Fourier series: lub keeb kwm thiab cawv ntawm cov zauv mechanism rau txoj kev loj hlob ntawm science

Fourier series - qhov no saib arbitrarily xaiv zog rau lub sij hawm nyob rau hauv ib tug uake. Nyob rau hauv nqe lus dav dav, cov tshuaj no yog hu ua tus expansion caij rau ib tug orthogonal hauv paus. Qhov kev nthuav zog nyob rau hauv Fourier series yog ib tug heev haib daws ntau yam teeb meem vim lub zog ntawm cov transformation nyob rau hauv kev koom ua ke, ntau yam, raws li tau zoo raws li ib tug ua haujlwm nyob rau hauv lub sib cav qhia thiab convolution.

Ib tug neeg uas tsis paub nrog lej uas siab dua, raws li tau zoo raws li nrog tej hauj lwm ntawm lub Fabkis tus paub txog Fourier, feem ntau yuav yuav tsis to taub dab tsi lub "ibyam" thiab lawv ua dab tsi. Tsis tau no pom yog heev nrees nkag mus hauv peb lub neej. Nws yog siv tsis tau tsuas yog kev kawm txog zauv, tab sis kuj physicists, chemists, cov kws kho mob, astronomers, seismologists, oceanographers thiab lwm tus neeg. Cia peb kuj coj ib los ze zog saib nrog rau cov tej hauj lwm ntawm tus poj Fabkis tus paub txog tus uas tau cov discovery, ua ntej ntawm nws lub sij hawm.

Tus txiv neej thiab cov Fourier transform

Fourier series yog ib tug ntawm cov kev (nrog rau kev tsom xam thiab lwm tus) ntawm lub Fourier transform. Qhov no txoj kev yuav siv sij hawm qhov chaw txhua txhua lub sij hawm ib tug neeg hnov tej suab. Peb pob ntseg txiav converts lub suab yoj. Oscillatory zog ntawm elementary hais nyob rau hauv ib tug elastic nruab nrab yog nthuav rau hauv lub series (lub spectrum) successive ntim qhov tseem ceeb rau tones ntawm txawv heights. Tom ntej no, lub paj hlwb converts no cov ntaub ntawv mus rau hauv cov suab rau peb. Tag nrho cov no yog nyob rau hauv tas li ntawd mus rau peb cov muaj siab los yog nco qab nws tus kheej, tab sis nyob rau hauv thiaj li yuav to taub cov txheej txheem uas yuav coj ob peb lub xyoo los kawm kev kawm leb siab.

Nyeem ntxiv txog qhov kev Fourier transform

Lub Fourier transform yuav nqa tawm analytical, numerals thiab lwm txoj kev. Fourier series yog tus nab npawj txheej txheem rau decomposing tej oscillatory dab - los ntawm lub hiav txwv tides thiab tsis muaj teeb hnub ci mus (thiab lwm yam astronomical khoom) ua si. Siv cov zauv hom kev kawm, nws yog ua tau rau lauj cov nuj nqi, cov sawv cev tej oscillatory dab nyob rau hauv ib tug xov tooj ntawm sinusoidal cheebtsam uas mus los ntawm yam tsawg kawg nkaus rau nyiaj pab ntau tshaj thiab vice versa. Lub Fourier transform yog ib tug muaj nuj nqi piav txog cov theem thiab amplitude ntawm sinusoids coj mus rau ib tug kev zaus. Qhov no txoj kev yuav siv tau rau kev daws ib tug heev sib npaug uas piav qhia txog tus dynamic dab tshwm sim nyob rau hauv qhov kev txiav txim ntawm tshav kub, lub teeb los yog hluav taws xob. Tsis tas li ntawd, lub Fourier series siv los paub qhov txawv DC cheebtsam hauv complex waveforms, ua rau nws ua tau rau kom txhais tau cov kev sim pom nyob rau hauv cov tshuaj, Science News for KIDS thiab astronomy.

keeb kwm ntaub ntawv

Cov nrhiav leej txiv ntawm no ziag no yog lub Fabkis mathematician Zhan Batist Zhozef Fure. Nws lub npe tom qab thiab no transformation tau raug hu ua. Chiv, zaum siv ib tug txheej txheem kawm thiab piav txog cov mechanisms ntawm thermal conductivity - Thaum tshav kub kub tawm nyob rau hauv khib nyiab. Fourier tswv yim hais tias tus thawj lus muab faib rau cov thermal yoj yuav decomposed rau hauv yooj yim sinusoid, txhua tus uas yuav muaj nws cov kub yam tsawg kawg nkaus thiab siab tshaj plaws, raws li zoo raws li nws cov theem. Yog li txhua xws tivthaiv yuav tsum tau ntsuas los ntawm yam tsawg kawg nkaus rau nyiaj pab ntau tshaj thiab vice versa. Cov zauv muaj nuj nqi uas piav txog lub Upper thiab qis peaks ntawm tus nkhaus, raws li zoo raws li lub sij hawm ntawm txhua harmonic, hu ua lub Fourier transform ntawm lub kub faib kev qhia. Tus sau lub hom phiaj ntawm txo zuag qhia tag nrho kev faib nuj nqi uas yog ib qhov nyuaj rau zauv hauj lwm, nyob rau hauv ib tug heev yooj yim mus ko ib tug xov tooj ntawm periodic zog ntawm sine thiab cosine, nyob rau hauv tus nqi ntawm cov muab thawj zaug tis.

