ON THE EQUIVALENCE OF SOME CONVOLUTIONAL EQUALITIES IN SPACES OF SEQUENCES

The problem of the equivalence of two systems with $n$ convolutional equalities arose in investigation of the conditions of similarity in spaces of sequences of operators which are left inverse to the $n$-th degree of the generalized integration operator. In this paper we solve this problem. Note that we first prove the equivalence of two corresponding systems with $n$ equalities in the spaces of analytic functions, and then, using this statement, the main result of paper is obtained. Let $X$ be a vector space of sequences of complex numbers with K$\ddot{\rm o}$the normal topology from a wide class of spaces, ${\mathcal I}_{\alpha}$ be a generalized integration operator on $X$, $\ast$ be a nontrivial convolution for ${\mathcal I}_{\alpha}$ in $X$, and $(P_q)_{q=0}^{n-1}$ be a system of natural projectors with $\displaystyle x = \sum\limits_{q=0}^{n-1} P_q x$ for all $x\in X$. We established that a set $(a^{(j)})_{j=0}^{n-1}$ with $$ \max\limits_{0\le j \le n-1}\left\{\mathop{\overline{\lim}}\limits_{m\to\infty} \sqrt[m]{\left|\frac{a_{m}^{(j)}}{\alpha_m}\right|}\right\}<\infty $$ and a set $(b^{(j)})_{j=0}^{n-1}$ of elements of the space $X$ satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} a^{(k)}) \ast {(P_{k}b^{(j)})}, \quad j = 0, 1, ... \, , \, n-1, $$ if and only if they satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} b^{(k)}) \ast {(P_{k}a^{(j)})}, \quad j = 0, 1, ... \, , \, n-1. $$ Note that the assumption on the elements $(a^{(j)})_{j=0}^{n-1}$ of the space $X$ allows us to reduce the solution of this problem to the solution of an analogous problem in the space of functions analytic in a disc.

Convenient antiderivatives for differential linear categories

Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

Weighted adams type theorem for the riesz fractional integral in generalized morrey space

We prove the boundedness of the Riesz fractional integration operator from a generalized Morrey space

Fractional integration operator on some radial rays and intertwining for the Dunkl operator

In this paper we consider the differential-difference reflection operator associated with a finite cyclic group,It is to emphasize that both hyper–Bessel operators and the so-called Poisson–Dimovski transformation (transmutation) are typical examples of the operators of generalized fractional calculus [