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On Equivalence of Moduli of Smoothness
Abstract
It is known that if f is an element of W-p(k), then omega(m)(f, t)(p) less than or equal to t omega(m-1) (f', t)(p) less than or equal to .... Its inverse with any constants independent off is not true in general. Hu and Yu proved that the inverse holds true for splines S with equally spaced knots, thus omega(m)(S, t)(p) similar to t omega(m-1) (S', t)(p) similar to t(2)omega(m-2)(S ", t)(p) .... In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper. (C) 1999 Academic Press.
... Remark 1.6. For m = k − 1, (1.4) was proved by Hu and Yu [12] and by Hu [10] for uniform and quasi-uniform partitions, respectively. In fact, if we set m = k − 1, then Corollary 1.5 (with an additional restriction k ≤ r) becomes Theorem 1 in [10] (and so, in the case k ≤ m + 1 ≤ r, it follows from [10, Theorem 1]). ...
... For m = k − 1, (1.4) was proved by Hu and Yu [12] and by Hu [10] for uniform and quasi-uniform partitions, respectively. In fact, if we set m = k − 1, then Corollary 1.5 (with an additional restriction k ≤ r) becomes Theorem 1 in [10] (and so, in the case k ≤ m + 1 ≤ r, it follows from [10, Theorem 1]). Also, in the case k = r + 1 and m = r − 1, Corollary 1.5 follows from Theorem 2 of [10] (where (1.4) was proved for all k ≥ r + 1 in the case m = r − 1). ...
... In fact, if we set m = k − 1, then Corollary 1.5 (with an additional restriction k ≤ r) becomes Theorem 1 in [10] (and so, in the case k ≤ m + 1 ≤ r, it follows from [10, Theorem 1]). Also, in the case k = r + 1 and m = r − 1, Corollary 1.5 follows from Theorem 2 of [10] (where (1.4) was proved for all k ≥ r + 1 in the case m = r − 1). The weighted Ditzian–Totik (DT-) kth modulus of smoothness of a function f ∈ L p [−1, 1], 0 < p ≤ ∞, is defined by ...
Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any 1leq kleq r+1 , 0leq mleq r-1 , and 1leq pleqinfty , the inequality n^{-nu} omega_{k-nu }(s^{(nu)}, n^{-1})_p sim omega_{k} (s, n^{-1})_p , 1leq nu leq min\{ k, m+1\} , is satisfied, where sin mathbb{C}^m[-1,1] is a piecewise polynomial of degree leq r on a quasi-uniform (i.eE, the ratio of lengths of the largest and the smallest intervals is bounded by a constant) partition of an interval. Similar results for Chebyshev partitions and weighted Ditzian-Totik moduli of smoothness are also obtained. These results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines.
... Re-use and distribution is strictly not permitted, except for Open Access articles. [27]) in the case 1 ≤ p < ∞. It is easy to see that the same also holds in the case 0 < p < 1. ...
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.
... for m \ k, where C > 0 is some constant independent of t for small t. This kind of works began from a result of Yu and Zhou [5], and was investigated by Hu [2] and Hu and Yu [3]. As a whole, all these results indicate that for splines with arbitrary (fixed) knots, the inequality (1) holds in general L p spaces for small t. ...
The present paper investigates polynomials for which the inverse inequality for moduli of smoothness holds. The technique for approach is different from the previous works for splines and is elegantly organized.
... A few years later, Y.-K. Hu [6] generalized (1.6) to splines with any (fixed) knot sequence, and further to principal shiftinvariant spaces and wavelets under certain conditions. The equivalence (1.6) for splines has played key roles in shape-preserving spline and polynomial approximation repeatedly, (see [7, 8, 10, 11]), which motivates us to investigate further along the line. ...
It is well known that ωr(f,t)p⩽tωr-1(f′,t)p⩽t2ωr-2(f″,t)p⩽⋯ for functions f∈Wpr, 1⩽p⩽∞. For general functions f∈Lp, it does not hold for 0p1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the Lp space for 0p⩽∞. We then show its analogues for the Ditzian–Totik modulus of smoothness ωϕr(f,t)p and the weighted Ditzian–Totik modulus of smoothness ωϕr(f,t)w,p for polynomials with ϕ(x)=1-x2.
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: the first based on geometric properties of Banach spaces and the second on Littlewood-Paley and H\"{o}rmander type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the K-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on , , , nonlinear wavelet approximation, etc.
In this paper, we introduce a new concept of constrained approximation, which we call Log-convex polynomial approximation in log. We obtained a direct theorem of log-convex polynomial approximation in log for twice differentiable functions on [-1, 1], in term of the ordinary moduli of continuity.
In this paper, we present an approach to shape-preserving approximation based on interpolation space theory. In particular, we prove the corresponding approximation result related to the intersection property of the cone of nonnegative functions with respect to the couple (Lp,Bα,∞p).