Lub hauv paus ntsiab lus ntawm hloov dua siab tshiab thiab cov views ntawm contemporaries

Contemporaries ntawm tus paub txog - qhov ua daim ntawv thom nug ntawm thaum ntxov thib kaum cuaj caug xyoo - tsis kam txais no ziag no. Lub ntsiab pom yog pom zoo los ntawm Fourier hais tias cov discontinuous muaj nuj nqi piav txog ib tug ncaj kab los yog nkhaus yog nraus, nws yuav sawv cev raws li ib tug zaum ntawm sinusoidal kab zauv uas yog tas mus li. Raws li ib qho piv txwv, xav txog ib tug "kauj ruam" Heaviside: nws tus nqi yog pes tsawg mus rau sab laug ntawm cov kis thiab ib tug nyob rau ntawm txoj cai. Qhov no muaj nuj nqi qhia txog cov dependence ntawm hluav taws xob tam sim no nyob rau hauv lub sij hawm nce mus nce los rau lub kaw saw. Kawm kev tshawb xav ntawm lub sij hawm uas, tsis tau ces yuav tsum xws li ib tug teeb meem no, thaum ib tug discontinuous qhia yuav tau piav los ntawm ib tug ua ke ntawm tas mus li, cov kev khiav dej num, xws li exponential, sine, linear los yog quadratic.

Yuav ua li cas bothered lub Fabkis mathematicians nyob rau hauv cov kev tshawb xav ntawm Fourier?

Tom qab tag nrho, yog ib tug mathematician yog txoj cai mus hais lus cam, ces, summing ib txhis trigonometric Fourier series, nws yog ua tau kom tau ib tug muaj tseeb sawv cev ntawm cov kauj ruam ntawm kev hais lus, txawm yog hais tias nws muaj ib tug txheej ntawm zoo xws li cov kauj ruam. Nyob rau hauv thaum ntxov thib kaum cuaj xyoo pua, qhov no daim ntawv qhia ciali absurd. Tab sis txawm tag nrho cov xav phem cais tsis, ntau daim ntawv thom nug tau txhab rau tau txais ntawm txoj kev tshawb ntawm no tshwm sim, nws tsiv mus nyob tshaj cov thermal conduction kev tshawb fawb. Txawm li cas los, feem ntau zaum txuas ntxiv mus raug kev txom nyem rau lo lus nug: "Lub sum ntawm cov sine yoj series converges rau lub caij nyoog muaj nuj nqis ntawm ib tug discontinuous muaj nuj nqi?"

Convergence ntawm Fourier series: Piv txwv li

Qhov teeb meem ntawm convergence ntawm lub nce txhua lub sij hawm koj xav tau cov summation ntawm ib tug infinite series ntawm cov zauv. xav txog ib tug classic piv txwv rau qhov kev nkag siab ntawm no phenomenon. Yuav koj puas tau mus txog lub phab ntsa, yog tias txhua kauj ruam yog ib nrab yav dhau los? Puas xav koj yog ob meters los ntawm lub hom phiaj, tus thawj kauj ruam los ze zog mus nyob ib ncig ntawm ib nrab txoj kev, tom ntej no - lub cim ntawm ib tug peb-quarters, thiab tom qab lub thib tsib, koj yuav kov yeej yuav luag 97 feem pua ntawm txoj kev. Txawm li cas los, tsis muaj teeb meem ntau npaum li cas cov kauj ruam uas koj tau ua tej nuj nqis, cov npaj lub hom phiaj koj mus txog nyob rau hauv ib tug nruj zauv kev txiav txim zoo. Siv zauv suav, peb yuav ua pov thawj tias nyob rau hauv lub kawg tej zaum yuav los ze zog mus rau ib qho arbitrarily me me muab kev ncua deb. Qhov no yog sib npaug rau ib tug pov thawj qhia tau tias tag nrho cov nqi ntawm ib tug ib nrab, ib tug plaub, thiab li ntawm. E. Yuav yuav sib sau.

