Non-symmetric approximations of functional classes by splines on the real line

Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$. We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$. These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.

Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives

In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

Partial best approximations and the absolute Cesaro summability of multiple Fourier series

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1–4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f ∈ Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ- summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1 = ... = βs = 1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f ∈ Lq(I2) are obtained in the cases β1 = 1/q', −1 < β2 < 1/q'(in the second theorem), 1/q'< β1 < +∞, β2 = 1/q', (in the third theorem), −1 < β1 < 1/q', 1/q' < β2 < +∞ (in the fourth theorem).

On the value of the widths of some classes of functions from L_2

In this paper we find sharp inequalities of Jackson-Stechkin type between the best approximations of periodic differentiable functions by trigonometric polynomials and generalized moduli of continuity of m-th order in the space L_2. The exact values of various n-widths of classes of functions from L_2 defined by the generalized modus of continuity of the $r$-th derivative of the function f are calculated.

Approximate calculation of 8-point DCT for various scenarios of practical applications

In this paper, based on the parametric model of the matrix of discrete cosine transform (DCT), and using an exhaustive search of the parameters’ space, we seek for the best approximations of 8-point DCT at the given computational complexities by taking into account three different scenarios of practical usage. The possible parameter values are selected in such a way that the resulting transforms are only multiplierless approximations, i.e., only additions and bit-shift operations are required. The considered usage scenarios include such cases where approximation of DCT is used: (i) at the data compression stage, (ii) at the decompression stage, and (iii) both at the compression and decompression stages. The obtained results in effectiveness of generated approximations are compared with results of popular known approximations of 8-point DCT of the same class (i.e., multiplierless approximations). In addition, we perform a series of experiments in lossy compression of natural images using popular JPEG standard. The obtained results are presented and discussed. It should be noted, that in the overwhelming number of cases the generated approximations are better than the known ones, e.g., in asymmetric scenarios even by more than 3 dB starting from entropy of 2 bits per pixel. In the last part of the paper, we investigate the possibility of hardware implementation of generated approximations in Field-Programmable Gate Array (FPGA) circuits. The results in the form of resource and energy consumption are presented and commented. The experiment outcomes confirm the assumption that the considered class of transformations is characterized by low resource utilization.

Are climate models that allow better approximations of local meteorology better for the assessment of hydrological impacts? A statistical analysis of droughts

This work studies the benefit of using more reliable local climate scenarios to analyse hydrological impacts. It assumes that more reliable local scenarios are defined with the statistically corrected Regional Climate Model (RCM) simulations when they provide better approximations to the historical basic and drought statistics. The paper analyses if the best solutions in terms of their approximation to the local meteorology also provide the best hydrological assessments. A classification of the corrected RCM simulations attending to both approximations is performed. It has been applied in the Cenajo Basin (southeast Spain), where we demonstrate that the best approximations of the historical meteorological statistics provide also the best approximations of the hydrology ones. The selected RCMs were used to generate future (2071–2100) local scenarios under the RCP 8.5 emission scenario. The two selected RCMs predict significant changes of mean precipitation (−31.6 and −44.0 %) and mean temperature (+26.0 and +32.2 %). They also predict higher frequency (from 5 events in the historical period to 20 and 22 in the future), length (4.8 to 7.4 and 10.5 months), magnitude (2.53 to 6.56 and 9.62 SPI) and intensity (0.48 to 1.00 and 0.94 SPI) of extreme meteorological droughts.

On Best Approximations in Hyperconvex Spaces

In this manuscript, we present further extensions of the best approximation theorem in hyperconvex spaces obtained by Khamsi.