Schoenberg, I. J., Cardinal Spline Interpolation, vol. 12, Philadelphia, PA: Conference Board of the Mathematical Sciences, SIAM, 1973. Tim N. T. Goodman Dept. of Mathematics The University Dundee DD1 4HN Scotland, U.K. tgoodman@maths.dundee.ac.uk  Home/Search   Document Not in Database   Summary   Related Articles   Check

....kt = F n 1 (t) for which U 2 (F n ) U R (B 2 ) 10 was already computed explicitly in [13] and which also can be obtained from (7.1) for r = 1. The trigonometric kernels obtained by sampling cardinal B splines in the integers date back to Schoenberg [14] Polya frequency functions (see [15] and the references therein) as generalizations of B splines could also serve as generating functions in this context, although this idea is not pursued any further in this paper. More recently, B spline kernels have been used in preconditioning Toeplitz matrices, see [3] and [11] and in ....

Schoenberg, I.J. (1973), Cardinal Spline Interpolation, SIAM.

....is following in a Lagrangian fashion the flow map. The initial flow field quantities are interpolated on the particle locations, and all the flow quantities can be reconstructed by a linear superposition of the flow quantities carried by the particles as weighed by a smooth interpolation kernel [8, 9]. The discrete equations are obtained from continuum equations by expressing the flow quantities as a linear superposition of the physical quantities that are being carried by the particles. SPH belongs to a class of Lagrangian methods called particle methods. The key advantage of all particle ....

....field quantities are interpolated on the particle locations and all the flow quantities can be reconstructed by a linear superposition of the flow quantities carried by the particles as weighed by a smooth interpolation kernel. This interpolation is based on the theory of integral interpolants [8, 9] so that the interpolated value of any function A at position r is expressed as A(r # )W (r r # , h) dr , 12) where the integration is over the computational domain, W is an interpolation function, and h is a characteristic distance between the particles which is closely related to the ....

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I. J. Schoenberg, Cardinal Spline Interpolation (Soc. for Industr. & Appl. Math., Philadelphia, 1973).

....that the assumed underlying signal x(t) satisfies some desired properties. Taking signal interpolation as an example, the choice of #(t) is often some smooth function so that the resulting interpolant is visually pleasing. In the case of spline interpolation [9, 10] #(t) is taken to be a B spline [3] of a particular order, determining the degree of smoothness, whereas in the case of least squares approximation it is the basis function of the approximation subspace. In order to understand how interpolation is performed, consider once more the model (1) To interpolate signal x(n) by an ....

I. J. Schoenberg, Cardinal Spline Interpolation. SIAM, 1973.

....of several other equalization methods including the minimum mean squared error (MMSE) equalizer. In Sec. 4 we consider another application of FBPs, namely the interpolation of signals described by oversampled models. This method is a modification of the well known spline interpolation technique [4, 12] which requires the use of non causal IIR filters. E#cient implementation of this filtering is treated in [18] Here we show that by assuming even a slightly oversampled model for the signal, exact spline interpolation is possible using only FIR filters. This approach is thus di#erent from another ....

....for the signals admitting the model (21) is shown in Fig. 13, with #K (t) # = #(t K ) and B(e ) # =1 #(e ) While in principle #(t) can be chosen to be just about any function, various researchers have traditionally used continuously di#erentiable interpolating functions such as B splines [4, 12] to insure some smoothness properties of the resulting interpolant. The mth order B spline is given by the m fold convolution of the unit pulse function p(t) 1 for t [0, 1) 0 otherwise with itself. The important property of B splines is that they span the space of continuously ....

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I. J. Schoenberg, Cardinal Spline Interpolation. SIAM, 1973.

....a long history of application in this context [1] since they provide a flexible representation (e.g. see [2] and by simply choosing the order of these approximation functions it is possible to control the smoothness of the resulting continuous signal. Thanks to a classical result by Schoenberg [3], it is enough to consider a very special class of piecewise polynomials, namely B splines, since all other polynomial splines are obtained as a linear combination of shifted B splines. The work by Unser et al. 5] 6] 7] represents a major contribution in B spline signal processing. They ....

