case despite the huge oscillations of the latter. Finally, we show that the uncertainty product (known as Heisenberg’s uncertainty product) converges to 1 in case of the bandlimited Bernstein wavelet. 2. Preliminaries. In the following we adopt the notation from the monograph [9]. The letters N, N0 and R denote the set of positive integers, non-negative integers and real numbers. We write x, y to represent the elements of the three-dimensional space R3 endowed with the Euclidian canonical basis {ε1 , ε2 , ε3 }. Then the inner P3 product is denoted by x · y = i=1 xi yi . The corresponding norm of x is given by √ |x| = x · x. The unit sphere is denoted by Ω, elements of it are usually written by Greek letters, e.g., ξ or η. As customary the space of all real, square-integrable functions F on Ω is called L2 (Ω). L2 (Ω) is a Hilbert space endowed with the inner product Z hF, GiL2 (Ω) = F (ξ)G(ξ) dω(ξ), Ω and associated norm kF k L2 (Ω) = Z Ω 1/2 F (ξ) dω(ξ) . 2 2 It is well known that L (Ω) possesses an orthonormal basis consisting of the realvalued spherical harmonics {Yn,k }n=0,...;k=−n,...,n of degree n and order k as defined, for example, in [14, 15]. Then the spherical addition theorem n X 2n + 1 Pn (ξ · η) = Yn,k (ξ)Yn,k (η) 4π k=−n connects the spherical harmonics Yn,k of degree n and the Legendre polynomial Pn (for more details see [1, 9, 13]). Scaling Functions and Wavelets. First of all we recapitulate the definition of a so-called generator of a scaling function. As a matter of fact, the choice of this generator determines all properties of the spherical scaling function and its corresponding wavelet. THE SPHERICAL BERNSTEIN WAVELET 3 Definition 2.1 (Generator of a Scaling Function). A family {{Φ∧ J (n)}n∈N0 }J∈N0 is called a generator of a scaling function, if it satisfies the following requirements: 1. For all J ∈ N0 Φ∧ J (0) = 1, (2.1) 2. for all J, J 0 ∈ N0 with J ≤ J 0 and all n ∈ N ∧ 0 ≤ Φ∧ J (n) ≤ ΦJ 0 (n), (2.2) lim Φ∧ J (n) = 1. (2.3) 3. for all n ∈ N J→∞ For fixed J ∈ N0 the sequence {Φ∧ J (n)}n∈N0 is called the symbol of the corresponding scaling function ΦJ of scale J. According to [9] this scaling function of scale J is defined by ΦJ (ξ, η) = ∞ X Φ∧ J (n) n=0 2n + 1 Pn (ξ · η), 4π while the (linear) wavelet of scale J reads as follows ΨJ (ξ, η) = ∞ X Ψ∧ J (n) n=0 2n + 1 Pn (ξ · η), 4π ∧ ∧ ∧ where Ψ∧ J (n) is given via the linear refinement equation ΨJ (n) = ΦJ+1 (n) − ΦJ (n), n ∈ N0 . The whole set {ΦJ }J∈N0 is called a scaling function, the set {ΨJ }J∈N0 is called ∧ a wavelet. If for all scales J the symbol {Φ∧ J (n)}n∈N0 , respectively {ΨJ (n)}n∈N0 , is different from zero only for finitely many values of n, we call the corresponding scaling function {ΦJ }J∈N0 , respectively the wavelet {ΨJ }J∈N0 , bandlimited. The most simple representant of a scaling function is the well-known Shannon scaling function (see [5]). Definition 2.2 (Shannon Scaling Function). Let J ∈ N0 and η ∈ Ω be fixed. Then the Shannon scaling function of scale J is given by ΦSJ (ξ, η) = J 2X −1 ΦS∧ J (n) n=0 2n + 1 Pn (ξ · η), 4π where ( 1 ΦS∧ J (n) = 0 if n ∈ [0, 2J ), else. Obviously, its generator {{ΦS∧ J (n)}n∈N0 }J∈N0 fulfills all three conditions of Definition 2.1 and the Shannon scaling function is bandlimited. Space Localization. We start by formulating the space localization property mathematically: Suppose that F is real-valued and of class L2 (Ω). Moreover, assume without loss of generality, that kF kL2 (Ω) = 1. Then, we follow [5, 9, 16] and associate 4 M. J. FENGLER, W. FREEDEN, M. GUTTING to F the normal (radial) field η 7→ ηF (η), η ∈ Ω. This allows us to introduce the expectation value of F in the space domain by Z eF = ηF (η)2 dω(η), (2.4) Ω 3 such that eF ∈ R . In that sense, F (η)2 can be interpreted as probability density function with respect to the measure dω(η). Immediately, we observe that |eF | ≤ 1. Additionally, the variance in the space domain σF2 is given by Z 2 σF = (η − eF )2 F (η)2 dω(η). (2.5) Ω Hereby, σF can be regarded as the standard deviation of F . Let us clarify its geometrical meaning: Assume that eF 6= 0, then F localizes in the spherical cap C = {η ∈ Ω|1 − η · eF /|eF | ≤ 1 − |eF |}, with radius σF . Clearly, we say F is “space localized” if the ratio σF ∆F = |eF | (2.6) between the diameter σF of the spherical cap C and the absolute value of the expectation