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.
We consider approximation of functions of several variables by continuous linear splines interpolating the given function in the knots of a rectilinear lattice. For function classes defined in terms of a modulus of continuity, we give an exact es-timate for the error of approximation. In the particular case when the modulus of continuity is concave and the distance between points in Rd is measured in the l(p)-norm with 1 <= P <= 3, we calculate an explicit value of the exact approximation error on the class. Surprisingly, the behavior changes dramatically if P > 3. We show that the our estimate is no longer true, in general, when P > 3. We also consider approximation of a first derivative of a function by the corre-sponding derivative of the linear continuous spline and obtain an upper estimate for the error of approximation for an arbitrary modulus of continuity, all 1 <= P <= infinity, and triangulations of the staircase type.
In this paper we obtain a Bernstein inequality for polynomial differential operators and polynomial integral operators on Orlicz spaces. Let Phi : [0, +infinity) -> [0, +infinity] be an arbitrary Young function, K be an arbitrary compact set in R and P(x) be a polynomial. Then there exists a constant C independent of Phi such that IIPm(D)fII((Phi)) <= Cm sup(x is an element of K) |P-m(x)|IIfII((Phi)) for all m is an element of N and all f is an element of L-Phi,L-K, where L-Phi,L-K =( ){f is an element of L-Phi(R) : supp (f)over bar subset of K}, (f)over bar is the Fourier transform of f and II.II((Phi) )is the Luxemburg norm. The corresponding result for polynomial integral operators and an application are also given.
Given scattered data values of a piecewise smooth function f within a domain 52, we look for a piecewise adaptive approximation to f. Approximation techniques for scattered data approximation, as Radial Basis Function (RBF) or Moving Least -Squares (MLS), achieve reduced approximation orders near the boundary of the domain and near the unknown curves of jump singularities of the function or its derivatives. The idea used here is that the approximation errors near the bound-aries, and near a singularity curve, fully characterize the behavior of the function at these locations. We refer to these approximation error values as the signature of f. In this paper, we aim at using these values in order to define the approximation. Assuming smoothness of the singularity curve, we suggest using a signed-distance approach to construct an approximation of the singularity curve. Now we find approximations to the different smooth segments of f, based upon matching the signatures of the approximant to the signature of f. The resulting approximation captures the singularity of the given data. As a result, the error in this first stage approximation is smooth. Hence, a second stage corrected improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.
Inspired by the work of L. Feje & acute;r and some of his followers, we present inequalities for various sine polynomials. Among others, we offer a new extension of the classical Feje & acute;r-Jackson inequality and we prove that the inequality Sigma;(n)(k =1) (n - k + c n-k)(-1) sin(kx) < 0 (c is an element of R \ {-1, -2, ...}) holds for all n >= 2 and x is an element of (0, pi) if and only if c is an element of [-3/2, 4/3]. As an application of this result we obtain that the function x bar right arro x (1-2F(1)(1, 1; a; x)/x(2)+ bx+1) (-1/2 <= a <= -1/3; -2 < b < 2) is absolutely monotonic on (0, 1). Here, F-2(1) denotes the Gaussian hypergeometric function.
For the functions f, which can be represented in the form of the convolution f(x)=a_0/2+1/π∫_-π^π∑_k=1^∞e^-α k^rcos(kt-βπ/2)φ(x-t)dt, φ⊥1, α>0, r∈(0,1), β∈ℝ, we establish the Lebesgue-type inequalities of the form f-S_n-1(f)_C≤ e^-α n^r(4/π^2lnn^1-r/α r + γ_n) E_n(φ)_C. These inequalities take place for all numbers n that are larger than some number n_1=n_1(α,r), which constructively defined via parameters α and r. We prove that there exists a function, such that the sign "≤" in given estimate can be changed for "=".
When it comes to Machine Learning, without a doubt one of the most important topics is the method for learning functions. Prior, one aspect of the hypercircle inequality (Hi) in the context of kernel-based machine learning was introduced by Kannika Khompurngson and Charles A. Micchelli. However, the material on Hi only applies to the case of accurate data. Our previous work which was motivated by this limited said data has extended the hypercircle inequality to circumstances for which there is known data error. In this paper, our special interest is focused on a detailed analysis of the hypercircle inequality for data error (Hide) measured with l & INFIN; norm. Furthermore, the result is applied to a problem about the learning of the value of a function in the Hardy space of square-integrable function on the unit circle which is well-known in reproducing kernel Hilbert spaces.