Qhov teeb meem ntawm convergence: lub thib ob los, los yog kev ntsuas ntawm tus Tswv Kelvin

Pheej rau lo lus nug sawv nyob rau hauv lub lig thib kaum cuaj xyoo pua, thaum lub Fourier series tau sim siv twv seb rau kev siv ntawm cov ebbs thiab ntws. Nyob rau ntawm lub sij hawm, Tswv Kelvin yog invented ntaus ntawv yog ib qho analog computer uas tso cai rau cov neeg tsav nkoj rog thiab ua lag luam marine saib yog ib tug natural phenomenon. Qhov no mechanism txhais txheej theem thiab amplitudes ntawm lub rooj qhov siab ntawm lub tides thiab cov coj lub sij hawm lub sij hawm, ua tib zoo ntsuas nyob rau hauv qhov chaw nres nkoj thoob plaws hauv lub xyoo. Txhua parameter yog ib tug sinusoidal tivthaiv qhia tom tsim heights thiab yog ib tug ntawm cov kev Cheebtsam. Cov kev ntsuas tau yog tswv yim mus rau lub xam ntaus Tswv Kelvin, synthesizing nkhaus uas kwv yees qhov siab ntawm cov dej raws li ib tug muaj nuj nqi ntawm lub nram qab no xyoo. Heev sai sai no, cov curves twb kos npe rau tag nrho cov harbours ntawm lub ntiaj teb no.

Thiab yog hais tias tus txheej txheem yuav tawg discontinuous muaj nuj nqi?

Nyob rau ntawm lub sij hawm, nws ciali pom tseeb hais tias ntaus ntawv twv ib tug tidal yoj, nrog rau ntau cov ntsiab ntawm cov account yuav xam ib tug loj tus naj npawb ntawm theem thiab amplitudes, thiab thiaj li muab ib tug ntau yog twv ua ntej. Cuaj kaum, nws muab tawm hais tias qhov no yeej yog ib tsis cai nyob rau hauv tus neeg mob qhov twg lub tidal qhia tias yuav tsum tsim, muaj ib tug ntse dhia, uas yog, yog discontinuous. Nyob rau hauv cov kev tshwm sim hais tias tus apparatus mus nkag rau cov ntaub ntawv los ntawm ib lub rooj ntawm lub sij hawm lub ntsiab lus, nws laij ob peb Fourier coefficients. Recovering tus thawj muaj nuj nqi vim lub sinusoidal tivthaiv (nyob rau hauv raws li cov pom coefficients). Lub sib xws ntawm tus thawj thiab cov reconstructed qhia yuav tsum ntsuas ntawm tej taw tes. Thaum lub rov suav thiab kev sib piv yuav pom hais tias tus nqi ntawm cov loj tshaj yuam kev yog tsis txo. Txawm li cas los, lawv yog laus nyob rau hauv lub cheeb tsam coj mus rau lub point ntawm kev sib tawg, thiab lwm yam taw tes yuav pes tsawg. Nyob rau hauv 1899, qhov no tshwm sim twb paub tseeb hais tias raws Yausua Willard Gibbs ntawm Yale University.

Convergence ntawm Fourier series thiab cov kev loj hlob ntawm kev kawm txog zauv raws li ib tug tag nrho

Fourier tsom xam tsis siv rau kab zauv uas muaj ib tug infinite tus naj npawb ntawm bursts ntawm ib tug tej yam luv. Nyob rau hauv kev Fourier series, yog hais tias tus thawj muaj nuj nqi yog sawv cev los ntawm cov kev tshwm sim ntawm lub sij lub cev ntsuam, yeej ib txwm sib. Yog muaj lus nug ntawm convergence ntawm tus txheej txheem no rau tej chav kawm ntawm kev khiav dej num tau coj mus rau tshiab ceg ntawm kev kawm txog zauv, xws li lub hom phiaj ntawm generalized zog. Nws yog txuam nrog cov npe xws li Schwartz, J .. Mikusiński thiab J. tuam tsev. Nyob rau hauv no txoj kev xav, ib tug ntshiab thiab leej theoretical lub hauv paus rau tej kev qhia tau tsim muaj raws li cov Dirac delta muaj nuj nqi (nws qhia txog cov cheeb tsam ntawm ib cheeb tsam, feeb meej nyob rau hauv ib tug infinitesimal zej zog ntawm tus taw tes) thiab "kauj ruam" Heaviside. Los ntawm no ua hauj lwm Fourier series ua muaj feem xyuam rau kev daws sib npaug thiab teeb meem, uas koom nrog intuitive lub tswv yim: point xwb, taw tes loj, magnetic dipoles, thiab cov concentrated load rau hauv lub beam.

