On mean convergence of Fourier-Jacobi series

We establish the order of growth of modified Lebesgue constants of Fourier-Jacobi sums in $L_{p,w}$ spaces.

On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients

In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find and classify a solution by virtue of an asymptotic of some relation containing the Jacobi series coefficients of the right-hand side. The main results are the following—the conditions imposed on the parameters, under which the Abel equation has a unique solution represented by the series, are formulated; the relationship between the values of the parameters and the solution smoothness is established. The independence between one of the parameters and the smoothness of the solution is proved.

Riemann–Liouville Operator in Weighted Lp Spaces via the Jacobi Series Expansion

In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces.

Determination of a jump by conjugate Fourier-Jacobi series

We prove the equiconvergence related to conjugate Fourier-Jacobi series and differentiated Fourier-Jacobi series for functions of harmonic bounded variation. A jump of a such function is determined by the partial sums of its conjugate Fourier-Jacobi series.

KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.