value |eF | is small (a table with examples can be found in [2]). For a concise description of the space localization as well as the uncertainty principle on the sphere we refer the reader to [5, 9, 16]. According to [9] we can simplify the expression for the expectation value eF if F is a radial basis function. In this case, we assume without loss of generality that F is localized at ε3 = (0, 0, 1)T . Then (2.4) reads as Z 1 eF = 2π tF (t)2 dt ε3 . (2.7) −1 and following [9] the variance in (2.5) can be rewritten as σF2 = 1 − |eF |2 . (2.8) Localization in Momentum. In order to define the localization in momentum we follow the concepts of [5] and [16]. Thus, we consider the expectation value in the momentum domain which is in fact the expectation value of the surface curl operator L∗ on the sphere Ω. Suppose again that F is real valued with kF kL2 (Ω) = 1, but this time F is an element of a space of Sobolev type H2 (Ω), i.e., for F ∈ H2l (Ω) with l ∈ N there exists a function G ∈ L2 (Ω) whose Fourier coefficients G∧ (n, k) fulfill G∧ (n, k) = (−n(n + 1))l F ∧ (n, k) for all n = 0, 1, . . . ; k = −n, . . . , n. For a concise introduction to spherical Sobolev spaces see, e.g., [5, 6, 9]. The expectation value in the momentum domain of the function F is given by Z ∗ (2.9) eL = (L∗η F (η))F (η) dω(η), F Ω ∗ 3 which by definition leads to the value eL F = 0 ∈ R . Its variance in the momentum L∗ 2 domain σF is defined as 2 ∗ 2 Z ∗ ∗ )F (η) = σFL dω(η) (Lη − eL F Ω Z (2.10) = (−∆∗η F (η))F (η) dω(η), Ω THE SPHERICAL BERNSTEIN WAVELET 5 where the last equation easily results from the surface theorem of Stokes. Again we can simplify this expression in the case of radial basis functions assuming without loss of generality that F is localized in ε3 Z 1 ∗ 2 F (t)Lt F (t) dt, (2.11) = −2π σFL −1 d 2 d dt (1−t ) dt where Lt = denotes the Legendre operator. The localization in momentum ∗ ∗ L∗ ∆L is measured by the square root of the variance, i.e., ∆L F F = σF . In consequence, we are able to state the uncertainty principle for spherical functions (cf. [5, 9, 16]) which says that sharp localization in the space domain as well as in the momentum domain is not possible. Theorem 2.3 (Uncertainty Principle). Let F ∈ H2 (Ω) with kF kL2 (Ω) = 1. Then ∗ 2 2 (σF ) σFL ≥ |eF |2 and (if eF 6= 0) ∗ ∆ F ∆L F ≥1 (2.12) The left hand side in (2.12) is called uncertainty product. To avoid that the expectation value in momentum domain is vector valued and equal to 0 for all functions we start over considering the negative Beltrami operator −∆∗ instead of L∗ . Therefore, we have to assume that F ∈ H4 (Ω) with kF kL2 (Ω) = 1. This leads to the expectation value of −∆∗ Z ∗ 2 ∗ e−∆ −∆∗η F (η) F (η) dω(η) = σFL = (2.13) F Ω which turns out to be the same as the variance of L∗ (see (2.10)). The variance with respect to −∆∗ is defined as Z 2 2 ∗ ∗ 2 (−∆∗ )2 −∆∗ F (η) − e (2.14) dω(η) = e = σF−∆ (−∆∗η ) − e−∆ F F F Ω and the localization in momentum in terms of these expressions reads as follows (if −∆∗ eF 6= 0) σF−∆ ∗ ∆−∆ = F (−∆∗ )2 eF ∗ −(e−∆ F −∆ eF ∗ ∗ 2 ) 12 12 ∗ ∗ = ∆L = e−∆ F . F (2.15) Note that this quantity does not change. Thus, the uncertainty principle (2.12) is left unmodified. 3. The Bernstein Wavelet. In this section we introduce the Bernstein scaling function and the corresponding Bernstein wavelet. The name Bernstein is motivated J −1 by the fact that it is proportional to the Bernstein polynomial B22J −1 on the interval [−1, 1] (see, for instance, [17]). Later on, we also derive certain properties of this kernel. Definition 3.1 (Bernstein Scaling Function). Let η ∈ Ω be fixed. The Bernstein scaling function of scale J ∈ N0 is given by 2J −1 2J−2 1 + ξ · η B ΦJ (ξ, η) = . (3.1) π 2 6 M. J. FENGLER, W. FREEDEN, M. GUTTING As usual in linear wavelet theory, we introduce the Bernstein wavelet of a certain scale by differences of scaling functions of two consecutive scales. Definition 3.2 (Bernstein Wavelet). Let η ∈ Ω be fixed. The Bernstein wavelet of scale J ∈ N0 is given by B B ΨB J (ξ, η) = ΦJ+1 (ξ, η) − ΦJ (ξ, η). Clearly, we are able to understand ΦB J as bandlimited radial basis function with , such that symbol {ΦB∧ (n)} n∈N0 J ΦB J (ξ, η) = ΦB J (t) = J 2X −1 ΦB∧ J (n) n=0 2n + 1 Pn (t), 4π t = ξ · η. Interestingly, we find an explicit representation of the symbol by the following theorem. Theorem 3.3 (Symbol of the Bernstein Scaling Function). Suppose that J ∈ N0 . Then the scaling function ΦB J (·, η) of scale J possesses the symbol given by ( (2J )!