With s = T • s ( nT ), this Fourier series can now be recognized as a form of the Poisson summation formula.

With optical imaging devices, for example, it is the Fourier transform of the point spread function ( hence a function of spatial frequency ) i. e. the intensity distribution caused by a point object in the field of view.

With the help of the convolution theorem and the fast Fourier transform, the complexity of the convolution can be reduced to O ( n log n ).

With a FFT based spectrum analyzer, the frequency resolution is, the inverse of the time T over which the waveform is measured and Fourier transformed.

With Fourier transform analysis in a digital spectrum analyzer, it is necessary to sample the input signal with a sampling frequency that is at least twice the bandwidth of the signal, due to the Nyquist limit.

With these dualities, the version of the Fourier transform to be introduced next coincides with the classical Fourier transform on R.

With the substitution, and the Fourier transform property, ( for P > 0 ), becomes:

With another definition, and the transform property becomes a periodic summation ( with period P ) and its equivalent Fourier series:

With a passionate interest they read and commented on the theories of Étienne Cabet, Charles Fourier, Proudhon, and finally, listened with delight to Belinsky's letter to Gogol.

With the advent of near-infrared tunable lasers, particularly the Ti-sapphire laser ( which has a range of ~ 700-1100 nm ), Fourier Transform Resonance Raman Spectrometers may soon be commercially available.

With the help of Ziggy Elman, also in the band, he transformed a traditional Jewish melody into a popular song, `` And The Angels Sing ''.

With the announcement that Lansing Township had been made the capital, the small village quickly transformed into the seat of state government.

With the help of standard procedures, MPEG-4 data can be interpreted and transformed into other signal types compatible with any available network.

For example, Bill Haley's incompletely bowdlerized cover of " Shake, Rattle and Roll " transformed Big Joe Turner's humorous and racy tale of adult love into an energetic teen dance number, while Georgia Gibbs replaced Etta James's tough, sarcastic vocal in " Roll With Me, Henry " ( covered as " Dance With Me, Henry ") with a perkier vocal more appropriate for an audience unfamiliar with the song to which James's song was an answer, Hank Ballard's " Work With Me, Annie ".

With the approach of the early ' 90s and Rush's characteristic sound still intact, the band transformed their style once again to harmonize with the alternative rock movement.

With the construction of its current building, designed by Alessandro Mendini, the museum has been transformed into one of the most modern and innovative of its kind in the Netherlands.

With a group of investors, Balin purchased a former pizza parlor on Fillmore Street, which he transformed into a music club, The Matrix, and began searching for members for his group.

With the decline of these industries in the 1960s, Sussex County was transformed into a bedroom community that absorbed population shifts from New Jersey's urban areas.

With the completion of Tempe Town Lake, commercial and high-rise development along the reservoir quickly transformed the cityscape of Mill Avenue and the skyline of downtown Tempe.

With its ability to make X-ray images of star clusters, supernova remnants, galactic eruptions, and collisions between clusters of galaxies, in its first decade of operation it has transformed astronomer's view of the high-energy universe.

With the decline of the lumbering industry in the early 1900s, railways that had been used to transport lumber were transformed to passenger lines to bring outdoorsmen to fish, hunt, and experience the beauty of the " north country.

With the introduction of legal gambling in Mississippi in the 1990s, Biloxi was again transformed.

With the coming of the petrochemical industry in the early 1900s, Clinton again transformed into a refinery center.

With the crowd gone, Porco and Curtis agree to delay the air force together, whilst Curtis asks to see Porco's face, suggesting perhaps he may have been transformed back into a human.

* With his School of American Ballet, New York City Ballet, and 400 choreographed works, Balanchine transformed American dance and created modern ballet, developing a unique style with his dancers highlighted by brilliant speed and attack.

With the tumultuous years of 1789-1815, European culture was transformed by revolution, war and disruption.

With ongoing decentralization, they were accorded more powers in the 1997 Kapodistrias reform of local and regional government, and were finally transformed into fully separate entities by the 2010 Kallikratis plan ( Law 3852 / 2010 ), which entered into effect on 1 January 2011.

