Convolution of sequences

Fibonacci Convolution Sequences

(1-2) F<„'> £ FWFi , 1=0 However, there are some easier methods of calculation. Let the Fibonacci polynomials Fn(x) be defined by (1.3) Fn+2(x) = xFn+1(x) + Fn(x), Fo(x)~0, F7(x) = 1 . Then, since Fn(1)= Fn, the recursion relation for the Fibonacci numbers, Fn+2= Fn+i + Fn, follows immediately by taking x = I In a similar manner we may write recursion relations for {Fff^} . From (1.3), taking t...

Convergence of Sequences of Convolution Operators

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 ...

Ascent sequences and the binomial convolution of Catalan numbers

In this paper, we consider two sets of pattern-avoiding ascent sequences: those avoiding both 201 and 210 and those avoiding 0021. In each case we show that the number of such ascent sequences is given by the binomial convolution of the Catalan numbers. The result for {201, 210}-avoiders completes a family of results given by Baxter and the current author in a previous paper. The result for 002...

Weighted Convolution Measure Algebras Characterized by Convolution Algebras

The weighted semigroup algebra Mb (S, w) is studied via its identification with Mb (S) together with a weighted algebra product *w so that (Mb (S, w), *) is isometrically isomorphic to (Mb (S), *w). This identification enables us to study the relation between regularity and amenability of Mb (S, w) and Mb (S), and improve some old results from discrete to general case.

Convolution and Homogeneous Spaces