We consider quasi-interpolation with a main application in radial basis function approximations and compression in this article. Constructing and using these quasi-interpolants, we consider wavelet and compression-type approximations from their linear spaces and provide convergence estimates. The results include an error estimate for nonlinear approximation by quasi-interpolation, results about compression in the space of continuous functions and a pointwise convergence estimate for approximands of low smoothness.
In this paper we solve the problem of approximating functionals (phi(A)x, f) (where phi(A) is some function of self-adjoint operator A) on the class of elements of a Hilbert space that is defined using another function psi(A) of the operator A. In addition, we obtain a series of sharp Taikov-type additive inequalities that estimate vertical bar(phi(A)x, f)vertical bar with the help of parallel to psi(A)x parallel to and parallel to x parallel to. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in Hormander theorem on comparison of operators in the case when operators are acting in a Hilbert space and are functions of a self-adjoint operator. Second, we obtain Taikov-type inequalities for functions of the operator 1/i d/dt in the spaces L-2(R) and L-2(T), as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.
Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike polynomials being described in the unit disk, this method is particularly effective in solving PDEs over a circular region. Further, the proposed method can solve PDEs with discontinuous Dirichlet and Neumann boundary conditions, and as these discontinuous functions cannot be defined at some of the Chebyshev or Gauss-Lobatto points, the much acclaimed pseudo-spectral methods are not directly applicable to such problems. Solving such PDEs is also a new application of Zernike polynomials as so far the main application of these polynomials seem to have been in the study of optical aberrations of circularly symmetric optical systems. In the present method, the given PDE is converted to a system of linear equations of the form Ax = b which may be solved by both l(1) and l(2) minimization methods among which the l(1) method is found to be more accurate. Finally, in the expansion of a function in terms of Zernike polynomials, the rate of decay of the coefficients is given for certain classes of functions.
In this paper, the authors present some monotonicity, convexity-concavity, log -convexity, log-concavity, subadditivity and superadditivity properties of certain combinations defined in terms of the function u(x, y) = J(y)(x) = e(x) - (1 + x/y)(y) which is a solution to the partial differential equation (x + y)u(yx) - y(uy) + u(x) - u = 0 for x, y is an element of (0, infinity). By these properties, sharp double inequalities are obtained for the functions J(y)(x) and Rn+1(x) (sic) J(n+1)(x)(2)/[J(n)(x)J(n+2)(x)] for x, y is an element of (0, infinity) and n is an element of N, including the sharp Tur & aacute;n-type inequalities for R-n(x) (n >= 2), thus perfecting the related results proved recently by H. Alzer and solving his open problem on the best possible upper bound of R-n(x).
In a recent paper, construction of fractal surfaces is viewed as a framework to associate a parameterized family of self-referential functions with a prescribed continuous bivariate function on a rectangular region. In this study, first we develop a closely related theme that provides a family of self-referential functions corresponding to a bivariate Lebesgue integrable function. Some approximation theoretic aspects of these self-referential functions that are close to the germ function are considered. Our methods are based in part on similar results established by the first author when the germ function is univariate. Further, we study the bivariate fractal operator that maps the germ function to its self-referential coun-terpart. These results on fractal operator are perhaps of independent interest as the previous works on univariate counterparts in Lebesgue spaces had been obtained only in the setting of bounded linear maps.
