Associative and commutative distribution algebra with multipliers, and generalized solutions of nonlinear equations
The author constructs an associative-commutative algebra \(E^*\) with unit and without divisors of zero generated by the distributions from \[ E=\text{span}\{\delta^{(m-1)}(x- c_k), P(x-c_k)^{-m} x^{m-1}; m=1,2,\dots;c_k\in\mathbb{R}; k=1,2,\dots,s\}\subset{\mathcal S}'. \] If \(S_*\) is the space of test functions of asymptotic form \[ \varphi_*(x)= \sum^\infty_{n=-\infty} \varphi_n(x)y^n,\quad y^n\to 0_+, \] \(\varphi_n\in{\mathcal S}\) with only a finite number of components different from zero (each \(\varphi_*\) can be represented as a vector \(\varphi_*(x)=(\dots,\varphi_{-n}(x),\varphi_{-n-1}(x),\dots);\) \(\varphi_{\pm\infty}=0\)), the elements of \(E^*\) are vector valued distributions on \(S_*\), \(f^*(x)= (\dots,f_n(x),f_{n+1}(x),\dots)\); \(f_n\in E\), and can also be represented in the asymptotic form \(f^*(x)=\sum^\infty_{n=-\infty} f_n(x)y^n\); \(y\to 0_+\), \(f_{-\infty}=0\). On \(E^*\) one defines the derivatives, the antiderivatives, the value at a point, the Fourier transform and, of course, an associative-commutative product which allows to solve problems connected with the product, particularly, to find the general solution of a nonlinear equation as a linear combination of elements from \(E^*\).
generalized solutions of nonlinear equations
scientific article; zbMATH DE number 857382
scientific article