With Cecil King ( Rothermere's nephew ) in charge of the paper's finances and Guy Bartholomew as editor, during the late-1930s the Mirror was transformed from a conservative, middle class newspaper into a left-wing paper for the working class.

Image: Young-refugee-delhi1947. jpg |" With the tragic legacy of an uncertain future, a young refugee sits on the walls of Purana Qila, transformed into a vast refugee camp in Delhi.

With the assistance of the famous architect, Pieter Post of Haarlem, he transformed Recife by building a new town adorned with splendid public edifices and gardens, which was called after his name, Mauritsstad.

With a journey time of 20 minutes from Kilmarnock to Glasgow ( roughly half that of the existing train service ), the M77 motorway has transformed the link between Glasgow and Kilmarnock.

With facts mainly in his mind, he was often acute in the matter of style, and he said, `` The young who have as yet nothing to say will try larks with initial letters and broken lines.

With the initial encouragement of Russian agents, a series of agreements was concluded between Serbia and Bulgaria in March 1912.

With the initial peace treaty articles ratified in April, a recently formed Congressional committee under Hamilton, was considering needs and plans for a peacetime army.

Icon of the Melanesian Brotherhood Martyrs at Canterbury Cathedral ( Anglican Communion ) With the Reformation, after an initial uncertainty among early Lutherans, who painted a few " icon "- like depictions of leading Reformers, and continued to paint scenes from Scripture, Protestants came down firmly against icon-like portraits, especially larger ones, even of Christ.

* With r between 0 and 1, the population will eventually die, independent of the initial population.

* With r between 3 and ( approximately 3. 45 ), from almost all initial conditions the population will approach permanent oscillations between two values.

* With r between 3. 45 and 3. 54 ( approximately ), from almost all initial conditions the population will approach permanent oscillations among four values.

* With r increasing beyond 3. 54, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc.

With the exception of initial forced bets, money is only placed into the pot voluntarily by a player who, at least in theory, rationally believes the bet has positive expected value.

With Tesla's salary of $ 18 per week, the payment would have amounted to over 53 years ' pay and the amount was equal to the initial capital of the company.

With 12, 000 Confederate troops at Fort Donelson, Foote's initial approach by Union naval ships were repulsed by Donelson's guns.

With the exception of the buboes, the initial symptoms of plague are very similar to many other diseases, making diagnosis difficult.

With the end of the Civil War, the competing railroads coming from Missouri took advantage of their initial strategic advantage for a building boom.

With the initial detection of an extrasolar X-ray source, the first question usually asked is " What is the source?

With the initial phase of construction completed, it is easy to see why the budget for the exhibition was going to be larger than anyone expected.

With a high foreign debt ( originally $ 5. 7 billion at 1998 net present value ) and a good track record on economic reform, Mozambique was the first African country to receive debt relief under the initial Heavily Indebted Poor Country ( HIPC ) Initiative.

With both the ΔT and the ΔV charging methods, both manufacturers recommend a further period of trickle charging to follow the initial rapid charge.

With an analogous construction, the cokernel of ƒ can be seen as an initial object of a suitable category.

With an initial target date of 1 May 1944, the infantry attack was conceived as a joint assault by five divisions transported by landing craft, constituting the largest amphibious operation in military history.

With the initial targets unaccomplished, the second and larger wave of assault landings brought in reinforcements, support weapons and headquarter elements at 07: 00 only to face nearly the same difficulties as had the first.

With initial goals of selling 50, 000 simputers, the project had sold only about 4, 000 units by 2005, and has been called a failure by news sources.

With the Roman advance in Campania, Irna began to lose its importance, being supplanted by the new Roman colony ( 194 BC ) of Salernum, developing around an initial castrum.

With perfect knowledge of the initial conditions and of the context of an action, the course of this action can be predicted in chaos theory.

With the Liberals gaining the initial momentum, a successful counterattack by Allan Gregg resulted in the PCs being re-elected with a solid but reduced majority and 43 percent of the popular vote.

With Kristen Nygaard, he produced the initial ideas for object-oriented ( OO ) programming in the 1960s at the Norwegian Computing Center ( NR ) as part of the Simula I ( 1961 – 1965 ) and Simula 67 ( 1965 – 1968 ) simulation programming languages.