Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

We define generalized standard triples $\boldsymbol{X}$, $\boldsymbol{Y}$, and $L(z) = z\boldsymbol{C}_{1} - \boldsymbol{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\boldsymbol{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\boldsymbol{X}(z \boldsymbol{C}_{1}~-~\boldsymbol{C}_{0})^{-1}\boldsymbol{Y}~=~\boldsymbol{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\boldsymbol{P}$. This representation can be used in constructing so-called algebraic linearizations for matrix polynomials of the form $\boldsymbol{H}(z) = z \boldsymbol{A}(z)\boldsymbol{B}(z) + \boldsymbol{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$. This can be done even if $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\boldsymbol{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases.

An Efficient Polynomial Chaos Method for Stiffness Analysis of Air Spring Considering Uncertainties

Traditional methods for stiffness analysis of the air spring are based on deterministic assumption that the parameters are fixed. However, uncertainties have widely existed, and the mechanic property of the air spring is very sensitive to these uncertainties. To model the uncertainties in the air spring, the interval/random variables models are introduced. For response analysis of the interval/random variables models of the air spring system, a new unified orthogonal polynomial expansion method, named as sparse quadrature-based interval and random moment arbitrary polynomial chaos method (SQ-IRMAPC), is proposed. In SQ-IRMAPC, the response of the acoustic system related to both interval and random variables is approximated by the moment-based arbitrary orthogonal polynomial expansion. To efficiently calculate the coefficient of the interval and random orthogonal polynomial expansion, the sparse quadrature is introduced. The proposed SQ-IRMAPC was employed to analyze the mechanic performance of an air spring with interval and/or random variables, and its effectiveness has been demonstrated by fully comparing it with the most recently proposed orthogonal polynomial-based interval and random analysis method.

Nonnegative and Strictly Positive Linearization of Jacobi and Generalized Chebyshev Polynomials

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $$(P_n(x))_{n\in \mathbb {N}_0}$$ ( P n ( x ) ) n ∈ N 0 satisfies nonnegative linearization of products, i.e., the product of any two $$P_m(x),P_n(x)$$ P m ( x ) , P n ( x ) is a conical combination of the polynomials $$P_{|m-n|}(x),\ldots ,P_{m+n}(x)$$ P | m - n | ( x ) , … , P m + n ( x ) . Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. Gasper (Can J Math 22:582–593, 1970) was able to determine the set V of all pairs $$(\alpha ,\beta )\in (-1,\infty )^2$$ ( α , β ) ∈ ( - 1 , ∞ ) 2 for which the corresponding Jacobi polynomials $$(R_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( R n ( α , β ) ( x ) ) n ∈ N 0 , normalized by $$R_n^{(\alpha ,\beta )}(1)\equiv 1$$ R n ( α , β ) ( 1 ) ≡ 1 , satisfy nonnegative linearization of products. Szwarc (Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol 2, Nova Sci. Publ., Hauppauge, NY pp 103–139, 2005) asked to solve the analogous problem for the generalized Chebyshev polynomials $$(T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( T n ( α , β ) ( x ) ) n ∈ N 0 , which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure $$(1-x^2)^{\alpha }|x|^{2\beta +1}\chi _{(-1,1)}(x)\,\mathrm {d}x$$ ( 1 - x 2 ) α | x | 2 β + 1 χ ( - 1 , 1 ) ( x ) d x . In this paper, we give the solution and show that $$(T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( T n ( α , β ) ( x ) ) n ∈ N 0 satisfies nonnegative linearization of products if and only if $$(\alpha ,\beta )\in V$$ ( α , β ) ∈ V , so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper’s original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper’s one.

On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.