Fourier jerin: tarihi da tasiri na ilmin lissafi inji ga ci gaban kimiyya

Fourier jerin - wannan ra'ayi siddan zaba ayyuka zuwa lokaci a jere. A general sharuddan, wannan bayani ne ya kira fadada kashi a wani orthogonal akai. The fadada ayyuka a Fourier jerin ne quite mai iko kayan aiki domin warware matsaloli daban-daban saboda da kaddarorin da canji a cikin hadewa, bambantawa, kazalika da motsa a cikin shaida magana da convolution.

A mutum wanda ba saba da hakan lissafi, kazalika da ayyukan Faransa masanin kimiyya Fourier, mafi m ba zai fahimci abin da "darajõji" da kuma abin da suke aikatãwa. Amma duk da haka wannan canji da aka quite da tabbaci shiga rayuwarmu. Ana amfani ba kawai lissafi, amma kuma lissafin kimiyya, chemists, likitoci, Taurari, Seismologists, oceanographers da sauransu. Bari mu ma dauki mukalli da ayyukan da babban Faransa masanin kimiyya ne da sanya gano, gaba a lokacinsa.

Mutumin da Fourier fasalin

Fourier jerin ne daya daga cikin hanyoyin da (tare da bincike da kuma wasu) daga cikin Fourier canza. Wannan tsari faruwa duk lokacin da wani mutum ya ji wani sauti. Our kunne ta atomatik sabobin tuba cikin sauti kalaman. Oscillatory motsi na na farko barbashi a wani na roba matsakaici aka fadada a cikin jerin (bakan) m girma dabi'u ga sautunan na daban-daban Heights. Next, kwakwalwa sabobin tuba da wannan data cikin saba sauti gare mu. Duk wannan shi ne ban da muradinmu ko sani kanta, amma domin fahimtar matakai cewa dauki shekaru da dama don nazarin mafi girma lissafi.

Read more game da Fourier fasalin

A Fourier fasalin za a iya za'ayi analytical, numerals da kuma wasu hanyoyin. Fourier jerin ne adadi tsari ga saki wani oscillatory matakai - daga teku tides, kuma tãguwar ruwa ta da haske ga hasken rana hawan keke (da kuma sauran astronomical abubuwa) aiki. Amfani da wadannan ilmin lissafi dabaru, yana yiwuwa kwakkwance da aiki, wakiltar wani oscillatory matakai a yawan sinusoidal aka gyara cewa tafi daga m zuwa iyakar da mataimakin versa. A Fourier fasalin ne mai aiki kwatanta lokaci da mawadãta daga sinusoids m zuwa wani mita. Wannan tsari za a iya amfani da shi don warware wata sosai hadaddun lissafai wanda bayyana tsauri matakai abin da ke faruwa a karkashin mataki na zafi, haske ko lantarki da makamashi. Har ila yau, Fourier jerin amfani da rarrabe DC gyara a hadaddun waveforms, sa shi yiwuwa a daidai fassara gwaji lura a magani, sunadarai da ilmin taurari.

tarihi bayanai

Da kafa uba na wannan ka'ida Faransa lissafi Zhan Batist Zhozef Fure. Sunansa daga baya kuma wannan canji da aka kira. Da farko, da masana kimiyya amfani da wata dabara don nazarin da bayyana sunadaran da thermal watsin - zafi yaduwa a daskararru. Fourier da shawarar cewa na farko sababbu rarraba da thermal kalaman za a iya bazu zuwa cikin sauki sinusoid, kowanne daga abin da zai yi da zazzabi m da kuma iyakar, kazalika da ta zamani. Kamar haka kowane irin bangaren da za a auna daga m zuwa iyakar da mataimakin versa. A ilmin lissafi aiki da ya bayyana manya da ƙananan kololuwa da kwana, kazalika da lokaci na kowane masu jituwa, da ake kira Fourier fasalin na da yawan zafin jiki rarraba magana. Marubucin na da ka'idar rage overall rarraba aiki da cewa da wuya a ilmin lissafi description, a wani sosai sauki rike lamba na lokaci-lokaci ayyuka na ba tare da kuma cosine, a cikin adadin ba da farko rarraba.

A qa'ida ta yi hira da da ra'ayoyi da Sahaban

Sahaban na masanin kimiyya - manyan lissafi na farkon karni na sha tara - bai yarda da wannan ka'ida. Babban ƙin yarda ya yardar Fourier cewa discontinuous aiki kwatanta wani madaidaiciya line ko kwana ta kasance an tsãge, shi za a iya wakilta a matsayin mai Naira Miliyan Xari sinusoidal maganganu da suke ci gaba da. Kamar yadda wani misali, ka yi la'akari da "mataki" Heaviside: da darajar ne sifili zuwa hagu na da rata da kuma daya a dama. Wannan aiki ya bayyana dogaro da wutar lantarki a halin yanzu a lokacin m ga ƙulli sarkar. Littafi ka'idar a wannan lokaci, ya taba ci karo da irin wannan halin da ake ciki, a lokacin da wani discontinuous magana zai iya bayyana ta a hade da m, na kowa ayyuka, kamar karuwa da sauri, ba tare da, mikakke ko quadratic.