: A complete characterization is given of closed shift-invariant subspaces of L 2 (IR d ) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace. AMS (MOS) Subject Classifications: 41A25, 41A63; 41A30, 41A15, 42B99, 46E30 Key Words and phrases: approximation order, Strang-Fix conditions, shift-invariant spaces, radial basis functions, orthogonal projection. Authors' affiliation and address: 1 Center for Mathematical Sciences University of Wisconsin-Madison 610 Walnut St. Madison WI 53705 and 2 Department of Mathematics University of South Carolina Columbia SC 29208 This work...
A complete characterization is given of closed shift-invariant subspaces of L2(Rd) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.
We prove that if a function f ∈ C [0, 1] changes sign finitely many times, then for any n large enough the degree of copositive approximation to f by quadratic spliners with n−1 equally spaced knots can be estimated by Cω2(f, 1/n), where C is an absolute constant. We also show that the degree of copositive polynomial approximation to f ∈ C1[0, 1] can be estimated by Cn−1ωr(f′, 1/n), where the constant C depends only on the number and position of the points of sign change. This improves the results of Leviatan (1983, Proc. Amer. Math. Soc.88, 101-105) and Yu (1989, Chinese Ann. Math.10, 409-415), who assumed that for some r ≥ 1, f ∈ Cr[0, 1]. In addition, the estimates involved Cn−rω(fr, 1/n) and the constant C dependended on the behavior of f in the neighborhood of those points. One application of the results is a new proof to our previous ω2 estimate of the degree of copositive polynomia approximation of f ∈ C[0, 1], and another shows that the degree of copositive spline approximation cannot reach ω4, just as in the case of polynomials.
The order of positive and copositive spline approximation in the Lp-norm, 1 ≤ p < ∞, is studied; the main results are 1.(1) the error of positive approximation by splines is bounded by Cω2(f, 1n)p if f has a nonnegative extension;2.(2) the order deteriorates to ω1 if f does not have such an extension;3.(3) the error of copositive spline approximation is bounded by Cω(f, 1n)p;4.(4) if f is also continuous, the error in (3) can be estimated in terms of the third τ-modulus τ3(f, 1n)p.All constants in the error bounds are absolute.
We characterize functions with a given degree of nonlinear approximation by linear combinations with n terms of a function φ, its dilates and their translates. This gives a unified viewpoint of recent results on nonlinear approximation by spline functions and give their extension to functions of several variables. Our approach is formulated in terms of wavelet decompositions.
This classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.
Let wf k (f,d),f(x): = Ö{1 - x2 } ,\omega _\phi ^k (f,\delta ),\phi (x): = \sqrt {1 - x^2 } ,
, are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a functionf
| f(x) - pn (x) | \leqslant Cw3 (f,n - 1 Ö{1 - x2 } + n - 2 ),x Î [ - 1,1]; || f - pn ||¥ \leqslant Cwf 3 (f,n - 1 ); || f - pn ||p \leqslant Ct3 (f,n - 1 )p . \begin{gathered} \left| {f(x) - p_n (x)} \right| \leqslant C\omega _3 (f,n^{ - 1} \sqrt {1 - x^2 } + n^{ - 2} ),x \in [ - 1,1]; \hfill \\ \left\| {f - p_n } \right\|_\infty \leqslant C\omega _\phi ^3 (f,n^{ - 1} ); \hfill \\ \left\| {f - p_n } \right\|_p \leqslant C\tau _3 (f,n^{ - 1} )_p . \hfill \\ \end{gathered}
As a consequence, for a functionf
|| f - pn* ||¥ \leqslant Cn - 1 w2 (f¢,n - 1 ),\left\| {f - p_n^* } \right\|_\infty \leqslant Cn^{ - 1} \omega _2 (f\prime ,n^{ - 1} ),
wheren2 andC is an absolute constant.
We give sufficient conditions on a single function [^(j)]\hat \varphi
and are satisfied even when does not decay quickly at infinity.
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. We study the behavior of moduli of smoothness of splines s of order r with equally spaced knots fx i g, x i+1 Gamma x i = h. The main results are (1) For each 0 m ! r, all quantities h j !mGammaj (s (j) ; h)p , 0 j m, are equivalent and can be measured by a discrete norm of the mth differences of the B-spline coefficients of s, which we call the mth discrete modulus of smoothness of s. (2) All quantities h j !mGammaj (s (j) ; h)p , m r, 0 j r Gamma 1, are equivalent to !r (s; h)p , which can be measured by the rth discrete modulus of s. (3) When h is replaced by t, 0 t h, in the results above, all the quantities can still be measured by the corresponding discrete modulus multiplied by a power of t=h. The results generalize the notion of discrete norm of B-spline series in case of equal spacing. As an application, we use these results to prove that ! 3 is the best rate of convex approximation by such splines. Key words. Splines with equally spaced knots, Modulu...
An introduction to wavelets Academic Press, 1992. 44 I. Daubechies, Ten lectures on wavelets
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K. Chui, An introduction to wavelets, Academic Press, 1992.
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