Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm inequalities for the partial sum operators $S_n$, their maximal operator $S^*$ and the commutator $[M_b, S_n]$, where $M_b$ denotes the operator of pointwise multiplication by b \in \BMO. We also prove some norm inequalities for $S_n$ when $\nu$ is a sum of a Laguerre weight on $\R^+$ and a positive mass on $0$

On the Invalidity of Fourier Series Expansions of Fractional Order

The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions $e^{in\omega x}$ are replaced by the Mittag-Leffler functions $E_\alpha \left (i (n\omega x)^\alpha\right) ,$ over the interval $[0, M_\alpha/ \omega]$ where $0< \omega<\infty$ and $M_\alpha$ is the period of the function $E_\alpha \left( ix^\alpha\right),$ i.e., $E_\alpha \left( ix^\alpha\right)=E_\alpha \left( i(x+M_\alpha)^\alpha\right).