Direct and Inverse Approximation Theorems for Local Trigonometric Bases

We study best approximations of polynomials with respect to multiplicative systems in the Lp-spaces with Muckenhoupt weights. Using Jackson’s and Bernstein’s inequalities, we obtain the direct and inverse approximation theorems in terms of the K-functional and the inverse theorem of the Timan-Besov type. In the case of a power weight, we give a criterion for the membership of a function in the weighted Lp-space in terms of the Fourier coefficients with respect to multiplicative systems.

One considers the spectral problem with boundary conditions , , for functions on . It is assumed that is a linear bounded operator from the Hölder space , , into and the are bounded linear functionals on with . Let be the linear span of the root functions of the problem , , , corresponding to the eigenvalues with , and let . An estimate of is obtained in terms of the -functional for (the direct theorem) and an estimate of this -functional is obtained in terms of for (the inverse theorem). In several cases two-sided bounds of the -functional are found in terms of appropriate moduli of continuity, and then the direct and the inverse theorems are stated in terms of moduli of continuity. For the spectral problem with periodic boundary conditions these results coincide with Jackson's and Bernstein's direct and inverse theorems on the approximation of functions by a trigonometric system.

The Hölder classes L p α ( L ) L_p^{\alpha } (L) in the L p ( L ) L_p(L) norm on a chord-arc curve L L in R 3 \mathbb {R}^3 are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the L p ( L ) L^p(L) norm, the direct theorem is proved for a certain subclass of L p α ( L ) L^\alpha _p(L) and the inverse theorem covers the entire Hölder class.

Many inverse theorems have been proven using the ordinary modulus of smoothness but no one proved the invers theorems for approximation by Bernstein’s Durrmeyer operator in terms 2nd modulus of smoothness.

This is the second of a series devoted to the direct and inverse approximation theorems of the p-version of the finite element method in the framework of the weighted Besov spaces. In this paper, we combine the approximability of singular solutions in the Jacobi-weighted Besov spaces, which were analyzed in the previous paper,4 with the technique of partition of unity in order to prove the optimal rate of convergence of the p-version of the finite element method for elliptic boundary value problems on polygonal domains.

This thesis is concerned with Local Discriminant Basis (LDB) algorithm, its properties, optimization and applications in feature extraction and classification. LDB algorithm targets features extraction from redundant dictionaries such as wavelet packets or local trigonometric bases at low computational complexity. As the main contribution of this thesis, an optimization process is introduced to further improve the accuracy of the overall scheme in applications when a region of interest can be specified by the experts in the field of application (based on LDB selected features) to further characterize signal classes in smaller regions. Audio signal and textured image classifications are practical applications that are studied in this thesis to test the efficiency of optimally weighted local discriminant basis algorithm (OLDB) as a feature extraction scheme. Various properties of the algorithm such as noise behavior and stability analysis are studied from an engineering perspective. The implementation aspects of the algorithm in one dimension are reviewed as well as in two dimensions that serve as implementation guidelines.

A genetic algorithm for the selection of wavelet packet bases for various tasks is proposed. We introduce a transform operator to convert the constrained combinatorial basis selection problem to an unconstrained numerical optimization problem. This algorithm offers a great deal of flexibility for the selection of wavelet packet bases in terms of different criteria for different tasks. To demonstrate the performance of this algorithm, we present two examples respectively for signal approximation and signal classification. Although the algorithm is described with wavelet packets, it can also be used for the selection of other tree structured local bases, such as local trigonometric bases.

In this work we prove some inverse and simultaneous approximation theorems in the weighted variable exponent Lebesgue space and discuss the constructive description problems in the generalized Lipschitz subclasses.

In this paper, we consider linear combinations of modified Baskakov operators. By use of several lemmas, we obtain the inverse theorem (Theorem 1.1) and the saturation (Theorem 1.2) for those linear combinations.

In this paper, we introduce the Bezier variant of the Jakimovski–Leviatan–Păltănea operators based on Appell polynomials. We establish some local results, a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness and also study the rate of convergence for the functions having a derivative of bounded variation for these operators.

ABSTRACTThis work introduces ‐Bézier–Bernstein‐integral type operators, along with local approximation results, a direct approximation theorem that leverages the Ditzian–Totik modulus of smoothness, and a quantitative Voronovskaja‐type theorem using the Ditzian–Totik modulus of continuity. Additionally, we derive the convergence rate for differential functions with bounded variation in their derivatives. Ultimately, we demonstrate numerical examples that validate the theoretical results and showcase the effectiveness of the defined operators.

Direct and converse theorems are established for the q-Bernstein polynomials introduced by G. M. Phillips. The direct approximation theorems are given for the second-order Ditzian–Totik modulus of smoothness. The converse results are theorems of Berens–Lorentz type.

Polynomial approximation with doubling weights having finitely many zeros and singularities

We obtain an instance of the multidimensional analogs of the classical direct and converse Jackson and Bernshtein-Vallee-Poussin theorems (a version stronger than those obtained by S. M. Nikol’skii) and significant generalizations of these theorems to polynomial approximations of functions of real variables from Nikol’skii and Besov spaces on bounded domains with Lipschitzian boundary. The direct theorem concerning the Nikol’skii spaces was previously obtained by Yu. A. Brudnyi and generalized to joint polynomial approximations of functions and their derivatives by B. N. Konovalov.

In this paper we give the direct approximation theorem, the inverse theorem, and the equivalence theorem for Szász-Durrmeyer-Bézier operators in the space ( ) with Ditzian-Totik modulus. MSC:41A25, 41A27, 41A36.