Fourier txoj kev

Fourier series, nyob rau hauv raws li cov ntsiab cai ntawm cuam tshuam, pib nrog cov decomposition ntawm complex ntaub ntawv mus rau hauv yooj yim. Piv txwv li, ib tug kev hloov nyob rau hauv kub khiav vim nws cov zaj ntawm lub ntau yam teeb meem ntawm lub tshav kub insulating cov ntaub ntawv uas cov lus zoo los yog hloov hauv av nto - muaj av qeeg, ib tug kev hloov nyob rau hauv lub orbit ntawm lub vaj xilethi-aus lub cev - tus ntawm lub ntiaj teb. Feem ntau, cov equations piav txog tej yam yooj yim classical system elementary solved rau txhua tus neeg wavelength. Fourier lawm hais tias tej yam yooj yim kev daws teeb meem yuav tau summed li rau ntau yam dej num. Nyob rau hauv cov lus ntawm kev kawm txog zauv, Fourier series - ib tug vib this kev lub cuav ntawm kev qhia tawm ntawm cov harmonic - cosine thiab sine tsis. Yog li ntawd, qhov no tsom xam yog tseem paub tias hauv lub npe "harmonic tsom xam".

Fourier series - ib qho zoo tagnrho txoj kev mus rau lub "computer uas muaj hnub nyoog"

Ua ntej mus rau lub creation ntawm computer technology Fourier yog cov zoo tshaj plaws riam phom nyob rau hauv lub arsenal zaum ua hauj lwm nrog nthwv dej xwm ntawm peb lub ntiaj teb no. Fourier series nyob rau hauv txoj daim ntawv tso cai rau koj mus tsis tau tsuas yog daws tau yooj yim teeb meem uas yog amenable mus ncaj daim ntawv thov ntawm Newton txoj kev cai mechanics, tab sis kuj yuav tsum muaj sib npaug. Feem ntau ntawm cov discoveries ntawm Newtonian science ntawm lub puas tawm xyoo los ua tau xwb vim lub Fourier txoj kev.

Fourier series hnub no

Nrog rau txoj kev loj hlob ntawm Fourier transform computers tau nce mus rau ib tug tshiab. Cov txheej txheem no yog nrees entrenched nyob rau hauv yuav luag tag nrho cov teb ntawm science thiab technology. Raws li ib qho piv txwv, ib cov suab thiab video. Nws yuav ua raws li twb tau ua tau tsuas yog ua tsaug rau qhov kev tshawb xav tsim los ntawm qhov Fabkis mathematician ntawm thaum ntxov thib kaum cuaj caug xyoo. Yog li, lub Fourier series nyob rau hauv txoj daim ntawv tau tso cai rau ua ib tug txhob nyob rau hauv txoj kev tshawb no ntawm txheej chaw. Nyob rau hauv tas li ntawd, nws tau cuam tshuam rau txoj kev kawm ntawm cov physics ntawm semiconductor cov ntaub ntawv thiab ntshav, microwave acoustics, oceanography, radar, seismology.

Trigonometric Fourier series

Nyob rau hauv kev kawm txog zauv, ib tug Fourier series yog ib txoj kev uas sawv cev rau arbitrary complex zog raws li ib tug zaum ntawm sob. Nyob rau hauv kev mob, tus xov tooj ntawm kab zauv tej zaum yuav infinite. Qhov ntau dua tus xov tooj suav nyob rau hauv lub xam, cov ntau yog zaum kawg tshwm sim yog tau. Cov feem ntau siv yooj yim trigonometric cosine los yog sine muaj nuj nqi. Nyob rau hauv cov ntaub ntawv no, lub Fourier series yog hu ua trigonometric, thiab qhov kev txiav txim ntawm tej kab zauv - harmonic decomposition. Qhov no txoj kev plays ib qho tseem ceeb luag hauj lwm nyob rau hauv kev kawm lej. Ua ntej ntawm tag nrho cov, cov trigonometric series muab ib tug txhais tau rau cov duab, raws li zoo raws li txoj kev tshawb no ntawm kev khiav dej num, nws yog ib lub ntsiab chav tsev ntawm cov kev tshawb xav. Nyob rau hauv tas li ntawd, nws tso cai rau peb mus daws tau ib tug xov tooj ntawm cov teeb meem nyob rau hauv lej physics. Thaum kawg, qhov no kev tshawb xav tau pab mus rau txoj kev loj hlob ntawm zauv tsom xam, nws muab sawv mus rau ib tug xov tooj ntawm cov tseem ceeb heev ceg ntawm zauv science (kev tshawb xav ntawm integrals, lub hom phiaj ntawm periodic zog). Nyob rau hauv tas li ntawd, cov starting taw tes rau txoj kev loj hlob ntawm cov hauv qab no theories: poob lawm, kev khiav dej num ntawm ib tug tiag tiag nce mus nce los, haumxeeb tsom xam, thiab kuj nteg lub lub hauv paus rau harmonic tsom xam.