On a regular form defined by a pseudo-function (Q1911453)

In an earlier paper [Ann. Numer. Math. 2, No. 1-4, 123-143 (1995; Zbl 0837.42010)] the author has found representations for the inverses of the Chebychev forms \[ \langle {\mathcal T}, f \rangle={1 \over \pi} \int^1_{-1} {f(x) dx \over \sqrt {1-x^2}} \text{ and }\langle {\mathcal U}, f \rangle={2 \over \pi} \int^1_{-1} \sqrt {1-x^2} f(x) dx \] of the first and second degree, respectively. These inverses were given in terms of the pseudo-function given by \[ \begin{multlined} \left \langle P f{Y(1-x^2) \over x^2}, f \right \rangle= \\ =Pf \int^\infty_{- \infty} {Y(1-x^2) \over x^2} f(x) dx :=\lim_{\varepsilon \downarrow 0} \left( \int^{- \varepsilon}_{-1} {f(x) \over x^2} dx + \int^1_\varepsilon {f(x) \over x^2} dx-{2 \over \varepsilon} f(0) \right),\end{multlined} \] where \(f\) is a polynomial and \(Y(x)=\begin{cases} 0,\;x \leq 0 \\ 1,\;x > 0. \end{cases}\). In the present paper the author studies forms \(u\) defined in terms of a pseudo-function \[ \langle u,f \rangle=Pf \int^\infty_{- \infty} {V(x) \over x^2} f(x) dx \] where \(v\) given by \(\langle v,f \rangle=\int^\infty_{- \infty} V(x) f(x) dx\) is regular. Conditions on \(v\) are given such that \(u\) is also regular. In the last section the author studies the special case \(V(x)=(1/2) Y(1-x^2)\), where \(v=L\) is the Legendre form in more details.