Cauchy product

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin Louis Cauchy. C n = ∑ i = 0 n a n − i ( B i − B ) + A n B . {displaystyle C_{n}=sum _{i=0}^{n}a_{n-i}(B_{i}-B)+A_{n}B,.}     (1) | B n − B | ≤ ε / 3 ∑ k ∈ N | a k | + 1 {displaystyle |B_{n}-B|leq {frac {varepsilon /3}{sum _{kin {mathbb {N} }}|a_{k}|+1}}}     (2) | a n | ≤ ε 3 N ( sup i ∈ { 0 , … , N − 1 } | B i − B | + 1 ) . {displaystyle |a_{n}|leq {frac {varepsilon }{3N(sup _{iin {0,dots ,N-1}}|B_{i}-B|+1)}},.}     (3) | A n − A | ≤ ε / 3 | B | + 1 . {displaystyle |A_{n}-A|leq {frac {varepsilon /3}{|B|+1}},.}     (4) In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin Louis Cauchy. The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, it is by abuse of language: they actually refer to discrete convolution. Convergence issues are discussed in the next section. Let ∑ i = 0 ∞ a i {displaystyle extstyle sum _{i=0}^{infty }a_{i}} and ∑ j = 0 ∞ b j {displaystyle extstyle sum _{j=0}^{infty }b_{j}} be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows: Consider the following two power series with complex coefficients { a i } {displaystyle {a_{i}}} and { b j } {displaystyle {b_{j}}} . The Cauchy product of these two power series is defined by a discrete convolution as follows: Let (an)n≥0 and (bn)n≥0 be real sequences.If ∑ n = 0 ∞ a n = A {displaystyle extstyle sum _{n=0}^{infty }a_{n}=A} and ∑ n = 0 ∞ b n = B {displaystyle extstyle sum _{n=0}^{infty }b_{n}=B} , and at least one of them converges absolutely, then their Cauchy product converges to AB. (Rudin, Principles of Mathematical Analysis, Theorem 3.50) Let (an)n≥0 and (bn)n≥0 be real or complex sequences. It was proved by Franz Mertens that, if the series ∑ n = 0 ∞ a n {displaystyle extstyle sum _{n=0}^{infty }a_{n}} converges to A and ∑ n = 0 ∞ b n {displaystyle extstyle sum _{n=0}^{infty }b_{n}} converges to B, and at least one of them converges absolutely, then their Cauchy product converges to AB. It is not sufficient for both series to be convergent; if both sequences are conditionally convergent, the Cauchy product does not have to converge towards the product of the two series, as the following example shows: