Sparse-grid interpolation provides good approximations to smooth functions in high dimensions based on relatively few function evaluations, but in standard form is expressed in Lagrange polynomials and requires function values at all points of a sparse grid. Here we give a block-diagonal factorization of the matrix for changing basis from a Lagrange polynomial formulation of a sparse-grid inter...

With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an m-bonacci orbital angular momentum (OAM) p...

Many types of radial basis functions, such as multiquadrics, contain a free parameter. In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge. Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials. In the 1-D case, we prove that with s...

4 Discretization using Polynomial Interpolation Consider a function to be a continuous function defined over and let represent the values of the function at an arbitrary set of points in the domain Another function, say in that assumes values exactly at is called an interpolation function. Most popular form of interpolating functions are polynomials. Polynomial interpolation has many important ...

Inspired by the advancements in (fully) homomorphic encryption recent decades and its practical applications, we conducted a preliminary study on underlying mathematical structure of corresponding schemes. Hence, this paper focuses investigating challenge deducing bivariate polynomials constructed using operations, namely repetitive additions multiplications. To begin with, introduce an approac...

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a 1/r singularity. For hydrogenic atoms, numerically exact ...

As known, the problem of choosing ‘‘good’’ nodes is a central one in polynomial interpolation. While the problem is essentially solved in one dimension (all good nodal sequences are asymptotically equidistributed with respect to the arc-cosine metric), in several variables it still represents a substantially open question. In this work we consider new nodal sets for bivariate polynomial interpo...

Abstract We consider the problem of construction determinant formulas for partition function six-vertex model with domain wall boundary conditions that depend on two sets spectral parameters. In pioneering works Korepin and Izergin a formula was proposed proved using recursion relation. later works, equivalent were given by Kostov rational case Foda Wheeler trigonometric case. Here, we develop ...

Denote by Ln the projection operator obtained by applying the Lagrange interpolation method, weighted by (1 − x), at the zeros of the Chebyshev polynomial of the second kind of degree n + 1. The norm ‖Ln‖ = max ‖f‖∞≤1 ‖Lnf‖∞, where ‖ · ‖∞ denotes the supremum norm on [−1, 1], is known to be asymptotically the same as the minimum possible norm over all choices of interpolation nodes for unweight...

Abstract. We consider a Lagrange–Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r−1) derivatives, at the points ±1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted L-spaces, 1 < p < ∞, proving a Marcinkiewicz inequality involving the derivative of the polynomial at ...