Piecewise-linear product integration for a half-order diagonal singularity

Let $X:=C([0,1],\mathbb{R})$ and let $a:[0,1{]}^2\to \mathbb{R}$ be continuous. Use $S_x$ from The half-order diagonal singularity defines a bounded improper integral to define $(Ku)(x):=S_x(y\mapsto a(x,y)u(y))\mathrm{.}$ For an integer $n\ge 1$, set $h:=1\mathrm{/}n$ and $y_j:=jh$ for $j=0,\dots ,n$. Define the nonnegative piecewise-linear hat functions by ${\varphi }_{n,0}(y):=\max (1-y\mathrm{/}h,0),{\varphi }_{n,n}(y):=\max (1-(1-y)\mathrm{/}h,0),$ and, for $1\le j<n$, ${\varphi }_{n,j}(y):=\max (1-\mathrm{\mid }y-y_j\mathrm{\mid }\mathrm{/}h,0)\mathrm{.}$ Their piecewise-linear interpolant is $I_{n}g:={\sum }_{j=0}^{n}g(y_j){\varphi }_{n,j}\mathrm{.}$ The product weights and product-integration operator are $w_{n,j}(x):=S_x({\varphi }_{n,j}),(K_{n}u)(x):={\sum }_{j=0}^n{w}_{n,j}(x)a(x,y_j)u(y_j)=S_x(I_n(y\mapsto a(x,y)u(y)))\mathrm{.}$ The kernel factor $s(x,y)=\mathrm{\mid }x-y{\mathrm{\mid }}^{-1\mathrm{/}2}$ is integrated in the weights; the smooth factor $a(x,y)u(y)$ is interpolated. This item defines the pointwise formulas; the subsequent results on this page establish their mapping properties, stability, and error estimates.