# zbMATH — the first resource for mathematics

Dunkl’s theory and best approximation by entire functions of exponential type in $$L_{2}$$-metric with power weight. (English) Zbl 1304.41020
Summary: In this paper, we study the sharp Jackson inequality for the best approximation of $$f \in L_{2,{\kappa}}(\mathbb{R}^{d})$$ by a subspace $$E_{\kappa}^{2}(\sigma)\, (SE_{\kappa}^{2}(\sigma))$$, which is a subspace of entire functions of exponential type (spherical exponential type) at most $$\sigma$$. Here $$L_{2,\kappa}(\mathbb{R}^{d})$$ denotes the space of all $$d$$-variate functions $$f$$ endowed with the $$L_{2}$$-norm with the weight $$v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi)} }$$, which is defined by a positive subsystem $$R_{+}$$ of a finite root system $$R \subset \mathbb{R}^d$$ and a function $$\kappa(\xi): R \to \mathbb{R}_{+}$$ invariant under the reflection group $$G(R)$$ generated by $$R$$. In the case $$G(R) = \mathbb{Z}_{2}^{d}$$, we get some exact results. Moreover, the deviation of best approximation by the subspace $$E_{\kappa}^{2}(\sigma)\, (SE_{\kappa}^{2}(\sigma))$$ of some class of the smooth functions in the space $$L_{2,{\kappa}}(\mathbb{R}^{d})$$ is obtained.
##### MSC:
 41A50 Best approximation, Chebyshev systems 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Full Text:
##### References:
 [1] Arestov, V V; Popov, V Yu, Jackson inequalities on a sphere in $$L$$\^{2}, Izv. Vyssh. Uchebn. Zaved. Mat., 399, 13-20, (1995) · Zbl 0860.41015 [2] Babenko, A G, Sharp Jackson-Stechkin inequality in $$L$$\^{2} for multidimensional spheres, Mat. Zametki, 60, 333-355, (1996) · Zbl 0903.41014 [3] Babenko, A G, Exact Jackson-Stechkin inequality in the space $$L$$\^{2}(ℝ\^{m}) (in Russian), Trudy Inst. Mat. i Mekh. UrO RAN, 5, 182-198, (1998) · Zbl 1076.41503 [4] Berdysheva, E E, Two interrelated extremal problems for entire functions of several variables, Mat. Zametki, 66, 336-350, (1999) [5] Jeu, M F E, The Dunkl transform, Invent. Math., 113, 147-162, (1993) · Zbl 0789.33007 [6] Jeu, M F E, Paley-Wiener theorems for the Dunkl transform, Trans. Amer. Math. Soc., 358, 4225-4250, (2006) · Zbl 1160.33010 [7] Dunkl, C F, Integral kernels with reflection group invariance, Canadian J. Math., 43, 1213-1227, (1991) · Zbl 0827.33010 [8] Dunkl, C F, Hankel transforms associated to finite reflection groups, Contemp. Math., 138, 123-138, (1992) · Zbl 0789.33008 [9] Ibragimov, I I; Nasibov, F G, Estimation of the best approximation of a summable function on the real axis by entire functions of finite degree, Dokl. Akad. Nauk SSSR, 194, 1013-1016, (1970) · Zbl 0224.41004 [10] Ivanov, V I; Chertova, D V; Liu, Y P, The sharp Jackson inequality in the space $$L$$\^{2} on the segment [−1, 1] with the power weight, Proc. Steklov Inst. Math., 264, 133-149, (2009) · Zbl 1312.41016 [11] Ivanov, A V; Ivanov, V I, Dunkl’s theory and jackson’s theorem in the space $$L$$_{2}(ℝ\^{d}) with power weight, Proc. Steklov Inst. Math., 273, 86-98, (2011) · Zbl 1229.42022 [12] Li, J; Liu, Y P, The Jackson inequality for the best $$L$$\^{2}-approximation of functions on [0, 1] with the weight $$x$$, Numerical Mathematics: Theory, Methods and Applications, 1, 340-356, (2008) · Zbl 1174.41338 [13] Li, J; Liu, Y P; Su, C M, Jackson inequality and widths of function classes in $$L$$\^{2}([0, 1], $$x$$\^{2ν+1}), J. Complexity, 28, 582-596, (2012) · Zbl 1258.41006 [14] Liu, S. S., Liu, S. D.: Special Functions (in Chinese), 2nd ed. China Meteorological Press, Beijing, 2003 [15] Liu, Y. P.: Higher monotonicity properties of normalized Bessel functions. Int. J. Wavelets, Multiresolut. Inf. Process., to appear · Zbl 1302.33005 [16] Moskovskii, A V, Jackson theorems in the spaces $$L$$_{p} (ℝ\^{n}) and $$L$$_{p,l} (ℝ_{+}), Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform., 3, 44-70, (1997) [17] Nikolskii, S. M.: Approximation of Functions of Several Variables and Imbedding Theorems, Nauka, Moscow 1969; 2nd ed., 1977; English transl. of 1st ed., Springer-Verlag, New York, 1975 [18] Popov, V Yu, On the best mean square approximations by entire functions of exponential type (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat., 6, 65-73, (1972) [19] Popov, V Yu, Exact constants in Jackson inequalities for best spherical mean square approximations (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat., 12, 67-78, (1981) [20] Rösler, M.: Dunkl Operators, Theory and Applications (Springer-Verlag, Berlin, 2003), Ser. Lecture Notes in Math., 1817, 93-135 · Zbl 1029.43001 [21] Rösler, M, Positivity of dunkl’s intertwining operator, Duke Math J., 98, 445-463, (1999) · Zbl 0947.33013 [22] Rösler, M, Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys., 192, 519-542, (1998) · Zbl 0908.33005 [23] Thangavelu, S; Xu, Y, Convolution operator and maximal function for Dunkl transform, Journal D’analyse Mathématique, 97, 25-55, (2005) · Zbl 1131.43006 [24] Triméche, K, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transform. Spec. Funct., 13, 17-38, (2002) · Zbl 1030.44004 [25] Watson, G. N.: A Treatise on the Theory of Bessel Functions, Cambridge university Press, New York, 1944 · Zbl 0063.08184
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.