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Aganippe76

Aganippe76

Answered question

2022-07-10

Suppose that λ 1 ( A ) λ 2 ( A ) λ n ( A ) are (real) eigenvalues of a Hermitian matrix A and denote the empirical measure by L A := 1 n i = 1 n δ λ i ( A ) . Let F A to be distribution function related to the (counting) measure L A . We have the following integration by parts formula
g d L A g d L B = ( F A F B ) d g ,
in which g   : R R is a bounded function with bounded total variation. I am curious whether the integration by parts in this case is connected to Abel's summation by parts, and if so, can anyone explicitly present the connection?

Answer & Explanation

Jamiya Costa

Jamiya Costa

Beginner2022-07-11Added 18 answers

Sure, just take g to be a distribution function of some discrete probability distribution L g (which is always bounded w/ bounded TV). Let S be the union of all the supports of L g , L A , L B , and let I = [ n ] be the index for the set S, so that you can rewrite the LHS as:
i [ n ] g ( s i ) ( δ s i S A δ s i S B )
and the right hand side is
i [ n ] ( g ( s i ) g ( s i 1 ) ) ( F A F B ) ( s i )
This is basically summation by parts formula (where we note δ s i S A = F A ( s i ) F A ( s i 1 ) ) , except the boundary terms, but that is fine, since ( F A F B ) ( s n ) = 0, and g ( s 0 ) = 0, where s 0 is some other number smaller than the smallest number in S.
The wikipedia page on summation by parts explains it as well I think.

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