The discrete Tchebycheff transformation in certain spaces of sequences and its applications (Q2770595)

The finite Tchebycheff transformation NEWLINE\[NEWLINE (T\phi)(n)=\Phi(n)=\int^1_{-1}(1-x^2)^{-1/2}T_n(x)\phi (x)dx \tag{*} NEWLINE\]NEWLINE where \(T_n(x)\) denotes the well-known Tchebycheff polynomial of the first kind of degree \(n=0,1,2,\dots\), has been investigated by different authors from a classical point of view, see \textit{I. N. Sneddon} [The use of integral transforms. McGraw Hill Book Co., New York (1972; Zbl 0237.44001)]; \textit{P. L. Butzer} and \textit{R. L. Stens} [Funct. Approximation Comment. Math. 5, 129-160 (1977; Zbl 0375.44003)], among others, as well as in distributional spaces, see \textit{A. H. Zemanian} [Generalized integral transformations, Pure Applied Math. 18, Interscience Publishers, New York (1968; Zbl 0181.12701)] reprinted by [Dover Publications, New York (1987; Zbl 0643.46029)]. In this paper the authors analyse the discrete version NEWLINE\[NEWLINE (T^*\Phi)(x)=\phi(x)=\frac 1 \pi \sum^\infty_{n=-\infty } \Phi(n) T_n (x) \tag{**} NEWLINE\]NEWLINE on certain space of sequences. Notice that \((**)\) is the inversion formula of \((*)\). NEWLINENEWLINENEWLINEIt is proved that the discrete Tchebycheff transformation defines an isomorphism from the space \(T(\mathbb Z)\) of testing-sequences of rapid descent into the space \(T(-1,1)\) of testing-functions with slow growth at the end points 1 and \(-1\). This result is extended to spaces of generalized sequences and distributions. The discrete translation operator and the discrete convolution are also studied. Finally, the operational calculus generated is applied in solving certain finite-difference equations.