When (φn) is a sequence of positive functions in L1(R) such that the convolutions φn ? f converge in L1-norm to f for all f ∈ L1(R), then these convolutions may or may not converge almost everywhere too. There is always a subsequence (φnk ) such that at least φnk ? f converges almost everywhere to f for all f ∈ Lp(R), 1 < p < ∞. A special case of this is when the functions φn are the dilations ...

This paper will look at the connection of operators and frames respectively related sequences from different viewpoints. We will give a survey of the classification of frames and similar sequences by the properties of the associated operators. Furthermore an important part will be the investigation of how to find a matrix representation of operators with Bessel sequences, frames and Riesz bases...

the theory of c-frames and c-bessel mappings are the generalizationsof the theory of frames and bessel sequences. in this paper, weobtain several equivalent conditions for dual of c-bessel mappings.we show that for a c-bessel mapping $f$, a retrievalformula with respect to a c-bessel mapping $g$ is satisfied if andonly if $g$ is sum of the canonical dual of $f$ with a c-besselmapping which  wea...

We present a self-learning approach for synthesizing programs from integer sequences. Our method relies on tree search guided by learned policy. system is tested the On-Line Encyclopedia of Integer Sequences. There, it discovers, its own, solutions 27987 sequences starting basic operators and without human-written training examples.