Quotient Of Polynomials Research Articles

Discrete rational 𝐿_{𝑝} approximation

In this paper, the problem of approximating a function defined on a finite subset of the real line by a family of generalized rational functions whose numerator and denominator spaces satisfy the Haar conditions on some closed interval [a, b] containing the finite set is considered. The pointwise closure of the family restricted to the finite set is explicitly determined. The representation obtained is used to analyze the convergence of best approximations on discrete subsets of [a, b] to best approximations over the whole interval (as the discrete subsets become dense) in the case that the function approximated is continuous on [a, b] and the rational family consists of quotients of algebraic polynomials. It is found that the convergence is uniform over [a, b] if the function approximated is a so-called normal point. Only L p {L_p} norms with 1 ⩽ p > ∞ 1 \leqslant p > \infty are employed.

Schrödinger equation and quantum state codons in discrete transform space

Quantum states of simple systems are shown to have composite root-pole structure in the complex transform plane. The Schrödinger condition becomes the inverse of a continuity equation expressing invariance of the Ψ-transform codon under discrete displacement. Four distinct quotient polynomial solutions model the Legendre, Hermite, Laguerre, and Jacobi polynomial families. Schrödinger coefficients are identified with pole strengths of quotient polynomials and are geometrically interpreted in terms of universal root and pole interactions.

On quotients of exponential polynomials

On quotients of exponential polynomials

The quotient of exponential polynomials

The quotient of exponential polynomials