Abin da ya dame Faransa lissafi da ka'idar Fourier?

Bayan duk, idan wani lissafi da yake daidai ga jayayya, sa'an nan, Summing wani iyaka trigonometric Fourier jerin, yana yiwuwa a samu wani cikakken wakilci na mataki na magana, ko idan yana da wani sa na kama da matakai. A farkon karni na sha tara, wannan sanarwa da jũna m. Amma duk da dukan shakka, da yawa lissafi sun kumbura da ikon yinsa, daga cikin nazarin wannan sabon abu, motsi da shi bayan da thermal madugu karatu. Duk da haka, mafi masana kimiyya ci gaba da fama da tambaya: "Can da Naira Miliyan Xari da ba tare da kalaman jerin converges zuwa ainihin darajar da discontinuous aiki?"

Haduwa da Fourier jerin: misali

A batun na haduwa yakan duk lokacin da ka bukata da summation na wani iyaka jerin lambobi. la'akari da wani classic misali ga fahimtar wannan sabon abu. Za ka iya taba isa bangon, idan kowane mataki ne rabin baya? Yi tsammani kai ne mita biyu daga burin, da farko mataki kusa da a kusa da rabin hanya, na gaba - da alamar wata uku-bariki, da kuma bayan na biyar, zã ku rinjaya kusan kashi 97 cikin dari na hanya. Duk da haka, ko ta yaya mutane da yawa matakai ka yi ba, da nufin manufa ka isa a wani m ilmin lissafi hankali. Amfani lamba lissafin, za mu iya tabbatar da cewa a karshen iya zama kusa da wani siddan kananan ba nesa. Wannan daidai ne da wani dalĩli nuna cewa jimlar darajar daya rabin, daya karo na hudu, da sauransu. E. Za ayan hadin kai.

A batun na haduwa: na biyu zuwan, ko kayan aiki na Ubangiji Kelvin

Akai-akai da tambaya tashi a cikin marigayi karni na sha tara, a lokacin da Fourier jerin yi kokarin amfani da su hango ko hasashen da tsanani da ebbs da gudana. A wannan lokaci, Ubangiji Kelvin da aka kirkiro na'urar ne wani analog kwamfuta da yarda matuƙan sojojin ruwa da kuma m marine duba ne na halitta sabon abu. Wannan inji ayyana kafa manzilõli, kuma amplitudes na tebur tsawo daga cikin tides da kuma daidai lokacin da lokacin, a hankali auna a cikin harbor ko'ina cikin shekara. Kowane siga ne sinusoidal bangaren magana tide Heights kuma ya kasance daya daga cikin na yau da kullum da aka gyara. The ji sakamakon ne shigar da sarrafa kwamfuta na'urar Ubangiji Kelvin, rairaya kwana cewa annabta tsawo daga cikin ruwa kamar yadda wani aiki na da wadannan shekara. Bada jimawa ba, wadannan masu lankwasa da aka kõma sama domin dukan harbors na duniya.

Kuma idan aiwatar da zai karye discontinuous aiki?

A wannan lokacin, kamar fili da cewa na'urar tsinkaya a tidal kalaman, da yawa abubuwa na asusun iya lissafta a manyan yawan manzilõli, kuma amplitudes, kuma haka samar da wani mafi m Hasashen. Duk da haka, shi ya juya fitar da cewa wannan abin kwaikwaya ba lura a lokuta inda tidal magana cewa za a iya hada, na dauke da kaifi Jump, wato, masu discontinuous. A cikin taron cewa na'ura shiga data daga wani tebur na lokaci maki, shi calculates 'yan Fourier coefficients. Murmurewa asali aiki saboda da sinusoidal bangaren (daidai da samu coefficients). Hilafa tsakanin asali da kuma kerata magana za a iya auna a kowane batu. Lokacin da sake lissafin da kwatancen za a iya gani cewa, darajar mafi girma kuskure ba rage. Duk da haka, suna sarrafa a yankin m ga batu na katsewa, da kuma wani batu ayan sifili. A 1899, wannan sakamakon da aka tabbatar a rubuce Joshuwa Willard Gibbs na Yale University.