....and references therein) Splines of order n are equal to polynomials of degree n on each interval between two knots, and those polynomials are connected in such a way that the overall function is (n 1) times continuously di#erentiable even at the knots ( 2] pp. 3 7) It was shown by Schoenberg [3] that every n th order spline with equidistant knots, f c (t) can be represented as c (t) k= # c(k)# (t k) #t # R. 1) Here c(k) is an l 2 sequence of reals and # (t) is a centered B spline of order n, which is obtained as an n fold convolution of the centered unit pulse with ....

I. J. Schoenberg, Cardinal Spline Interpolation. SIAM, 1973.

....to have been constructed in [1] for the purpose of demonstrating the linear independence over an interval of all B splines which do not vanish identically on that interval. Since then, such linear functionals have been constructed in various ways and for a variety of jobs [2] 7] 11] 13] [16], some of which are listed in Section 2. In particular, it was shown in [6] that there exists a smallest number D k so that, for all t and all i with t i t i k , an h i # IL # can be found with supp h i # [t i , t i k ] #h i # # # D k (t i k t i ) and # h i N j = # ij , all ....

....t i k 1 # b # t i k , there exists h i # IL # such that (4.8) supp h i # [a, b] #h i # # # D k (b a) # h i N j = # ij , all j. Then (# 2) k 2 # D k # 2k 9 k 1 . Proof: Only the first inequality still requires proof. For this, take Schoenberg s Euler spline [14] [16], E k (t) # k # # j= # ( j N j,k 1,ZZ # t k 1 2 # 7 with (4.9) # k = 1 # k 1 (#) # # 2 # k 1 # j # ( 1) j 2j 1 # k 1 # # # 2 # k 2 so chosen that E k (#) # , all # # ZZ. Then E (1) k (t) 2# k # j ( j N j,k,ZZ # t k ....

I. J. Schoenberg (1973), Cardinal Spline Interpolation, CBMS, SIAM (Philadelphia). 13

....to each other through a two scale relation. Some good examples of admissible central basis functions are the one sided power functions x n n with n integer. These can be localized using the backward (n 1) order di#erence operator to yield the causal B splines of degree n (cf. 31] [32]) # n (x) # n 1 x n n Fourier # 1 e j# j# n 1 (15) The B splines satisfy the three conditions in Definition 1; they constitute one of the earliest example of valid scaling functions [24] Localizing less simple central basis functions turns out to be more ....

I.J. Schoenberg, Cardinal spline interpolation, SIAM, Philadelphia, PA, 1973.

....equivalent to the invertibility of its symbol. This observation is very useful in any general study of Riesz basis of shift invariant subspaces which typically arises in wavelet analysis as well as in any investigation of concrete such spaces, for example, cardinal spline spaces (c.f. Schoenberg [1]) In a recent investigation [7] concerning the asymptotic behavior of Gram Schmidt orthonormalization procedure applied to the nonnegative integer shifts of a given function, the problem of determining whether or not such functions form a Riesz basis in L 2 (IR ) arose. In this case, the ....

Schoenberg I. J. 1973. Cardinal spline interpolation, CBMS-NSF Vol 12, SIAM Philadelphia.

....] # D # k max i k j#i k [t j , t j k ]f 0 with D # k some constant depending only on k. It seems likely that K(k) is much closer to its lower bound (9) # 2) k 1 # K(k) than to the rather fast growing upper bound (8) One obtains (9) with the aid of Schoenberg s Euler spline [6]: With t i = i, all i, the kth degree Euler spline E k (t) # k # i ( i M i,k 1 (t (k 1) 2) satisfies E k (i) i , all i, hence k [i, i k]E k = 2 k , 5 with # k : 1 ## j # sin(2j 1)# 2 (2j 1)# 2 # k 1 = # 2) k 1 ## j ( 1) j (2j ....

