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The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomials, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two
The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
Knuth's article titled "Convolution Polynomials" [9] defines a generalized class of convolution polynomial sequences by their special generating functions of the form = ( ()) = = (), for some analytic function F with a power series expansion such that F(0) = 1.
Convolution. Cauchy product –is the discrete convolution of two sequences; Farey sequence – the sequence of completely reduced fractions between 0 and 1; Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
Sequence transformations include linear mappings such as discrete convolution with another sequence and resummation of a sequence and nonlinear mappings, more generally. They are commonly used for series acceleration , that is, for improving the rate of convergence of a slowly convergent sequence or series .
The original series can be regained by ... The binomial transform of a sequence is just the nth forward differences of ... Their binomial convolution is defined by ...
A Fibonacci sequence of order n is ... (sequence A000073 in the OEIS) The series was ... A convolved Fibonacci sequence is obtained applying a convolution operation ...