Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.06766 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917747048316928 |
|---|---|
| author | Sahoo, Partiswari Maharana Sabita |
| author_facet | Sahoo, Partiswari Maharana Sabita |
| contents | In the present paper, the random series $\sum\limits_{m=0}^\infty c_m C_m(\varpi)q_m^{(ζ,η)}(u)$ in orthogonal Jacobi polynomials $q_m^{(ζ,η)}(u)$ is discussed. The scalars $c_m$ are Fourier--Jacobi coefficients of a function in the weighted space $\mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}),\mathrm{p}>1.$ The random variables $C_m(\varpi)$ are chosen to be the Fourier--Jacobi coefficients of symmetric stable process $Y_{ζ,η}(v,\varpi)$ of index $χ\in [1,2]$ for $ζ,η\geq 0,$ which are not independent. We prove that, under certain conditions on $\mathrm{p},ζ$ and $η,$ the random Fourier--Jacobi series converges in probability to the stochastic integral \begin{equation*} \int_{-1}^1 \mathfrak{g}(u,v)dY_{ζ,η}(v,\varpi). \end{equation*} We also establish the existence of this integral in the sense of probability for $\mathfrak{g} \in \mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06766 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Convergence of Random Fourier--Jacobi Series in weighted $\mathrm{L}_{[-1,1]}^{\mathrm{p},(ζ,η)}$ Space Sahoo, Partiswari Maharana Sabita Functional Analysis 60G99 40G15 In the present paper, the random series $\sum\limits_{m=0}^\infty c_m C_m(\varpi)q_m^{(ζ,η)}(u)$ in orthogonal Jacobi polynomials $q_m^{(ζ,η)}(u)$ is discussed. The scalars $c_m$ are Fourier--Jacobi coefficients of a function in the weighted space $\mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}),\mathrm{p}>1.$ The random variables $C_m(\varpi)$ are chosen to be the Fourier--Jacobi coefficients of symmetric stable process $Y_{ζ,η}(v,\varpi)$ of index $χ\in [1,2]$ for $ζ,η\geq 0,$ which are not independent. We prove that, under certain conditions on $\mathrm{p},ζ$ and $η,$ the random Fourier--Jacobi series converges in probability to the stochastic integral \begin{equation*} \int_{-1}^1 \mathfrak{g}(u,v)dY_{ζ,η}(v,\varpi). \end{equation*} We also establish the existence of this integral in the sense of probability for $\mathfrak{g} \in \mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}).$ |
| title | On the Convergence of Random Fourier--Jacobi Series in weighted $\mathrm{L}_{[-1,1]}^{\mathrm{p},(ζ,η)}$ Space |
| topic | Functional Analysis 60G99 40G15 |
| url | https://arxiv.org/abs/2305.06766 |