Browsing by Subject "summability"

Now showing 1 - 4 of 4

Access status: Open Access ,
Analytic continuation of solutions of some nonlinear convolution partial differential equations
(2015) Tahara, Hidetoshi
The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.
Access status: Open Access ,
Cesàro summability of Taylor series in higher order weighted Dirichlet-type spaces
(Wydawnictwa AGH, 2024) Ghara, Soumitra; Gupta, Rajeev; Reza, Md. Ramiz
For a positive integer $m$ and a finite non-negative Borel measure $\mu$ on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces $\mathcal H_{\mu, m}$. We show that if $\alpha\gt\frac{1}{2}$, then for any $f$ in $\mathcal H_{\mu, m}$ the sequence of generalized Cesàro sums $\{\sigma_n^{\alpha}[f]\}$ converges to $f$. We further show that if $\alpha=\frac{1}{2}$ then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer $m$.
Access status: Open Access ,
On the summability of divergent power series solutions for certain first-order linear PDES
(2015) Hibino, Masaki
This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.
Access status: Open Access ,
q-analogue of summability of formal solutions of some linear q-difference-differential equations
(2015) Tahara, Hidetoshi; Yamazawa, Hiroshi
Let $q\gt 1$. The paper considers a linear q-difference-differential equation: it is a $q$-difference equation in the time variable $t$, and a partial differential equation in the space variable $z$. Under suitable conditions and by using $q$-Borel and $q$-Laplace transforms (introduced by J.-P. Ramis and C. Zhang), the authors show that if it has a formal power series solution $\hat{X}(t,z)$ one can construct an actual holomorphic solution which admits $\hat{X}(t,z)$ as a $q$-Gevrey asymptotic expansion of order 1.