By similar arguments, it can be shown that the discrete convolution of sequences and is given by:

By the convolution theorem, this equation may be Fourier transformed to

By the convolution theorem, the FT of an arbitrary transparency function-multiplied ( or truncated ) by an aperture function-is equal to the FT of the non-truncated transparency function convolved against the FT of the aperture function, which in this case becomes a type of " Greens function " or " impulse response function " in the spectral domain.

Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their ( f and the reversed and shifted g ) product for each possible amount of sliding.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

By " satisfactory " one would mean at least the equivalent of Plancherel theorem.

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.

By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius.

By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.

By Rice's theorem, the 1-halting problem is undecidable.

By the corollary to the recursion theorem, there is an index such that returns.

By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.

By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.

By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.

By the central limit theorem, this distribution approaches the normal distribution as n increases.

By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.

By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once.

By the rank-nullity theorem, a system of n vectors in k dimensions ( where all dimensions are necessary ) satisfies a ( p = n − k )- dimensional space of relations.

By Tychonoff's theorem we have that is compact since is, so the closure of in is a compactification of.

By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.

By the divergence theorem Gauss's law can alternatively be written in the differential form:

By a general property of the Fourier transform, that symmetry means it is the complex conjugate of:

By making measurements of the signal at many discrete positions of the moving mirror, the spectrum can be reconstructed using a Fourier transform of the temporal coherence of the light.

By applying the inverse Fourier transform, we can write:

By the uniqueness theorem for Fourier expansions, we must then equate the Fourier coefficients term by term, giving

By 1844, the Brook Farmers adopted a societal model based on the socialist concepts of Charles Fourier and began publishing The Harbinger as an unofficial journal promoting Fourierism.

By the properties of the Fourier transform, ( it )< sup > r </ sup > ψ ( t ) is the Fourier transform of (− 1 )< sup > r </ sup > D < sup > r </ sup > ( x ), where D is the differential operator with respect to x.

By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Fourier.

By taking Fourier transforms this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above.

By 1834, he was in Paris, and began familiarizing himself with the work of Fourier, Owen and Cabet.

By means of the Fourier transform, pseudo-differential operators can be defined which allow for fractional calculus.

By the time Richardson writes Grandison, he transforms the letter writing from telling of personal insights and explaining feelings into a means for people to communicate their thoughts on the actions of others and for the public to celebrate virtue.

By looking into Mnemosyne's eyes he receives knowledge which transforms him fully into a god.

By this he meant that myth transforms concepts into beings and stories.

By making sarcastic comments to the rest of the team, he inadvertently transforms Spike's cynic personality into optimistic and super-positive, makes Wesley and Fred act extremely drunk despite very little alcohol, causes Gunn to urinate all over the Wolfram & Hart building ( he had told Gunn to " stake out his territory ," resulting in literally marking his territory in a manner similar to animals ), and influences Angel to have sex with Eve after sarcastically commenting on the sexual tension between the two.

By night she transforms into the mysterious ( yet magical ) thief, Saint Tail.

By doing so, the frail, grey-haired criminal transforms into Ibac, a brutal muscleman with a buzz cut.

By changing the number of protons, double electron capture transforms the nuclide into a new element.

By some mystical power, the dagger transforms into a great sword every time Flavio wields it.