(2J −1)! if n ∈ [0, 2J ), J −n−1)!(2J +n)! B∧ (2 (3.2) ΦJ (n) = 0 else. Before going into the details of the proof of Theorem 3.3, we borrow the following recursion formulae for the Legendre polynomials from [9, 13]: First, suppose that n ≥ 0 then (n + 1)Pn+1 (t) + nPn−1 (t) − (2n + 1)tPn (t) = 0, (3.3) for t ∈ [−1, 1] with P−1 (t) = 0. Second, let n ≥ 0, then 0 Pn0 (t) − Pn−2 (t) = (2n − 1)Pn−1 (t), (3.4) for t ∈ [−1, 1] with P−1 (t) = P−2 (t) = 0. Based on these recursion formulae we are able to verify Theorem 3.3. Proof. Formally, the symbol of ΦB J can be computed by solving the integral expression given by ΦB∧ J (n) = 2π Z 1 −1 2J−2 π 1+t 2 2J −1 Pn (t) dt. For the following considerations we use the following abbreviation Z 1 In,k = (1 + t)k Pn (t) dt. −1 Straightforward integration yields I0,k = 2k+1 , k+1 I1,k = I0,k k . k+2 From the recurrence formula (3.3) we obtain (n ≥ 1) (n + 1)In+1,k + nIn−1,k − (2n + 1)In,k+1 + (2n + 1)In,k = 0. (3.5) 7 THE SPHERICAL BERNSTEIN WAVELET 6 0 10 −5 10 5 −10 10 4 −15 10 3 −20 10 −25 10 2 −30 10 1 −35 10 −40 10 0 10 20 30 n 40 50 60 70 0 −4 Fig. 3.1. Non-zero elements of the B B symbol of ΦB 4 (stars), Φ5 (dots), and Φ6 (pluses). −3 −2 −1 0 θ 1 2 3 4 Fig. 3.2. Bernstein scaling funcB tion ΦB 4 (cos θ) (dashed), Φ5 (cos θ) (dotted), and ΦB (cos θ) (solid). 6 6 0 10 −10 5 10 −20 10 4 −30 10 3 −40 10 2 −50 10 1 −60 10 0 −70 10 −80 10 0 20 40 60 n 80 100 120 140 Fig. 3.3. Non-zero elements of the B B symbol of ΨB 4 (stars), Ψ5 (dots), and Ψ6 (pluses). −1 −4 −3 −2 −1 0 θ 1 2 3 4 Fig. 3.4. Bernstein wavelet B ΨB 4 (cos θ) (dashed), Ψ5 (cos θ) (dotted), B and Ψ6 (cos θ) (solid). Furthermore, we find by partial integration and (3.4) that (2n + 1)In,k+1 = −(k + 1)(In+1,k − In−1,k ). Combining these results we arrive at the following recursion formula: (n + k + 2)In+1,k = −(2n + 1)In,k + (k + 1 − n)In−1,k . By letting k = 2J − 1 this gives us the following result. ΦB∧ J (0) = 1, 2J − 1 , 2J + 1 B∧ J B∧ (n + 2J + 1)ΦB∧ J (n + 1) = −(2n + 1)ΦJ (n) + (2 − n)ΦJ (n − 1). ΦB∧ J (1) = By induction over n we obtain the desired result. These considerations also help us to compute an explicit representation of the L2 (Ω)norm of the Bernstein scaling function. 8 M. J. FENGLER, W. FREEDEN, M. GUTTING Lemma 3.4 (Norm of the Bernstein Scaling Function). Let J ∈ N0 and η ∈ Ω be fixed. Then the L2 (Ω)-norm of the Bernstein scaling function ΦB J of scale J is given by 2J−1 kΦB . J (·, η)kL2 (Ω) = p π(2J+1 − 1) (3.6) Proof. The squared norm of ΦB J can be evaluated by aid of 2 kΦB J (·, η)kL2 (Ω) 1 = 2π Z 1 = 2π Z −1 −1 2J −1 !2 2J−2 1 + t dt π 2 2J+1 −2 22J−4 1 + t P0 (t) dt. π2 2 Clearly, the latter integral is of the same type as in (3.5). Thus, we find by analogous considerations and k = 2J+1 − 2 that 2 kΦB J (·, η)kL2 (Ω) = 22J−2 . π(2J+1 − 1) We denote the L2 (Ω)-normalized Bernstein scaling function of scale J by Φ̃B J (·, η) = ΦB J (·, η) . B kΦJ (·, η)kL2 (Ω) Based on the results obtained so far, we are now able to prove that the Bernstein scaling function forms an approximate identity. Approximate Identity. Loosely spoken an approximate identity is a family of kernels depending on a parameter which converges to the Dirac distribution as this parameter tends to a limit value. In our case of scaling functions this parameter is the scale J which tends to infinity. The third property of the generator, i.e., (2.3) in Definition 2.1, ensures that any scaling function naturally forms an approximate identity. In conclusion, we have to show in the following that this limit holds true for the Bernstein scaling function. Theorem 3.5 (Approximate Identity). The Bernstein scaling function {ΦB J }J∈N0 forms an approximate identity. Proof. It is sufficient to prove that limJ→∞ ΦB∧ J (n) = 1 for fixed n ∈ N0 . To accomplish this we take the logarithm of (3.2). Then J log ΦB∧ J (n) = 2 X log k + k=1 = = J 2X −1 k=1 J 2 X log k − k=2J −n J n X log k=1 log k − J 2X +n 2J X −n−1 k=1 log k − J 2X +n log k k=1 log k k=2J 2 −k 2J + k . (3.7) THE SPHERICAL BERNSTEIN WAVELET 9 Since n ∈ N0 is fixed, the series in (3.7) is finite. Then we obtain lim J→∞ log ΦB∧ J (n) = lim J→∞ n X log k=1 2J − k 2J + k = 0, and, therefore, limJ→∞ ΦB∧ J (n) = 1. Monotonicity. Since the first property (2.1) constituting a scaling function is obviously fulfilled by ΦB∧ J (0), we are left with the monotonicity condition (2.2). Theorem 3.6 (Monotonicity). The generator of the Bernstein scaling function 0 0 {ΦB J }J∈N0 is monotonically increasing, i.e., for all J, J ∈ N0 with J ≤ J and all n ∈ N we have B∧ 0 ≤ ΦB∧ J (n) ≤ ΦJ 0 (n). Proof. From Theorem 3.3 it is clear that 0 ≤ ΦB∧ J (n) for all J ∈ N0 and all n ∈ N. Therefore, it remains to prove the inequality between two different scales. It suffices to show that, for fixed n ∈ N0 and arbitrary J, J 0 ∈ N0 with J ≤ J 0 , the following ratio holds ΦB∧ J (n) ≤ 1. ΦB∧ J 0 (n) Similar to Theorem 3.5 we consider the logarithms and obtain ! 0 n X (2J − k)(2J + k) ΦB∧ J (n) = log log B∧ (2J + k)(2J 0 − k) ΦJ 0 (n) k=1 which we want to be less or equal to 0. Thus, it is sufficient to show that the argument of each logarithm in this sum is smaller than 1. This can be easily seen since from 0 J ≤ J 0 follows 2J ≤ 2J and therefore, 0 0 0 2J+J − k 2 + k(2J − 2J ) (2J − k)(2J + k) ≤ 1. = 0 (2J + k)(2J − k) 2J+J 0 − k 2 + k(2J 0 − 2J ) Now all conditions of Definition 2.1 are guaranteed and we can rightly speak of the Bernstein scaling function. 4. Oscillations. Throughout this section we are concerned with the oscillations of the Bernstein and Shannon scaling function, respectively. Before going into the mathematical details we like to motivate our considerations by comparing in Fig. S B S 4.1 and Fig. 4.2 both normalized kernels Φ̃B 4 and Φ̃4 , and Φ̃6 and Φ̃6 , respectively. Although both kernels are bandlimited to the same degree in each plot, we notice how smoothly the Bernstein kernel decays to zero whereas the Shannon kernel oscillates heavily. Now, mathematically spoken, we understand oscillations as sign changes of the slope. In the Shannon case this consideration can be simplified if we prove that this kernel 10 M. J. FENGLER, W. FREEDEN, M. GUTTING 5 20 4 15 3 10 2 5 1 0 0 −1 −4 −3 −2 −1 0 θ 1 2 3 4 −5 −4 Fig. 4.1. The L2 (Ω)-normalized Shannon scaling function Φ̃S 4 (t) (solid) and the L2 (Ω)-normalized Bernstein scaling function Φ̃B 4 (t) (dashed). −3 −2 −1 0 θ 1 2 3 4 Fig. 4.2. The L2 (Ω)-normalized Shannon scaling function Φ̃S 6 (t) (solid) and the L2 (Ω)-normalized Bernstein scaling function Φ̃B 6 (t) (dashed). possesses only simple zeros which imply slope changes. To be more specific, we borrow from [12] an alternative representation of ΦSJ (t), i.e., ΦSJ (t) = J 2X −1 n=0 = ( 2n + 1 Pn (t) 4π 2J P2J (t)−P2J −1 (t) 4π t−1 22J 4π (4.1) for t ∈ [−1, 1) for t = 1. (4.2) Theorem 4.1 (Zeros of the Shannon Kernel). Let J ∈ N0 . Then the Shannon scaling function ΦSJ (t) possesses exactly 2J − 1 distinct, simple zeros on the interval [-1,1]. Proof. From (4.1) we see that ΦSJ (t) possesses at most 2J − 1 zeros. The case J ≤ 1 is trivial. For J ≥ 2 it is sufficient to show that the numerator NJ (t) = P2J (t) − P2J −1 (t) in (4.2) possesses 2J − 1 zeros in (-1,1). It is known that the Legendre polynomial Pn possesses n distinct (simple) zeros on (−1, 1) (see [1]). Thus, we denote by t1 , . . . , t2J the zeros of P2J in increasing order, and with s1 , . . . , s2J −1 the zeros of P2J −1 , respectively. Furthermore, the zeros of P2J −1 divide the zeros of P2J (see [1]). In detail, we have si ∈ (ti , ti+1 ). Clearly, the expression NJ (si ) = P2J (si ) − P2J −1 (si ) = P2J (si ) has the same alternating sign as P2J (si ), i = 1, . . . , 2J − 1. Therefore, we know from the Intermediate Value Theorem of classical analysis that NJ (t) possesses at least 2J − 2 zeros in (t1 , t2J ). Finally, we show that the remaining zero is placed in (−1, t1 ). This can be seen as follows: Keeping in mind that Pn (−1) = (−1)n we know that P2J has a negative sign in (t1 , t2 ) since P2J (−1) = 1 and t1 denotes the first zero. Thus, we obtain that NJ (s1 ) is negative and NJ (−1) = 2 is positive. Again, we can apply the Intermediate Value Theorem and arrive at the desired result. As an immediate consequence we find that the Shannon scaling function of scale J ≥ 2 possesses 2J −2 sign changes of its slope on (−1, 1), i.e., oscillations. Obviously, 11 THE SPHERICAL BERNSTEIN WAVELET the Bernstein scaling function as introduced in Definition 3.1 features no oscillations since -1 is a zero of multiplicity 2J − 1. 