In this paper we derive second and third order nonlinear difference equations for one of the recurrence coefficients in the three term recurrence relation of polynomials orthogonal with respect to a modified Laguerre weight. We show how these equations can be obtained from the B¨acklund transformations of the third Painlev´e equation. We also show how to use nonlinear difference equations to derive a few terms in the formal asymptotic expansions in n of the recurrence coefficients.
We study the log-concavity of the function a -> a(delta)J(a)(x), where J(a) (x) = e(x) - (1 + x/a)(a) and use our result to show that J(a)(x) satisfies certain functional inequalities. Among others, we prove that if a, b and lambda, mu are positive real numbers with a &NOTEQUexpressionL; b and lambda + mu = 1, then we have for all x > 0, a(lambda)b(mu)/lambda a + mu b < J(lambda a+mu b)(x)/J(a)(x)(lambda) J(b)(x)(mu). The lower bound is best possible.
We consider the classical problem of maximizing the value of the derivative of a polynomial at a given point x0 ∈ [−1, 1]. The corresponding extremal problem for general polynomials in the uniform norm was solved by V. Markov. In this paper, we consider the analog of this problem for k-absolutely monotone polynomials. As a consequence, we solve the analog of V. Markov’s problem, find the exact constant in Bernstein’s inequality and give a new proof of A. Markov’s inequality for monotone polynomials.
This article is devoted to results of rational approximation of the Markov function alpha circumflex expressionccent (z) = integral(F) d alpha(x)/z - x , where alpha is a positive Borel measure with support supp alpha = F = [a, b] subset of (0, infinity) and d alpha/dx > 0 a.e. on F (with respect to the Lebesgue measure). We study asymptotic properties of the best uniform rational approximation of Markov functions circumflex expressionccent alpha on point systems E-N subset of (-infinity, 0) when the number of points N in the set E-N and the degree of rational approximants n satisfy an asymptotic relation N/n -> theta > 2 as n -> infinity. The degree of rational approximation is described in terms of the solutions of certain logarithmic potential-theoretic problems, central among which is a minimal energy problem in the presence of an external field. We also investigate the limit distribution of poles of the best rational approximants and of points of Chebyshev alternance
In this work we address the problem of approximating a smooth function on a bounded domain from a set of data points on which we know the values of the objective function. While we can generally guarantee impressive approximations in the interior of the domain, the theory does not extend to the boundary of the domain. Indeed, numerical experiments present all forms of artifacts when performing approximations near the boundary of the domain. To achieve adequate approximations near boundaries, we will build upon our previous work, in which we have managed to construct high-order approximations to singular functions. By considering the boundary of the domain as a singularity, we show that we can similarly return high-order approximations to the objective function, even in the immediate vicinity of the boundary of the domain.
Let X be a Banach space, (I, mu) be a finite measure space and G be a closed subspace of X. In this paper, we study the problem of best coapproximation in the metric space Lp(I, X), 0 < p < 1, as a special case of the problem of coproximity of L-phi (I, G) in L-phi (I, X) whenever G is coproximinal in X, where phi is an increasing continuous subadditive function on [0, infinity) with phi (0) = 0, and L-phi(I, X), the space of all X-valued strongly measurable functions on I with integral(L) phi ||f(t)|| dt < infinity.
We shall present new characterizations of partially greedy and almost greedy bases. A new class of basis (which we call reverse partially greedy basis) arises naturally from these characterizations of partially greedy bases.
Let α∈(1,∞) and μ be a regular finite Borel measure on a locally compact abelian group. The paper deals with a general trigonometric approximation problem in L^α(μ), which arises in prediction theory of harmonizable symmetric α-stable processes. To solve it, a duality method is applied, which is due to Nakazi and was generalized by Miamee and Pourahmadi and in the sequel successfully applied by several authors. The novelty of the present paper is that we do not make any additional assumption on μ. Moreover, for α=2, multivariate extensions are obtained.
In this paper we study the behavior of best Lᵖ-approximations by rational functions to an analytic function on union of disks, when the measure of them tends to zero.
