This operation is an automorphism of the Schwartz space and, by duality, the space of tempereddistributions. This operation restricted to the dense subspace formed by the intersection of the L1 and L2spaces is a linear isometry into L2 according to the Plancherel theorem, which can be used to prove itsuniqueness. The integral over the reals of the squared modulus of a function f is equal to the integral overthe reals of the squared modulus of this operation applied to f, which is equivalent to the fact that thisoperation is unitary, according t
This operation is an automorphism of the Schwartz space and, by duality, the space of tempereddistributions. This operation restricted to the dense subspace formed by the intersection of the L1 and L2spaces is a linear isometry into L2 according to the Plancherel theorem, which can be used to prove itsuniqueness. The integral over the reals of the squared modulus of a function f is equal to the integral overthe reals of the squared modulus of this operation applied to f, which is equivalent to the fact that thisoperation is unitary, according to Parseval's relation. This transform is integrated over the reals and isdefined using a complex exponential. For 10 points, identify this operation that transforms a time-domainfunction to one in the frequency-domain, and which shares many properties with the Laplace transform.
This operation is an automorphism of the Schwartz space and, by duality, the space of tempereddistributions.
Fourier transform