5. Space Uncertainty. In this section we derive an explicit representation of the space uncertainty of the Bernstein scaling function. In fact, we are led to the surprising result that the Shannon scaling function possesses the same space uncertainty. Theorem 5.1 (Space Uncertainty of the Bernstein Scaling Function). Suppose that J ∈ N and η ∈ Ω is fixed. Then the space uncertainty of the L2 (Ω)-normalized Bernstein scaling function Φ̃B J (·, η) of scale J is given by √ ∆Φ̃B (·,η) = J 2J+1 − 1 , 2J − 1 where the expectation value and the variance in the space domain read as follows 2J − 1 η, 2J 2J+1 − 1 = . 22J eΦ̃B (·,η) = J 2 σΦ̃ B (·,η) J Proof. Without loss of generality we set η = ε3 . From (2.7) we obtain eΦ̃B (·,ε3 ) = J = 2π Z 1 −1 2 t Φ̃B (t) dt ε3 J 2π B kΦJ (·, ε3 )k2L2 (Ω) Z 1 ! 2 ΦB J (t) t −1 dt ε3 . B 3 Substituting ΦB J by (3.1) and kΦJ (·, ε )kL2 (Ω) by (3.6) we deduce eΦ̃B (·,ε3 ) = J = = J+2 (2 − 2)π 22J−2 2 2J+2 − 2 22J−2 Z 1 2J+1 − 1 2 Z 1 1 Z 2 t π −1 t 22J−4 −1 −1 J−2 1+t 2 1+t 2 1+t 2 2J −1 !2 2J+1 −2 2J+1 −2 ! dt ε3 dt ε3 ! P1 (t) dt ε3 . Again, we observe that the latter term is of the same type as studied in (3.5). By letting n = 1 and k = 2J+1 − 2 we find the desired result eΦ̃B (·,ε3 ) = J 2J − 1 3 ε . 2J As an immediate consequence we deduce from (2.8) for the variance 2 σΦ̃ B (·,ε3 ) = 1 − J 2J+1 − 1 (2J − 1)2 = , 22J 22J (5.1) 12 M. J. FENGLER, W. FREEDEN, M. GUTTING such that the space uncertainty is given by √ ∆Φ̃B (·,ε3 ) = J 2J+1 − 1 . 2J − 1 (5.2) Next we deal with the amazing result that the L2 (Ω)-normalized Shannon scaling function possesses the same space uncertainty as the normalized Bernstein scaling function. First, we need a preliminary result on the norm of the Shannon scaling function of an arbitrary scale. Lemma 5.2 (Norm of the Shannon Scaling Function of scale J). Suppose that J ∈ N0 and η ∈ Ω is fixed. Then the L2 (Ω)-norm of Shannon scaling function ΦB J (·, η) of scale J is given by 2J−1 kΦSJ (·, η)kL2 (Ω) = √ . π (5.3) Proof. This result follows immediately from the definition of the Shannon generator (see Definition (2.2)) and the following norm identity kΦSJ (·, η)k2L2 (Ω) = ∞ X 2n + 1 S∧ 2 ΦJ (n) . 4π n=0 Based on the latter result we are now in the position to investigate the space uncertainty of the Shannon scaling function. Theorem 5.3 (Space Uncertainty of the Shannon Scaling Function). Assume that J ∈ N and η ∈ Ω is fixed. Then the space uncertainty of the L2 (Ω)-normalized Shannon scaling function Φ̃SJ (·, η) of scale J is given by √ 2J+1 − 1 , ∆Φ̃S (·,η) = J 2J − 1 where the expectation value and the variance in the space domain read as follows 2J − 1 η, 2J J+1 2 −1 = . 2J 2 eΦ̃S (·,η) = J 2 σΦ̃ S (·,η) J Proof. We start with the explicit formula for the expectation value (see also [11]) ! ∞ X 1 ∧ 3 eΦ̃J (·,ε3 ) = (n + 1) Φ∧ J (n)ΦJ (n + 1) ε 2πkΦJ (·, ε3 )k2L2 (Ω) n=0 ∞ P ∧ 2 (n + 1)Φ∧ (n)Φ (n + 1) J J n=0 3 ε . = ∞ P 2 ∧ (2n + 1) (ΦJ (n)) n=0 13 THE SPHERICAL BERNSTEIN WAVELET 2 Without loss of generality we set η = ε3 . The variance σΦ̃ can then be computed J (·,η) as in (2.8) and for the space uncertainty we have ∆Φ̃J (·,η) σΦ̃J (·,η) = = |eΦ̃J (·,η) | ∞ P 2 1) (Φ∧ J (n)) (2n + n=0 ∞ P 2 n=0 ∧ (n + 1)Φ∧ J (n)ΦJ (n + 1) 2 12 − 1 . The results of the theorem are obtained by combining Definition 2.2 with these formulae. Obviously, the last result indicates together with Theorem 4.1 that the quantity ∆F for the space localization (see (2.6)) does not reflect the amount of oscillations. This is nicely illustrated in Fig. 4.1 and Fig. 4.2. 6. Uncertainty in Momentum and the Uncertainty Product. We now derive the explicit formulae for the uncertainty in momentum and for the uncertainty product of the Bernstein scaling function. We again compare these quantities with their corresponding values in the Shannon case. Theorem 6.1 (Uncertainty in Momentum of the Bernstein Scaling Function). Let J ∈ N and η ∈ Ω be fixed. Then the uncertainty in momentum of the L2 (Ω)normalized Bernstein scaling function Φ̃B J (·, η) of scale J is given by −∆∗ ∆Φ̃ B (·,η) J ∗ ∆L Φ̃B J (·,η) = √ = 2J − 1 √ 2 where the expectation value (which coincides with the variance with respect to the L∗ operator) and the variance in the momentum domain (with respect to the operator −∆∗ ) read as follows 2 2J − 1 L∗ = σΦ̃B (·,η) , J 2 J 2 J 2 J+1 ∗ (2 − 1) 3 · 2 −1 −∆ = σΦ̃ . B (·,η) J+1 4 2 −3 J ∗ −∆ eΦ̃ B (·,η) = ∗ 2 L Proof. Without loss of generality we set η = ε3 . For the calculation of σΦ̃ B (·,ε3 ) J we use formula (2.11) L∗ σΦ̃ B (·,ε3 ) J 2 = −2π = Z 1 B Φ̃B J (t)Lt Φ̃J (t) dt −1 −2π B kΦJ (·, ε3 )k2L2 (Ω) Z 1 −1 B ΦB J (t)Lt ΦJ (t) dt. 14 M. J. FENGLER, W. FREEDEN, M. GUTTING Using (3.1) and (3.6) we get L∗ σΦ̃ B (·,ε3 ) J 2J −1 2J −1 2J−2 1 + t 2J−2 1 + t Lt dt π 2 π 2 −1 2J −1 2J −1 Z 1+t −2π 2 (2J+1 − 1) 22J−4 1 1 + t Lt dt = 22J−2 π2 2 2 −1 2J −1 2J −1 Z 1+t −(2J+1 − 1) 1 1 + t Lt dt. (6.1) = 2 2 2 −1 −2π 2 (2J+1 − 1) = 22J−2 2 Z 1 Simple differentiation yields the result of the application of the Legendre operator Lt Lt 1+t 2 2J −1 2J −1 d 1+t 2 d = (1 − t ) dt dt 2 J 2 −2 2J − 1 1 + t = 2J − 2 − 2J t . 2 2 Substituting this in (6.1) we obtain by aid of the Legendre polynomials P0 (t) = 1 and P1 (t) = t ∗ L σΦ̃ B (·,ε3 ) J 2 2J −1 J 2J −2 1+t 2 −1 1+t 2J − 2 − 2J t dt 2 2 2 −1 J+1 Z 2 −3 −(2J+1 − 1)(2J − 1) 1 1 + t 2J − 2 − 2J t dt = 22 2 −1 2J+1 −3 Z 1 J+1 J −(2 − 1)(2 − 1) 1+t J P0 (t) = (2 − 2) dt 22 2 −1 2J+1 −3 ! Z 1 1+t J dt . −2 P1 (t) 2 −1 = −(2J+1 − 1) 2 Z 1 Once again we apply (3.5) to come up with L∗ σΦ̃ B (·,ε3 ) J 2 −(2J+1 − 1)(2J − 1) = 22 = 2J − 1 . 2 2(2J+1 − 3) −2 (2 − 2) J+1 2 −2 (2J+1 − 2)(2J+1 − 1) J 2 J ∗ From (2.13) we know that this also corresponds to the expectation value e−∆ . 3 Φ̃B J (·,ε ) The uncertainty in momentum is given by −∆∗ ∆Φ̃ B (·,ε3 ) J = ∗ ∆L 3 Φ̃B J (·,ε ) = L∗ σΦ̃ B (·,ε3 ) J = √ 2J − 1 √ . 2 15 THE SPHERICAL BERNSTEIN WAVELET 2 −∆∗ Finally, the variance σΦ̃ is computed via (2.14) B (·,η) J ∗ −∆ σΦ̃ B (·,η) J 2 2 (−∆∗ )2 −∆∗ = eΦ̃B (·,η) − eΦ̃ B (·,η) J J 2 J Z 2 −1 ∗ 2 B B = −∆ξ Φ̃J (ξ, η) Φ̃J (ξ, η) dω(ξ) − 2 Ω J 2 Z 2 −1 ∗ B (ξ, η) ∆ Φ̃ (ξ, η) dω(ξ) − = ∆∗ξ Φ̃B . J ξ J 2 Ω Again, this integral can be simplified as we are considering the structure as radial basis function. Thus, without loss of generality we set η = ε3 . Z Z 1 2 ∗ B 3 ∗ B 3 Lt Φ̃B dt ∆ξ Φ̃J (ξ, ε ) ∆ξ Φ̃J (ξ, ε ) dω(ξ) = 2π J (t) Ω −1 = 2π B kΦJ (·, ε3 )k2L2 (Ω) Z 1 −1 2 Lt ΦB dt. J (t) Inserting our known expressions for the norm (3.6) and for Lt ΦB J (t) Z Ω ∆∗ξ Φ̃B J 2 2π 2 (2J+1 − 1) (ξ, ε3 ) dω(ξ) = 22J−2 2J−2 π 2 Z 1 −1 Lt 1+t 2 !2 2J −1 !2 dt 2J −2 2J − 1 1 + t (2J − 2 − 2J t) dt 2 2 −1 2J+1 −4 Z 2 (2J+1 − 1)(2J − 1)2 1 1 + t = 2J − 2 − 2J t dt. 23 2 −1 2J+1 − 1 = 2 Z 1 The last integral can be decomposed into 3 integrals of type (3.5) which compute to (−∆∗ )2 eΦ̃B (·,ε3 ) = J 23 (2J − 1) (2J − 1)3 (2J+1 − 1)(2J − 1)2 = J+1 . 3 J+1 J+1 2 (2 − 3)(2 − 1) 2 −3 Therefore, we obtain −∆∗ σΦ̃ B (·,ε3 ) J 2 (2J − 1)3 = J+1 − 2 −3 2J − 1 2 2 = (2J − 1)2 3 · 2J+1 − 1 . 4 2J+1 − 3 For comparison we also calculate the uncertainty in momentum for the Shannon scaling function. Theorem 6.2 (Uncertainty in Momentum of the Shannon Scaling Function). Let J ∈ N and η ∈ Ω be fixed. Then the uncertainty in momentum of the L2 (Ω)normalized Shannon scaling function Φ̃SJ (·, η) of scale J is given by √ 22J − 1 −∆∗ L∗ √ ∆Φ̃ = ∆ = S (·,η) S (·,η) Φ̃ J J 2 where the expectation value (which coincides with the variance with respect to the L∗ operator) and the variance in the momentum domain (with respect to the operator 16 M. J. FENGLER, W. FREEDEN, M. GUTTING −∆∗ ) read as follows 2 22J − 1 L∗ = σΦ̃S (·,η) , J 2 J 2 J 2 J ∗ (2 − 1) (2 + 1)2 −∆ = σΦ̃ . S (·,η) 12 J ∗ e−∆ = Φ̃S (·,η) ∗ −∆ Proof. We start with the following representation of eΦ̃ (see also [11]) (·,η) J ∗ L σΦ̃ J (·,η) 2 = = ∞ X 1 2 (2n + 1)n(n + 1)(Φ∧ J (n)) 4πkΦJ (·, η)k2L2 (Ω) n=0 ∞ P n=0 2 (2n + 1)n(n + 1)(Φ∧ J (n)) ∞ P n=0 . (6.2) 2 (2n + 1)(Φ∧ J (n)) The uncertainty in momentum is then given by the square root of (6.2). Finally, the 2 −∆∗ variance σΦ̃ is deduced from (·,η) J ∗ −∆ σΦ̃ (·,η) J 2 = (−∆∗ )2 J (·,η) = eΦ̃ ∞ P n=0 2 ∗ − e−∆ Φ̃ (·,η) J 2 (2n + 1)n2 (n + 1)2 (Φ∧ J (n)) ∞ P n=0 2 (2n + 1)(Φ∧ J (n)) ∞ P 2 (2n + 1)n(n + 1)(Φ∧ J (n)) n=0 − ∞ P n=0 2 (2n + 1)(Φ∧ J (n)) 2 . The results of the theorem are obtained by combining Definition 2.2 with these formulae. Now we have all the tools to discuss the uncertainty products for the two scaling functions. It is of particular interest whether these products converge to the minimal value 1 as the scale J tends to infinity, since a value of 1 indicates that the functions are well localized in space and frequency. Theorem 6.3 (Uncertainty Products of the Bernstein and of the Shannon Scaling Function). The uncertainty product of the L2 (Ω)-normalized Bernstein scaling function Φ̃B J (·, η) of scale J is given by ∗ ∆Φ̃B (·,η) ∆L Φ̃B J J (·,η) = 2J+1 − 1 2J+1 − 2 12 , and for J −→ ∞ this product tends to 1. The uncertainty product of the L2 (Ω)-normalized Shannon scaling function Φ̃SJ (·, η) of scale J reads as follows L∗ ∆Φ̃S (·,η) ∆Φ̃ S (·,η) J J and it diverges for J −→ ∞. = (2J+1 − 1)(2J + 1) 2J+1 − 2 12 THE SPHERICAL BERNSTEIN WAVELET 17 Proof. The product for the Bernstein scaling function of scale J can easily be computed from Theorem 5.1 and Theorem 6.1. In the Shannon case the result follows from Theorem 5.3 and Theorem 6.2. In both cases the behavior for J −→ ∞ is obvious. 7. Concluding Remarks. In this article we have demonstrated that there exist bandlimited scaling functions without oscillations. Furthermore, the Bernstein scaling function represents an example of a family of kernels that minimizes its uncertainty product in the limit, but unlike the Gauß-Weierstraß kernels which also possess this property (cf. [5, 9]) this minimizer is bandlimited. In detail, the uncertainty product of the Bernstein scaling function tends to 1. To our knowledge it is the first bandlimited kernel presented in literature for which this property has been proven. An uncertainty product of 1 indicates that the space as well as frequency localization is well balanced. Note that the dyadic arrangement of the scales is not associated with the properties presented in this work. All of them hold true also for other types of scales. Finally, it should be emphasized that the Bernstein scaling function possesses a closed representation by elementary expressions. Especially the latter allows a fast and stable evaluation. The numerical properties of this new spherical scaling function and wavelet will be subject to further studies. REFERENCES [1] M. Abramowitz, I.A. Stegun: Handbook of Mathematical Functions. Dover Publications, Inc., New York, 1972. [2] M. J. Fengler: Vector Spherical Harmonic and Vector Wavelet Based Non-Linear Galerkin Schemes for Solving the Incompressible Navier-Stokes Equation on the Sphere. PhD-thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group (submitted), 2005. [3] M. J. Fengler, W. Freeden: A Nonlinear Galerkin Scheme Involving Vector and Tensor Spherical Harmonics for Solving the Incompressible Navier-Stokes Equation on the Sphere. SIAM J. Sci. Comput. (accepted), 2005. [4] M. J. Fengler, W. Freeden, V. Michel: The Kaiserslautern Multiscale Geopotential Model SWITCH-03 from Orbit Pertubations of the Satellite CHAMP and Its Comparison to the Models EGM96, UCPH2002 02 0.5, EIGEN-1s, and EIGEN-2. Geophys. J. Intern., Vol. 157, pp. 499-514, 2004. [5] W. Freeden: Multiscale Modelling of Spaceborne Geodata. B. G. Teubner, Stuttgart, Leipzig, 1999. [6] W. Freeden, V. Michel: Multiscale Potential Theory (With Applications to Earth’s Sciences). Birkhäuser Verlag, Boston, Basel, Berlin, 2004. [7] W. Freeden, M. Schreiner: New Wavelet Methods for Approximating Harmonic Functions. In: Geodetic Theory Today (F. Sanso, ed.), International Association of Geodesy Symposia, Vol. 114, Springer, pp. 112-121, 1995. [8] W. Freeden, U. Windheuser: Spherical Wavelet Transform and Its Discretization. Adv. Comput. Math., Vol. 5, pp. 51-94, 1996. [9] W. Freeden, T. Gervens, M. Schreiner: Constructive Approximation on the Sphere (With Applications to Geomathematics). Oxford Science Publications, Clarendon, Oxford, 1998. [10] W. Freeden, O. Glockner, M. Schreiner: Spherical Panel Clustering and Its Numerical Aspects. J. of Geodesy, Vol. 72, pp. 586-599, 1998. [11] N. Laı́n Fernández: Polynomial Bases on the Sphere. PhD-Thesis, University of Lübeck, Institute for Mathematics, 2003. [12] N. N. Lebedev: Spezielle Funktionen und Ihre Anwendung. B.I.-Wissenschaftsverlag, Mannheim, Wien, Zürich, 1973. [13] W. Magnus, F. Oberhettinger, R. P. Soni: Formulas and Theorems for the Special Functions of Mathematical Physics. Die Grundlagen der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 52, 1966. [14] C. Müller: Spherical Harmonics. Lecture Notes in Mathematics, Vol. 17, Springer, 1966. 18 M. J. FENGLER, W. FREEDEN, M. GUTTING [15] C. Müller: Foundation of the Mathematical Theory of Electromagnetic Waves. