Properties of integral transforms with \(\Lambda\)-multiplicative kernels (Q916948)

Let \(P=\{p_ n:n\in {\mathbb{Z}}'\}\) be a sequence of prime numbers where \({\mathbb{Z}}'={\mathbb{Z}}-\{0\}\). Define \(M=\{m_ n:n\in {\mathbb{Z}}\}\) by \(m_ 0=1\), \(m_ n=m_{n-1}/p_ n\), and \(m_{-n}=p_{-n}m_{1-n}\) (\(n\in {\mathbb{N}})\). Set \(x_ n=[x/m_ n]\) (mod \(p_ n)\), \(x_{-n}=[x/m_{1- n}]\pmod{p_{-n}}\) where \(x\in {\mathbb{R}}\), \(n\in {\mathbb{N}}\), and [\(\cdot]\) means the integral part. Then the expansion \(x=\sum^{-1}_{n=- \infty}x_ nm_{n+1}+\sum^{\infty}_{n=1}x_ nm_ n\) holds. Consider the dual sequence \(\hat P=\{\hat p_ n:n\in {\mathbb{Z}}'\}\) where \(\hat p_ n=p_{-n}\) and the corresponding expansion \(x=\sum^{- 1}_{n=-\infty}\hat x_ n\hat m_{n+1}+\sum^{\infty}_{n=1}\hat x_ n\hat m_ n.\) Finally, let \(\Lambda =\{\lambda_{nk}:n,k\in {\mathbb{Z}}'\) and \(k\geq n\}\) be an infinite triangular matrix whose entries are 0 or 1, while \(\lambda_{nn}=1\), \(n\in {\mathbb{Z}}'\). The integral transform indicated in the title is defined for any function \(f\in L_ 1[0,+\infty)\) by \(f(y)=\int^{+\infty}_{0}f(x)\overline{X_{\Lambda}(x,y)}dx\) where \[ X_{\Lambda}(x,y)=\prod '{}^{+\infty}_{n=-\infty}\prod '{}^{+\infty}_{k=n}\exp (2\pi i\lambda_{nk}x_ n\hat y_{-k}/p_ n...p_ k) \] and prime means that \(n,k\in {\mathbb{Z}}'\). The authors prove four theorems on the basic properties of this transform.