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We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.

Definition: Let $f(x)$ be integrable in $(a,b)$. If there exists a function $g(x)$ such that $f(x)g(x)$ is also integrable in $(a,b)$ and

$$ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx $$

then, $g(x)$ is the integral identity of $f(x)$ in the interval $(a,b)$.

Trivially the number 1 is an integral identity in all intervals. Given below is an example for a non trivial integral identity in the interval $(0,\infty)$. We have for $a > 0$ $$ \int_{0}^{\infty}\log\Big(1+\frac{a^2}{x^2}\Big)dx = \int_{0}^{\infty}\log\Big(1+\frac{a^2}{x^2}\Big)\log\Big(\frac{a}{x}\Big)dx = \pi a. $$

Questions:

  1. Is there any reference to such functions in literature?
  2. For what functions $f$ can we always find an integral identity in $(0,\infty)$ i.e how can we test if $f$ has an integral identity?
  3. For what functions $f$ can we always find an integral identity in all arbitrary intervals $(a,b)$?
  4. Can there be functions that have infinitely many integral identities in some intervals?
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1 Answer 1

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An alternate way to say it: "$1-g$ is orthogonal to $f$" $$ \int_a^b f(x)(1-g(x)) dx = 0 $$ Yes, there is extensive literature on orthogonal functions.

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