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Operational calculus and differential equations with infinitely smooth coefficients. (English) Zbl 0701.44006
The field $$M$$ of Mikusiński operators, provided with addition and convolution (denoted by $$+$$ and $$*$$) has been used in solving different problems. The inability to define a suitable product of a function and an operator has had a limiting effect, specially if we apply Mikusiński operators to differential equations. Many authors tried to introduce a product of an element belonging to $$M$$ and some special functions, as polynomials and exponential functions, but with restricted effects. The author of this paper introduces the product of a smooth function and an element of $$M^ k$$, subring of $$M$$.
Let $$C$$ be the ring of continuous functions on $$[0,\infty)$$ and $$\ell =\{1\}\in C$$. For $$k=0,1,...$$, let $$M^ k=\{x\in M$$, $$\ell^ k*x\in C\}$$. $$M^ k$$ is a Fréchet space with the family of seminorms: $$p_{k,m}(x)=\{\sup | (\ell^ k*x)(t)|$$, $$0\leq t\leq m\}$$, $$m\in N$$. $$M_ F$$ is the countable union space $$M_ F=\cup^{\infty}_{k=0}M^ k$$ with the corresponding convergent class. If $$x\in M_ F$$ it belongs to a $$M^ k$$. For $$f\in C^{\infty}$$ and $$x\in M^ k$$ the product is defined by $f\cdot x=\sum^{k}_{j=0}(- 1)^ j\left( \begin{matrix} k\\ j\end{matrix} \right)f^{(j)}(\ell^ k*x)/\ell^{k-j}\quad (g/\ell^ 0=g).$ The author proves properties of the defined product and applies it to differential equations proving that the corresponding equation in M to the differential equation $$y''+fy'+gy=0$$ with $$y(0)=\alpha$$, $$y'(0)=\beta$$, where $$\alpha$$,$$\beta\in {\mathbb R}$$ and $$f,g\in C^{\infty}$$ has only the classical solution.
Reviewer: B.Stanković
##### MSC:
 44A40 Calculus of Mikusiński and other operational calculi 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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