Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOrbit functions of SU(n) and Chebyshev polynomials
353
0
0.0
(
0
)
Added by
Maryna Nesterenko dr.
Publication date
2009
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Orbit functions of a simple Lie group/Lie algebra L consist of exponential functions summed up over the Weyl group of L. They are labeled by the highest weights of irreducible finite dimensional representations of L. They are of three types: C-, S- and E-functions. Orbit functions of the Lie algebras An, or equivalently, of the Lie group SU(n+1), are considered. First, orbit functions in two different bases - one orthonormal, the other given by the simple roots of SU(n) - are written using the isomorphism of the permutation group of n elements and the Weyl group of SU(n). Secondly, it is demonstrated that there is a one-to-one correspondence between classical Chebyshev polynomials of the first and second kind, and C- and $S$-functions of the simple Lie group SU(2). It is then shown that the well-known orbit functions of SU(n) are straightforward generalizations of Chebyshev polynomials to n-1 variables. Properties of the orbit functions provide a wealth of properties of the polynomials. Finally, multivariate exponential functions are considered, and their connection with orbit functions of SU(n) is established.
rate research
Read More
We shall give bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class and satisfying a second order differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polinomials, which are uniform in all the parameters.
An algebra denoted $mmathfrak{H}$ with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with Hahn polynomials is established. Representation bases corresponding to eigenvalue or generalized eigenvalue problems involving the generators are considered. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions thereby providing a unified description of these functions based on $mmathfrak{H}$. Models in terms of differential and difference operators are used to identify explicitly the underlying special functions as Hahn polynomials and rational functions and to determine their characterizations. An embedding of $mmathfrak{H}$ in $mathcal{U}(mathfrak{sl}_2)$ is presented. A Pade approximation table for the binomial function is obtained as a by-product.
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous Toda for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It also shown that the Kadomtesev-Petvishvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case the deformation do not satisfy a Pearson equation.
We show how polynomial mappings of degree k from a union of disjoint intervals onto [-1,1] generate a countable number of special cases of a certain generalization of the Chebyshev Polynomials. We also derive a new expression for these generalized Chebyshev Polynomials for any number of disjoint intervals from which the coefficients of x^n can be found explicitly in terms of the end points and the recurrence coefficients. We find that this representation is useful for specializing to the polynomial mapping cases for small k where we will have algebraic expressions for the recurrence coefficients in terms of the end points. We study in detail certain special cases of the polynomials for small k and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an expression for the discriminant for the case of two intervals that is valid for any configuration of the end points.
Given $n$ polynomials $p_1, dots, p_n$ of degree at most $n$ with $|p_i|_infty le 1$ for $i in [n]$, we show there exist signs $x_1, dots, x_n in {-1,1}$ so that [Big|sum_{i=1}^n x_i p_iBig|_infty < 30sqrt{n}, ] where $|p|_infty := sup_{|x| le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(sqrt{n})$ for the Chebyshev polynomials $T_1, dots, T_n$, and can be seen as a polynomial analogue of Spencers six standard deviations theorem.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
