ABSTRACT. Let M be a symmetric positive definite moment functional and let {PM n (ω)}n∈N be the family of orthonormal polynomials that corresponds to M. We introduce a family of linear differential operators K = (−i) P n (i d dt ), called the chromatic derivatives associated with M, which are orthonormal with respect to a suitably defined scalar product. We consider a Taylor type expansion of a...

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d = 4. Several examples of such scaling functions are provided in this paper....

The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in L2(R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L2(R) then the sequence is finite. In a related result, Shapiro ...

The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in L(R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L(R) then the sequence is finite. In a related result, Shapiro al...

Bochkariev's theorem states that for any uniformly bounded orthonormal system Φ, there is a Lebesgue integrable function such the Fourier series of it with respect to Φ diverges on set positive measure. In this paper, we extended some class variable exponent spaces. We characterized spaces Lp(⋅)[0;1], 1<p(x)<∞ a.e. [0;1], above mentioned Bochkarev's valid.

In a Freud-type weighted (w) space, introducing another weight (v) with infinitely many roots, we give a complete and minimal system with respect to vw, by deleting infinitely many elements from the original orthonormal system with respect to w. The construction of the conjugate system implies an interpolation problem at infinitely many nodes. Besides the existence, we give some convergence pro...