Conjugate function  From Encyclopedia of Mathematics Jump to:
navigation, search  A concept in the theory of functions which
is a concrete image of some involutory operator for the
corresponding class of functions.  1) The function conjugate to
a complex-valued function is the function whose values are the
complex conjugates of those of .  2) For the function conjugate
to a harmonic function see Conjugate harmonic functions.  3) The
function conjugate to a -periodic summable function on is given
by  it exists almost-everywhere and coincides almost-everywhere
with the -sum, , and the Abel–Poisson sum of the conjugate
trigonometric series.  4) The function conjugate to a function
defined on a vector space dual to a vector space (with respect
to a bilinear form ) is the function on given by   The conjugate
of a function defined on is defined in a similar way.  The
function conjugate to the function , , of one variable is given
by  The function conjugate to the function on a Hilbert space
with scalar product is t