I. J. Schoenberg, "Cardinal Spline Interpolation", CBMS, SIAM, Philadelphia, 1973. 8

....is to make the second derivative of f ## (x) 6 P i # i x x i vanish after the last sampling point which ensures that J(f) # and thus f # F . Note, that f is a piecewise cubic polynomial with continuous second derivatives; i.e. it is a cubic spline. This result is known, see [77, 130]. An example of a spline reconstruction (interpolation) is shown in Figure 6.4. For uniform sampling, the basis functions x x i 3 can be localized using digital filtering (with iterated finite di#erence filter) to obtain compactly supported uniform cubic B splines, which makes an ....

I.J. Schoenberg, Cardinal spline interpolation, SIAM, Philadelphia, PA, 1973.

....a curve whose components are spline functions. One particular spline function is the B spline (basic spline) so called because translates of this function form a basis for the space of spline functions. B splines were probably already known to Hermite, and certainly to Peano early this century [73] but were introduced in geometric modeling by Riesenfeld [68] and have become a popular device for curve and surface design. An easy introduction to B spline curves and surfaces is given by Bartels et al. 8] We are interested in the continuity of spline curves segments r(u) u min u umax and ....

I. J. Schoenberg. Cardinal Spline Interpolation. SIAM, 1973.

....1 (t) for which lim n 1 U 2 (F n ) UR (B 2 ) r 3 10 was already computed explicitly in [12] and which also can be obtained from (23) for r = 1. The trigonometric kernels obtained by sampling cardinal B splines in the integers date back to Schoenberg [13] Polya frequency functions (see [14] and the references therein) as generalizations of B splines could also serve as generating functions in this context, although this idea is not pursued any further in this paper. More recently, B spline kernels have been used in preconditioning Toeplitz matrices, see [3] and [10] and in ....

I. J. Schoenberg, Cardinal Spline Interpolation, SIAM, Philadelphia, 1973.

....(37) by replacing and with and , respectively. One of the most attractive nonorthonormal wavelets is NCSW. The two scaling functions of NCSW are defined from the uniform B spline of order , i.e. with supp XIANG AND LU: WAVELET MATRIX TRANSFORM APPROACH FOR SOLUTIONS OF INTEGRAL EQUATIONS 1213 [20]. NCSW has several remarkable features. For one thing, all the filter coefficients are dyadic rationales. Since division by two can be done very fast by computer, this makes them very suitable for fast computations. Another attractive property is that the scaling functions and wavelets are known ....

I. J. Schoenberg, Cardinal Spline Interpolation, CBMS-NSF Series in Appl. Math. #12. Philadelphia, PA: SIAM, 1973.

....] D 0 k max i k j i k j[t j ; t j k ]f 0 j with D 0 k some constant depending only on k. It seems likely that K(k) is much closer to its lower bound (9) 2) k 1 K(k) than to the rather fast growing upper bound (8) One obtains (9) with the aid of Schoenberg s Euler spline [6]: With t i = i, all i, the kth degree Euler spline E k (t) k X i ( i M i;k 1 (t (k 1) 2) satis es E k (i) i ; all i; hence k j[i; i k]E k j = 2 k ; 5 with k : 1 .X j sin(2j 1) 2 (2j 1) 2 k 1 = 2) k 1 .X j ( 1) j = 2j 1) ....

I. J. Schoenberg, \Cardinal Spline Interpolation", CBMS, SIAM, Philadelphia, 1973. 8

....We illustrate this result with the example of B splines. Recall that the n th degree B spline has a mask polynomial given by A(z) 1 2 n (1 z) n 1 ; z 2 C ; 2. 23) 9 cf. 9] The corresponding polynomial B of degree n Gamma 1 was called the Euler Frobenius polynomial by Schoenberg in [10] where several of them are given explicitly [10, p. 22] Here, we restrict ourselves to the cases n = 1; 2; 3. For n = 1, the linear case, the mask polynomial is 1 2 t z 1 2 Gamma t z 2 ; z 2 C ; Gamma 1 2 t 1 2 ; and the corresponding one parameter family ....

I.J. Schoenberg. Cardinal Spline Interpolation, volume 12 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1973.

....dx = N r # N r ] j r) N 2r (j r) j# Z, because N r is symmetric about its midpoint. 22 RONALD A. DEVORE AND BRADLEY J. LUCIER The polynomial # 2r (z) z r # j#Z #(j)z j is the Euler Frobenius polynomial of order 2r, which plays a prominent role in cardinal spline interpolation (see [Sch1]) It is well known that # 2r has no zeros on z = 1. Hence, the reciprocal 1 # 2r is analytic in a nontrivial annulus that contains the unit circle in its interior. One can easily find the coe#cients of reciprocals and square roots of Laurent series inductively. By finding the coe#cients of # ....

I. Schoenberg, Cardinal Spline Interpolation, Regional Conference Series in Applied Mathematics, vol. 12, SIAM, Philadelphia, 1973.

....equivalent to the invertibility of its symbol. This observation is very useful in any general study of Riesz basis of shift invariant subspaces which typically arises in wavelet analysis as well as in any investigation of concrete such spaces, for example, cardinal spline spaces (c.f. Schoenberg [1]) In a recent investigation [8] concerning the asymptotic behavior of Gram Schmidt orthonormalization procedure applied to the nonnegative integer shifts of a given function, the problem of determining whether or not such functions form a Riesz basis in L 2 (IR ) arose. In this case, the ....

Schoenberg I. J. 1973. Cardinal spline interpolation, CBMS-NSF Vol 12, SIAM Philadelphia.

....interpolation and spline functions VIII. The Budan Fourier Theorem for splines and applications Carl de Boor and I.J. Schoenberg Dedicated to M.G. Krein Introduction. The present paper is the reference [8] in the monograph [15], which was planned but not yet written when [15] appeared. The paper is divided into four parts called A, B, C, and D. We aim here at three or four di#erent results. The unifying link between them is that they all involve the sign structure of what one might call a Green s spline , i.e. a ....

....interpolation and spline functions VIII. The Budan Fourier Theorem for splines and applications Carl de Boor and I.J. Schoenberg Dedicated to M.G. Krein Introduction. The present paper is the reference [8] in the monograph [15] which was planned but not yet written when [15] appeared. The paper is divided into four parts called A, B, C, and D. We aim here at three or four di#erent results. The unifying link between them is that they all involve the sign structure of what one might call a Green s spline , i.e. a function which consists of two null splines pieced ....

[Article contains additional citation context not shown here]

I. J. Schoenberg, "Cardinal Spline Interpolation", Vol. 12, CBMS, SIAM, Philadelphia, 1973.

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Schoenberg, I. J., Cardinal Spline Interpolation, vol. 12, Philadelphia, PA: Conference Board of the Mathematical Sciences, SIAM, 1973. Tim N. T. Goodman Dept. of Mathematics The University Dundee DD1 4HN Scotland, U.K. tgoodman@maths.dundee.ac.uk

No context found.

Schoenberg, I.J., Cardinal spline interpolation, CBMS-NSF Series in Appl. Math., Vol. 12, SIAM Publ., Philadelphia, 1973.

No context found.

I. J. Schoenberg (1973), Cardinal Spline Interpolation, CBMS 12, SIAM, Philadelphia.

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I. J. Schoenberg, Cardinal Spline Interpolation, vol. 12 of Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1973.

No context found.

Schoenberg, I. J., Cardinal Spline Interpolation, SIAM, Philadelphia, Pennsylvania, 1973.

No context found.

I. J. Schoenberg, "Cardinal Spline Interpolation", CBMS, SIAM, Philadelphia, 1973.

No context found.

I.J. Schoenberg, Cardinal Spline Interpolation, volume 12 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1973.

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