Weighted orthogonal-polynomial Galerkin spaces and spectral scale

Let $\mu$ be a finite positive Borel measure on a compact interval $[a,b]$ with infinite relative support. For each integer $N\ge 0$, the moment data $(a,b,m=N+1,r=2N+1,\mu )$ satisfy the hypotheses of Positive-measure Gaussian quadrature moment data. Let $p_0,\dots ,p_N$ be the monic orthogonal polynomials supplied by Monic orthogonal polynomials exist uniquely through the quadrature order, and write

$$h_n={\int }_{[a,b]}{p}_n(x{)}^{2}d\mu (x)>0.$$

The finite-dimensional weighted Galerkin space is

$$V_N(\mu )=\mathrm{span}\{p_0,\dots ,p_N\}\subset L^2(\mu )\mathrm{.}$$

Here $L^2(\mu )$ means measurable functions with finite integral of the squared absolute value, modulo equality $\mu$-almost everywhere.

For a coefficient sequence $(c_n{)}_{n\ge 0}$ with ${\sum }_{n\ge 0}(1+n^2{)}^s\mathrm{\mid }{c}_n{\mathrm{\mid }}^2{h}_n<\mathrm{\infty }$, define its weighted spectral Sobolev norm by ${\parallel {\sum }_{n\ge 0}{c}_n{p}_n\parallel }_{H_{\mu }^s}^2={\sum }_{n\ge 0}(1+n^2{)}^s\mathrm{\mid }{c}_n{\mathrm{\mid }}^2{h}_n,s\ge 0.$ The real powers are those of Real powers for positive bases, with the zero-base positive-exponent convention; since every base $1+n^2$ is positive, The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$ makes every weight positive. The associated orthogonal Galerkin projection is ${\mathrm{\Pi }}_N({\sum }_{n\ge 0}{c}_n{p}_n)={\sum }_{n=0}^N{c}_n{p}_n$. This coefficient-defined scale is specific to the weight $\mu$ and the selected orthogonal-polynomial family; no equivalence with a derivative-based Sobolev norm is asserted here.