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Abstract—Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques. An important issue is the choice of adequate basis functions. We show that, contrary to the common belief, those that perform best are not interpolating. By opposition to traditional interpolation, we call their use generalized interpolation; they involve a prefiltering step when correctly applied. We explain why the approximation order inherent in any basis function is important to limit interpolation artifacts. The decomposition theorem states that any basis function endowed with approximation order can be expressed as the convolution of a B-spline of the same order with another function that has none. This motivates the use of splines and spline-based functions as a tunable way to keep artifacts in check without any significant cost penalty. We discuss implementation and performance issues, and we provide experimental evidence to support our claims. Index Terms—Approximation constant, approximation order, B-splines, Fourier error kernel, maximal order and minimal support (Moms), piecewise-polynomials. I.

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 two-scale 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 least-squares approximation error. Another contribution is the application of these results to B-splines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify ...

Abstract—This chapter presents a survey of interpolation and resampling techniques in the context of exact, separable interpolation of regularly sampled data. In this context, the traditional view of interpolation is to represent an arbitrary continuous function as a discrete sum of weighted and shifted synthesis functions—in other words, a mixed convolution equation. An important issue is the choice of adequate synthesis functions that satisfy interpolation properties. Examples of finite-support ones are the square pulse (nearest-neighbor interpolation), the hat function (linear interpolation), the cubic Keys' function, and various truncated or windowed versions of the sinc function. On the other hand, splines provide examples of infinite-support interpolation functions that can be realized exactly at a finite, surprisingly small computational cost. We discuss implementation issues and illustrate the performance of each synthesis function. We also highlight several artifacts that may arise when performing interpolation, such as ringing, aliasing, blocking and blurring. We explain why the approximation order inherent in the synthesis function is important to limit these interpolation artifacts, which motivates the use of splines as a tunable way to keep them in check without any significant cost penalty. I.

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 shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets 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 multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale 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 multi-wavelet expansions of a given order L.

Abstract—Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz bounds). We show that the exponential B-spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the th-order decay of the P-approximation error as a function of the knot spacing. Index Terms—Continuous-time signal processing, convolution, differential operators, Green functions, interpolation, modulation, multiresolution approximation, splines. I.

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Abstract. 1 The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well known form is Shannon’s uniform sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed.

. Under the appropriate definition of sampling density D # , a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D # # 1. This result is similar to Landau's result for the PaleyWiener space B 1/2 . If the shift invariant space consists of polynomial splines, then we show that D # < 1 is su#cient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B 1/2 . In the sampling problem one seeks to recover a function f from its samples {f(x j ) : j # ZZ}. Clearly, this problem is ill-posed, and becomes meaningful only after some a priori conditions on f . Until recently, the standard assumption is that f is band-limited, i.e., supp f # [-#, #] [7, 19, 29]. Although this is a reasonable assumption, in many applications di#erent hypothesis on the functions are desirable, e.g., for taking into account real acquisition and reconstruction devices, for obtaini...

Abstract- We investigate the functions of given ap-proximation order L that have the smallest support. Those are shown to be linear combinations of the B-spline of degree.L- 1 and its L- 1 first derivatives. We then show how to And the functions that minimize the asymptotic approximation constant among this finite di-mension space; in particular, a tractable induction re-lation is worked out. Using these functions instead of splines, we observe that the approximation error is dra-matically reduced, not only in the limit when the sam-pling step tends to zero, but also for higher values up to the Shannon rate. Finally, we show that those optimal functions satisfy a scaling equation, although less simple than the usual two-scale difference equation. I.