Some extremal properties of multivariate polynomial splines in the metric Lp(?d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (ℝ^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (ℝ^d in the metric L_p((ℝ^d).     

A Sharp Jackson Inequality in Lp(ℝd) with Dunkl Weight

A sharp Jackson inequality in the space L_p(?^d), 1 ≤ p < 2, with Dunkl weight is proved. The best approximation is realized by entire functions of exponential spherical type. The modulus of continuity is defined by means of a generalized shift operator bounded on L_p, which was constructed earlier by the authors. In the case of the unit weight, this operator coincides with the mean-value operator on the sphere.

     

A convolution metric in the space of measures and ε-isometries in Lp

For >0, 2,4,6,... on the set of those Borel measures on , such that x^pd(x)<, one introduces a metric, characterizing the nearness of the convolutions x^p* and x^p*. From the convergence of a sequence of probability measures in this metric there follows its convergence in the Kantorovich-Rubinshtein metric. From here one derives theorems on the approximation of -isometries: if H is a finite-dimensional subspace in L_p, then there exists a continuous function _H(), such that for any linear -isometric operator THL_p there exists a linear isometry UH L_p, such that T–U<_H().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 151–156, 1987.     

Some remarks concerning the Fourier transform in the space L 2(ℝ n )

For the Fourier transform in the space L ₂(?²) of square integrable multivariable functions, two practically useful estimates are proved in certain classes of functions characterized by a generalized continuity modulus.     

Limit theorems for the best polynomial approximations in the L∞ metric

A limiting equality is established between the best approximations in L of functions of several variables by algebraic polynomials and entire functions of exponential type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 336–342, March, 1991.     

Limit theorems for the best polynomial approximations in the L∞ metric

A limiting equality is established between the best approximations in L of functions of several variables by algebraic polynomials and entire functions of exponential type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 336–342, March, 1991.     

Some properties of polynomials orthogonal in a space with “intermediate” topology. Kernel function and extremal properties

A space of holomorphic functions is considered. A topology in this space is intermediate between the topology of uniform convergence and the topology of uniform convergence on compact sets. The properties of systems of orthonormal polynomials are studied in Hilbert spaces with this topology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 915–923, July, 1993.     

Generalized multiresolution structures in reducing subspaces of L 2(ℝ d )

In this paper, we introduce the notion of generalized multiresolution structure (GMS) in the set-ting of reducing subspaces of L2(Rd). For a general expansive matrix, we obtain a necessary and sufficient condition for GMS, and prove the existence of GMS in a reducing subspace. Using GMS, we obtain a pyramid decomposition and a frame-like expansion for signals in reducing subspaces.