Sobolev-type orthogonal polynomials on the unit ball (Q387367)

The Sobolev-type inner product on the unit ball \(B^d\subset\mathbb R^d\) is defined as NEWLINE\[NEWLINE(f,g)_S=(f,g)_{\mu}+\lambda\sum_{k=0}^N\frac{\partial f(s_k)}{\partial n}\frac{\partial g(s_k)}{\partial n},NEWLINE\]NEWLINE where \((f,g)_{\mu}=\omega_{\mu}\int_{B^d}f(x)g(x)(1-\| x\|^2)^{\mu-1/2}dx,\) \(\mu>-1/2,\) \(\omega_{\mu}\) is the normalizing constant such that \((1,1)_{\mu}=1,\) \(\frac{\partial}{\partial n}\) denotes the outward normal derivative on the unit sphere \(S^{d-1},\) \(s_k\in S^{d-1},k=0,\dots,N.\)NEWLINENEWLINEThe authors give an explicit formula for the \(n\)-th reproducing kernel associated with \((f,g)_S\) as a modification of the kernel function for \((f,g)_{\mu}\) and find asymptotics of the difference between these kernels on \(B^d.\)