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1.
The missing Wendland functions 总被引:1,自引:0,他引:1
Robert Schaback 《Advances in Computational Mathematics》2011,34(1):67-81
The Wendland radial basis functions (Wendland, Adv Comput Math 4:389–396, 1995) are piecewise polynomial compactly supported reproducing kernels in Hilbert spaces which are norm–equivalent to Sobolev spaces. But they only cover the Sobolev spaces and leave out the integer order spaces in even dimensions. We derive the missing Wendland functions working for half-integer k and even dimensions, reproducing integer-order Sobolev spaces in even dimensions, but they turn out to have two additional non-polynomial terms: a logarithm and a square root. To give these functions a solid mathematical foundation, a generalized version of the “dimension walk” is applied. While the classical dimension walk proceeds in steps of two space dimensions taking single derivatives, the new one proceeds in steps of single dimensions and uses “halved” derivatives of fractional calculus.
相似文献
$\label{eqstartrep} H^{d/2+k+1/2}({\mathbf{R}}^d),\;k\in {\mathbf{N}} $
(1)
2.
We prove that some multivariate linear tensor product problems are tractable in the worst case setting if they are defined as tensor products of univariate problems with logarithmically increasing smoothness. This is demonstrated for the approximation problem defined over Korobov spaces and for the approximation problem of certain diagonal operators. For these two problems we show necessary and sufficient conditions on the smoothness parameters of the univariate problems to obtain strong polynomial tractability. We prove that polynomial tractability is equivalent to strong polynomial tractability, and that weak tractability always holds for these problems. Under a mild assumption, the Korobov space consists of periodic functions. Periodicity is crucial since the approximation problem defined over Sobolev spaces of non-periodic functions with a special choice of the norm is not polynomially tractable for all smoothness parameters no matter how fast they go to infinity. Furthermore, depending on the choice of the norm we can even lose weak tractability. 相似文献
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4.
V. Ya. Yanchak 《Mathematical Notes》1970,8(4):718-721
Analogs of theorems of Jensen and Weierstrass are proven for functions of variable smoothness, defined on the real line.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 443–449, October, 1970.In conclusion I wish to express my gratitude to S. M. Nikol'skii and S. B. Stechkin for their valuable advice concerning this work. 相似文献
5.
We study d-variate approximation problems with varying regularity with respect to successive variables. The varying regularity is described by a sequence of real numbers {rk}k∈N satisfying We mainly consider algorithms that use finitely many continuous linear functionals. In the worst case setting we study approximation problems defined over suitable Korobov and Sobolev spaces. In the average case setting we study approximation problems defined over the space of continuous functions C([0,1]d) equipped with a zero-mean Gaussian measure whose covariance operator is given by an Euler or Wiener integrated process. We establish necessary and sufficient conditions on uniform weak tractability of those problems in terms of their regularity parameters {rk}k∈N. 相似文献
0≤r1≤r2≤r3≤?.
6.
S. G. Kobel’kov 《Mathematical Notes》2006,80(3-4):372-379
For a family of real-valued Gaussian processes ξ u (t), t ∈ [0, T], we obtain an exact asymptotics of the probability of crossing a level u as u → ∞ under certain conditions on the variance and correlation. This result is applied to the investigation of excursions of a stationary zero-mean process above a barrier increasing to infinity. 相似文献
7.
Heping Wang 《Journal of Approximation Theory》2005,137(2):264-276
In this note, we investigate the efficiency of the greedy algorithm for the classes of multivariate periodic functions with low mixed smoothness in Lq with regard to the wavelet-type basis. We find that the order of greedy approximation in the case of low smoothness is different for some range of parameters. 相似文献
8.
It is known that increasing powers of a continuous random variable converge in distribution to Benford’s law as the exponent approaches infinity. The rate of convergence has been estimated using Fourier analysis, but we present an elementary method, which is easier to apply and provides a better estimation in the widely studied case of a uniformly distributed random variable. 相似文献
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10.
Fractal and smoothness properties of space-time Gaussian models 总被引:1,自引:0,他引:1
Spatio-temporal models are widely used for inference in statistics and many applied areas. In such contexts, interests are
often in the fractal nature of the sample surfaces and in the rate of change of the spatial surface at a given location in
a given direction. In this paper, we apply the theory of Yaglom (1957) to construct a large class of space-time Gaussian models
with stationary increments, establish bounds on the prediction errors, and determine the smoothness properties and fractal
properties of this class of Gaussian models. Our results can be applied directly to analyze the stationary spacetime models
introduced by Cressie and Huang (1999), Gneiting (2002), and Stein (2005), respectively. 相似文献
11.
D. M. Bushev 《Ukrainian Mathematical Journal》2000,52(2):204-220
We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations
of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these
classes. 相似文献
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Stephen C Milne 《Advances in Mathematics》1980,35(2):129-157
14.
David C. Ullrich 《Arkiv f?r Matematik》1991,29(1):323-337
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David C. Ullrich 《Arkiv f?r Matematik》1991,29(1-2):323-337
17.
B. E. Klots 《Mathematical Notes》1977,21(1):12-19
The order of the quantity \(\delta (L) = \mathop {\sup }\limits_{\chi _{_{_{_1 } } } } \mathop {\inf }\limits_{\chi _2 } \parallel \chi _1 - \chi _2 \parallel L_S [0,2 = ]\) as L → ∞ is studied for the classes of periodic functionsx 1εW p n (I), andx 2 εW q m (L). Necessary and sufficient conditions under which the inequality $$\parallel x^{(n)} \parallel _{L_p } \leqslant C\parallel x\parallel _{L_q }^x \parallel x^{(m)} \parallel _{L_s }^\beta $$ with the constant independent of x holds for all periodic functions x(t) with \(\int_0^{2\pi } \chi (l)dl = 0\) andx (m) (t εL s [0, 2π] are found. 相似文献
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T. F. Ismagilov 《Moscow University Mathematics Bulletin》2013,68(5):215-220
A class of functions with a dominating mixed modulus of smoothness is introduced in the paper. Embedding theorems are proved for this class. 相似文献
20.
Necessary and sufficient conditions on functions and are found in order for the classes of functions
andH
+k
to coincide (k and r are natural numbers).These results are generalizations of [1–4].Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 785–794, December, 1977.The authors thank V. V. Salaev for his unflagging interest in their work. 相似文献