We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W . More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of s...

Based on the common Hermite-Gaussian modes, a general class of orthonormal Hermite-Gaussian-type modes is introduced. Such modes can most easily be defined by means of their generating function. It is shown that these modes remain in their class of orthonormal Hermite-Gaussiantype modes, when they propagate through first-order optical systems. A propagation law for the generating function is fo...

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider two new q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci polynomials and the recent works on the combinatorics of the Matrix Ansatz of the PASEP.

We investigate weighted Lp(0 < p <.) convergence of Hermite and Hermite– Fejér interpolation polynomials of higher order at the zeros of Freud orthogonal polynomials on the real line. Our results cover as special cases Lagrange, Hermite– Fejér and Krylov–Stayermann interpolation polynomials. © 2001 Academic Press

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.

Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the Triebel-Lizorkin and Besov spaces on R induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite Triebel-Lizorkin and Besov spaces are, in general, different from the respective classical sp...

The images of Hermite and Laguerre Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterised. These are used to characterise the image of Schwartz class of rapidly decreasing functions f on Rn and Cn under these semigroups. The image of the space of tempered distributions is also considered and a Paley-Wiener theorem for windowed ( short-time) Fourier tran...

Using the results on the coefficients of Hermite-Fejér interpolations in [5], we investigate convergence of Hermite and Hermite-Fejér interpolation of order m, m = 1, 2, . . . in Lp(0 < p < ∞) and associated product quadrature rules for a class of fast decaying even Erdős weights on the real line.

A Gaussian filter using the Hermite orthonormal series of functions is developed. The filter is compared with a similar filter using the HermiteRodriguez series on Doppler radar signals. The results indicate that a more compact filter can be achieved with the Hermite series compared to the Hermite-Rodriguez series.