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellung, Bd. 155, Springer, 1969. [16] F. J. Narcowich, J. D. Ward: Nonstationary Wavelets on the m-sphere for Scattered Data. Appl. Comp. Harm. Anal., Vol. 3, pp. 324-336, 1996. [17] H. R. Schwarz: Numerische Mathematik. B. G. Teubner, Stuttgart, 1997. !!" #%$'&( )*&,+.- $/-101- $/23-0456&)678:97;<& =?>@[email protected]@JIJRF>[email protected] GCUCVB,@B1UIW>XUBZY\[:P]CU^I [email protected]>SR,T I`PWPaLGRNB,EbGUcRFd,PBVeN #%$'&:f gh&i$/--j-k4,lX&)678$/-m]k1-$'& n [:[email protected]]UB:N1RP[@JIJRF>poq>CFP]rsN3IWNbdr t\GABU^IQZC,Pu\GvH[1>[email protected]>FN #%$'&Fw i &xyz:- $'4Mg{&i$/--j,- k4,lX&)678,$/-m]k:- $'& o}|,UB,B~o%P Y.RFU^[email protected]TRFU.[email protected]<GIQbvIW>,[email protected]B PBVB>@NOR>@ABOt\G AB1UB #%$'&M gh&i$/--j-k4sq&l m 781-3-,j1&] n [[email protected]`NQZC,PB n R,EB1P`I]>3YRT6o nX a%[email protected] #%$'&, &lOy:1- $ C [email protected] n REB1PWP`I]>3YR,[email protected]\G ABU^IQZCFP >KMB1U^NBOt*R[UQ3B UcRFd,PBV[email protected] o*G,[email protected]>@LRB,RVXC3YH>B,@JIWN<V !! #%$'&< lX&`&i,- k - $'4Mg{&i$/--j,- k4,lX& sz ^mWk ¡CU^N@B1PWP[>3YEB1Ne?UCK¢[email protected]@JIR>,N1TB1PEB1N [1>EDNB1IW>BUv[>£[email protected]>C,[email protected] n [:[email protected]<£3C,PB>@BQZA,>,I]£1B1> #%$'&¤ ¥&,lOym¦-$ C [email protected]a n IBa§BsG UB1NB>@[email protected]>,NOTRFU t*R1PB>FIREC,P¨[email protected]:PENe[email protected] o*G,[email protected]>FN @LRXRF>RNZG AB1U^IQ©B,Ra VXCsYH>B,@IQª¡[email protected] #%$'&:« q&,l m¦78:- §?BZY\[:P]CU^I _B,E n [[email protected][@JIRF> §BQsRKMBUr R,[email protected] n CFN3NO?B>,[email protected]¬[email protected]^I]d[,@IJRF> v UcRV [email protected]B1PWP`[email protected]B %[email protected]@AB [email protected]\ N ?UCK¢[email protected]@JIRF>,C,PvIB1PJE #%$'&<® gh&i$/--j-k4sq&l m 781- C [email protected][email protected]>po%>C,P]rsN3I`NOTRU t\GABU^IQZC,P:¯RE1IB1N #%$'&(M°XlX& z3 ^m]k4±%&l m¦78:-3-j1&J *R>@U^I]d[,@IJRF>,[email protected]B,[email protected] [email protected]²Ne?UcR[sG³6| uµ´\CFIWNBU^PWC[@BU>1³@LR @ABb¶>EXL>@B1U>[email protected]>C,P,p=b u·NBU RFU£sN<AR3G¸[email protected] tHo~a t§?L¹{vUC,[email protected]¦³ @C,P]r #%$'&((OlX&`&i,- k1¦-$'4sgh&i $/-,-j-k o{¹qRF>,P`I]>B1CU6CFPBU£sIW>et¬QZABVSB L> KZRPK¢IW>3Yº¨[email protected]>E|B>,N1RFU t\G AB1U^IQZC,P:¡CUVSRF>FIQ²NOTRUt*RPK¢IW>3Y @ABeL>QsRFVOG UB:N3N3I]d,PB ¹%CK¢IBUa/t¬@LRF£1B1N ·» [[email protected]>SRF>@ABOt\GABUB #%$'&(sfOgh&i$/-,-j- k4lX&1)678$/-mWk1- $ t\G CQ3BdRFU^>BS?UCK¢[email protected]@JIR>C,PvIB1PE B,@B1UVeI]>[email protected]>Xdr n BC,>,NORTF¼RQZC,P`P]r t¬[3G,[email protected]B,E CKMB:[email protected] #%$'&(:wi &xyz1- $'41)*&+H- $/-01- $/23-10 §B²Y\[1PWCU^I¦[email protected]>½[email protected][@ UB1P`I]VeI]>,CUr ´\>RPBFEMY\BeR,[email protected],>B1N3NbC>E UUcRFUH¯B1ACK¢IJRFU #%$'&(¾pgh&i$/--,j- k4~&,lOy11-$ n [[email protected]`NQZC,PBOtRP[@JIJRF>ST[email protected] n RPJREB1>,N<£3r UcRFd,PBVRF>¿§B²Y\[1PWCU |B1PWP[UcRIJEFC,P3t*[UcTCQ3B1N #%$'&(1¡gh&i$/--,j- k4À6&,Á-3Mt\GP`I]>BDVSR,EB1P`PWI]>sYRTMYB,[email protected]<GA,IQªTPR*Â @AB,[email protected],P<C,>E C,P YRFU^[email protected],VeIQ~C,NZGHB,[email protected] !!Ã #%$'&(M½lX&`&i,- k1¦-$'4±%&l m¦78:-J4sq&,l m¦781- %C,UVSRF>,IQ~t\GPWI]>Ba [email protected]>@AB Äa^E1I]VB>FN3IRF>CFP:¯·C,P`P<C>E¿@AB:I]U o*[email protected]>@LR¿@AB §B,QsRF>,[email protected][[email protected]>SRT @AB [email protected] N©B>[email protected]©¬[email protected]^I]d[@JIRF>STUcRFV ?UCK¢[email protected]@JIJRF>C,P<%[email protected]@²oqUd,[email protected],U^IWP]r tAC:[email protected]B1P`[email protected]BS=?Ud,[email protected] #%$'&(<¤¸i &xyz1- $ t\GP`[email protected]=¡[email protected]^NOTRFU6=?d,PWI » [B ¯R[1>EFCU^r½¨CFP[B UcRd,PB1VeN Å%Æ'ÇÈsÉOÊhÇËÆ/ÌÌ,ÍÌ ÎÏ,ÐXÇ1Ñ6ÒÓÆ/ÌÔWÎ1Ì Æ ÕÖ×ZØFÙ:ÚSÛ1ÙÜJÝ`Þ×ZØ,Ùß Ú¿Ö,àß1ÙWÙ`Ý]á3âãÖä å ß,ÖÝàØ,ÙæçáàÛ1ÙWØÜJÝÖFáFÞDäècÖé êß,äÙß×3ÜJÝÖFáFÞDÖ,äÜëßãìß1èÜÝ×ZØFÙ Å%Æ'ÇÈMíDÊhÇËÆ/ÌÌ,ÍÌ ÎÏ,îqÇ,Ð Ô¦ÒÓ:ÌïJÏsðqÇÐ Ô ÒÓ1Ìï ÕÖ×ZØ,Ù<ÚSÛ1ÙÜJÝWÞ×ZØFÙß~ñ*òòècÖMó,Ý]éXØÜÝJÖFá,Þ Öä å ß,Ö1ÞÜè'Ö<ò ë,Ý×bôÙJÖõ¬ö÷Fëß,ÖFèß,ÜJÝ×ZØ,Ù ø·Ø×ZùúâHècÖÛáà Øáàñ%ÞZòHß×3ÜJÞOÖä û ×²ÝßáÜÝJäÝ×Oü*ÖFéOòÛÜJÝ]á3â Å%Æ'Ç:ý,þ½ÐXÇ`ÿÇË,Ì Îï¦Ì Æ'ÏMÊhÇËÆ/ÌÌ,ÍÌ ÎÏÐXÇ ÔW Î ÷,ëß û ò ëß1è^Ý×ZØ,Ù:øßèá,ÞÜß1ÝW á ¸ Ø Mß1Ùß,Ü Informationen: Prof. Dr. W. Freeden Prof. Dr. E. Schock Fachbereich Mathematik Technische Universität Kaiserslautern Postfach 3049 D-67653 Kaiserslautern E-Mail: [email protected] [email protected]