Abel's inequality - Alchetron, The Free Social Encyclopedia

In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case.

Let {a₁, a₂,...} be a sequence of real numbers that is either nonincreasing or nondecreasing, and let {b₁, b₂,...} be a sequence of real or complex numbers. If {a_n} is nondecreasing, it holds that

| ∑ k = 1 n a k b k | ≤ max k = 1 , … , n ⁡ | B k | ( | a n | + a n − a 1 ) ,

and if {a_n} is nonincreasing, it holds that

| ∑ k = 1 n a k b k | ≤ max k = 1 , … , n ⁡ | B k | ( | a n | − a n + a 1 ) ,

where

B k = b 1 + ⋯ + b k .

In particular, if the sequence {a_n} is nonincreasing and nonnegative, it follows that

| ∑ k = 1 n a k b k | ≤ max k = 1 , … , n ⁡ | B k | a 1 ,

Abel's inequality follows easily from Abel's transformation, which is the discrete version of integration by parts: If {a₁, a₂,...} and {b₁, b₂,...} are sequences of real or complex numbers, it holds that

∑ k = 1 n a k b k = a n B n − ∑ k = 1 n − 1 B k ( a k + 1 − a k ) .

References

Abel's inequality Wikipedia

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