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49
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 207 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Quantitative Fourier Analysis of Approximation Techniques: Part II  Wavelets
 IEEE Trans. Signal Processing
, 1999
"... In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic ..."
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Cited by 63 (28 self)
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In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter. This is, in particular, true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asymptotically optimal upper bounds for the leastsquares approximation error. Another contribution is the application of these results to Bsplines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify ...
Quasiinterpolants And Approximation Power Of Multivariate Splines
, 1990
"... The determination of the approximation power of spaces of multivariate splines with the aid of quasiinterpolants is reviewed. In the process, a streamlined description of the existing quasiinterpolant theory is given. 1. Approximation power of splines I begin with a brief review of the approximation ..."
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Cited by 55 (8 self)
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The determination of the approximation power of spaces of multivariate splines with the aid of quasiinterpolants is reviewed. In the process, a streamlined description of the existing quasiinterpolant theory is given. 1. Approximation power of splines I begin with a brief review of the approximation power of univariate splines since the techniques for its investigation are also those with which people have tried to understand the multivariate setup. (That may in fact be the reason why we know so little about it.) I will then briefly discuss three examples to illustrate some basic limitations of the standard univariate approach. Let S := S k,t be the univariate space of splines of order k with knot sequence t. This means that S := span # with # := # # i # n i=1 , # i := M(t i , . . . , t i+k ) the normalized Bspline for the knots t i , . . . , t i+k , and t := (t j ) n+k j=1 a nondecreasing real sequence. This definition of a spline space is taylormade for the consideration of its approximation power, since the Bspline basis # : # # (n) # S : c ## #c := n # i=1 # i c(i) is so wellbehaved. (I have found it convenient to identify the sequence (# 1 , . . . , #n ) with the map c ## # i # i c(i).) We consider specifically approximation from S to X := C([a, b]), with [a, b] := [t k , t n+1 ] 1 supported by the National Science Foundation under Grant No. DMS8701275 and by the United States Army under Contract No. DAAL0387K0030 1 the interval of interest. In the corresponding norm #f# := {sup f(x) : a # x # b}, the basis (map) # satisfies ### = sup c ##c#/#c## = 1 (the result of the fact that the # i are nonnegative and sum to 1, i.e., form a partition of unity). Since any linear map Q on X into S has the form Q =: # i # i # ...
Approximation error for quasiinterpolators and (multi)wavelet expansions
 APPL. COMPUT. HARMON. ANAL
, 1999
"... We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wa ..."
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Cited by 48 (19 self)
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We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wavelets, as well as finite elements and multiwavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling step T. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multiwavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the twoscale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multiwavelet expansions of a given order L.
Fourier Analysis of the Approximation Power of Principal ShiftInvariant Spaces
, 1991
"... The approximation order provided by a directed set fs h g h?0 of spaces, each spanned by the hZZ d translates of one function, is analyzed. The "nearoptimal" approximants of [R2] from each s h to the exponential functions are used to establish upper bounds on the approximation order. These app ..."
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Cited by 28 (13 self)
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The approximation order provided by a directed set fs h g h?0 of spaces, each spanned by the hZZ d translates of one function, is analyzed. The "nearoptimal" approximants of [R2] from each s h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical StrangFix conditions are extended to bounded summable generating functions. The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale fs h g is obtained from s 1 by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions. AMS (MOS) Subject Classifications: primary 41...
The QuasiInterpolant as a Tool in Elementary Polynomial Spline Theory
, 1973
"... ll of the t j 's coincide. I leave unresolved any possible ambiguity when t = t j for some j, and concern myself only with left and right limits at such a point; i.e., I replace each t = t j by the "two points" t  j and t + j . As is well known, N ik > 0 on (t i , t i+k ), and N ik = 0 o# [t ..."
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Cited by 26 (10 self)
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ll of the t j 's coincide. I leave unresolved any possible ambiguity when t = t j for some j, and concern myself only with left and right limits at such a point; i.e., I replace each t = t j by the "two points" t  j and t + j . As is well known, N ik > 0 on (t i , t i+k ), and N ik = 0 o# [t + i , t  i+k ] so that (since t i < t i+k , by assumption) N ik is not identically zero, while on the other hand, no more than k of the N jk 's are nonzero at any particular point. Consequently, for an arbitrary a #<F1
Splines as linear combinations of Bsplines. A Survey
, 1976
"... This paper is intended to serve as a postscript to the fundamental 1966 paper by Curry and Schoenberg on Bsplines. It is also intended to promote the point of view that Bsplines are truly basic splines: Bsplines express the essentially local, but not completely local, character of splines; certai ..."
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Cited by 23 (3 self)
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This paper is intended to serve as a postscript to the fundamental 1966 paper by Curry and Schoenberg on Bsplines. It is also intended to promote the point of view that Bsplines are truly basic splines: Bsplines express the essentially local, but not completely local, character of splines; certain facts about splines take on their most striking form when put into Bspline terms, and many theorems about splines are most easily proved with the aid of Bsplines; the computational determination of a specific spline from some information about it is usually facilitated when Bsplines are used in its construction.
QUADRATIC SPLINE QUASIINTERPOLANTS ON BOUNDED DOMAINS OF R^d, d = 1, 2, 3
, 2003
"... We study some C¹ quadratic spline quasiinterpolants on bounded domains � ⊂ Rd, d = 1, 2, 3. These operators are of the form Q f (x) = ∑ k∈K (�) µk ( f)Bk(x), where K (�) is the set of indices of Bsplines Bk whose support is included in the domain � and µk ( f) is a discrete linear functional b ..."
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Cited by 14 (7 self)
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We study some C¹ quadratic spline quasiinterpolants on bounded domains � ⊂ Rd, d = 1, 2, 3. These operators are of the form Q f (x) = ∑ k∈K (�) µk ( f)Bk(x), where K (�) is the set of indices of Bsplines Bk whose support is included in the domain � and µk ( f) is a discrete linear functional based on values of f in a neighbourhood of xk ∈ supp(Bk). The data points x j are vertices of a uniform or nonuniform partition of the domain � where the function f is to be approximated. Beyond the simplicity of their evaluation, these operators are uniformly bounded independently of the given partition and they provide the best approximation order to smooth functions. We also give some applications to various fields in numerical approximation.
On local linear functionals which vanish at all Bsplines but one
, 1976
"... Introduction Let k # IN, t := (t i ) nondecreasing (finite, infinite or biinfinite) with t i < t i+k , all i, and let (N i ) be the sequence of Bsplines of order k for the knot sequence t. This means that N i = N i,k,t is the Bspline of order k with knots t i , . . . , t i+k , i.e., N i is give ..."
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Cited by 14 (4 self)
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Introduction Let k # IN, t := (t i ) nondecreasing (finite, infinite or biinfinite) with t i < t i+k , all i, and let (N i ) be the sequence of Bsplines of order k for the knot sequence t. This means that N i = N i,k,t is the Bspline of order k with knots t i , . . . , t i+k , i.e., N i is given by the rule (1.1) N i,k,t (t) := # [t i+1 , . . . , t i+k ]  [t i , . . . , t i+k1 ] # (  t)