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.

Lidstone–Euler interpolation and related high even order boundary value problem

We consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent polynomial sequence. Particularly, we consider the fourth-order case, arising in various physical models. Finally, we present some numerical examples and we compare the proposed method with a modified decomposition method for a tenth-order problem. The numerical results confirm the theoretical and computational ones.

Model theory and metric convergence II: Averages of unitary polynomial actions

We use model theory of metric structures to prove the pointwise convergence, with a uniform metastability rate, of averages of a polynomial sequence { T n } \{T_n\} (in Leibman’s sense) of unitary transformations of a Hilbert space. As a special case, this applies to unitary sequences { U p ( n ) } \{U^{p(n)}\} where p p is a polynomial Z → Z \mathbb {Z}\to \mathbb {Z} and U U a fixed unitary operator; however, our convergence results hold for arbitrary Leibman sequences. As a case study, we show that the non-nilpotent “lamplighter group” Z ≀ Z \mathbb {Z}\wr \mathbb {Z} is realized as the range of a suitable quadratic Leibman sequence. We also indicate how these convergence results generalize to arbitrary Følner averages of unitary polynomial actions of any abelian group G \mathbb {G} in place of Z \mathbb {Z} .