Haduwa da Fourier jerin da kuma ci gaban ilmin lissafi a matsayin dukan

Fourier analysis Bai shafi maganganu dauke da wani iyaka yawan bursts a wasu lokutta. A general Fourier jerin, idan asalin aikin da aka wakilta sakamakon da ainihin jiki ma'aunai, ko da yaushe converge. Tambayoyi da haduwa da wannan tsari don takamaiman azuzuwan na ayyuka sun jagoranci ga sabon rassan lissafi, kamar ka'idar jimlace ayyuka. An hade da sunaye kamar Schwartz, J .. Mikusiński kuma J. Haikalin Ubangiji. A karkashin wannan mahanga, a fili da kuma daidai ne msar tambayar dalilin wannan magana da aka kafa a matsayin Dirac Delta aiki (da shi ya bayyana yankin na guda yanki, mayar da hankali a wani infinitesimal unguwa na ma'ana) da kuma "mataki" Heaviside. Ta hanyar wannan aikin Fourier jerin zama m domin warware lissafai da matsaloli, wanda ya unsa da ilhama Concepts: aya cajin, ma'ana taro, Magnetic dipoles, da kuma mayar da hankali load a kan katako.

Fourier hanyar

Fourier jerin, daidai da ka'idojin kutse, za a fara da bazuwar na hadaddun siffofin cikin sauki. Alal misali, wani canji a cikin zafin rana ya kwarara saboda da nassi ta hanyar da daban-daban shingen zafi insulating abu na sababbu siffar ko canza ƙasa surface - wata girgizar kasa, wani canji a cikin sarari guda na cikin wani sarari suKe jiki - rinjayar da taurari. Yawanci, wadannan lissafai kwatanta sauki gargajiya tsarin na farko warware ga kowane mutum zango. Fourier ya nuna cewa sauki mafita za a iya taƙaice a matsayin don ƙarin hadaddun ayyuka. A harshen lissafi, Fourier jerin - a hanya domin biyayya da magana Naira Miliyan Xari da masu jituwa - cosine kuma ba tare da taguwar ruwa. Saboda haka, wannan bincike kuma aka sani a karkashin sunan "masu jituwa bincike".

Fourier jerin - wani manufa hanyar zuwa "kwamfuta shekaru"

Kafin halittar kwamfuta fasahar Fourier Hanyar ne mafi kyau makami a arsenal na masana kimiyya aiki tare da kalaman yanayin mu na duniya. Fourier jerin a hadaddun form ba ka damar ba kawai warware sauki matsalolin da suke amenable don kaitsaye aikace-aikace na Newton ta dokokin da makanikai, amma kuma da muhimman hakkokin lissafai. Mafi yawa daga cikin binciken Newtonian kimiyya na karni na sha tara ya zama zai yiwu ne kawai saboda da Fourier Hanyar.

Fourier jerin yau

Tare da ci gaban Fourier fasalin kwakwalwa sun tashi zuwa wani sabon matakin. Wannan dabara da aka daidaita kunshe a kusan dukkan fannonin kimiyya da fasaha. A matsayin misali, a dijital audio da video. Da aiwatar da aka sanya yiwu ne kawai godiya ga ka'idar ci gaba da Faransa lissafi na farkon karni na sha tara. Saboda haka, Fourier jerin a hadaddun form ya yarda ya yi nasara a cikin binciken na m sarari. Bugu da kari, shi ya shafi nazari da kimiyyar lissafi na semiconductor kayan da jini, obin na lantarki acoustics, oceanography, radar, seismology.

Trigonometric Fourier jerin

A lissafi, Fourier jerin ne a hanyar wakiltar sabani hadaddun ayyuka a matsayin Naira Miliyan Xari da sauki. A general lokuta, da yawan maganganu iya zama iyaka. A mafi girma da lambar kidaya a cikin lissafi, da mafi m karshe sakamakon da aka samu. Mafi na kowa yin amfani da sauki trigonometric cosine ko ba tare da aiki. A wannan yanayin, Fourier jerin ake kira trigonometric, da kuma shawarar da irin wannan bayanin - masu jituwa bazuwar. Wannan hanya tana taka muhimmiyar rawa a cikin ilmin lissafi. Da farko, da trigonometric jerin samar da wata hanya ga image, kazalika da nazari na ayyuka, shi ne babban naúrar ka'idar. Bugu da kari, shi ya yale mu mu shirya a yawan matsaloli a ilmin lissafi kimiyyar lissafi. A karshe, wannan ka'idar ya gudummawar da ci gaban na ilmin lissafi analysis, shi ya ba Yunƙurin zuwa lamba na da muhimmanci sosai rassan ilmin lissafi kimiyya (ka'idar integrals, ka'idar lokaci-lokaci ayyuka). Bugu da kari, masomin ci gaban da wadannan theories: sets, ayyuka na wani real m, aikin bincike, da kuma aza harsashin ginin for masu jituwa analysis.