Nazariyai funksiyaho

Nazariyai funksiyaho, yak fasli matematika buda, khosiyathoi umumii funksiyahoro meomuzad. Nazariyai funksiyaho az du shokhai mustaqil iborat ast: nazariyai funksiyahoi tagyiryoban­daash haqiqi va nazariyai funksiyahoi tagyiryobandaash kompleksi.

Dar tahlili matematikii «klas­siki» obekthoi asosii tadqiqot funksiyahoi dar fosilahoi dodashuda (bointiho yo beintiho) befosila va funksiyahoi to yak daraja sufta meboshand. Vale az nimai duyumi asri 19 cap karda inkishofi matematika zarurati omuzishi murattabi funk­siyahoi umumitarro ba miyon ovard. Sababi asosii in talabot on ast, ki hududi paydariayii funksiyahoi befosila funksiyai kanishdor budanash mumkin ast. Ba iborai digar, sinfi funksiyahoi befosila nisbat ba amali asosii tahlili matematiki — yoftani hudud ba khududguzari mahdud nabudaast. Az in jost, ki funksiyahoe, ki bo yorii katorhoi trigovometri barin usuli klassiki muayyan karda meshavavd, aksar vaqt funksiyahoi kanishdor va yo funk­siyahoi differensironidanashavanda meboshand. Ba monavdi hamin Go­splan funksiyahoi befosila funksiyai kanishdor shudanash mumkin ast. Nihoyat muodilahoi differensialie, ki hangomi hal kardani masalahoi fiziki ba vujud meoyand, bazan dar sinfi funksiyahoi ba qadri kofi sufta hal nadorandu dar sinfi vasetari funksiyaho halshavandaand. (Agar dar zimni in khudi mafhumi hal tamim doda shavad). Mahz hamin halhoi tamimyofta halli masalai fiziki mebo­shand. In va digar vaziyathoi ba in monand sababi asosii ba vujud omadani nazariyai funksiyahoi tagyi­ryobandaash haqiqi gardidaid.

Baze fakthoi juzii nazariyai funksiyahoi tagyiryobandaash haqiqi

(mavjudiyati qatorhoi funksiyahoi befosilai summaashon funksiyahoi kanishdor, misolhoi funksiyahoi befosilai ,dar hech yagon interval dif- feronsironidanashavanda va gayra) asri 19 kashf shudaand. Dar avvali asri 20 boshad hangomi ba sifati asosii tadqoqoti funksiyaho qabul shudani nazariyai majmuho, nazariyai funk­siyahoi tagyiryobandaash khaqiqi murattabona inkishof yoft.

Nazariyai funksiyahoi tagyiryoban­daash haqiqatro ba se raviya judo kardan mumkin ast: 1) Nazariyai funksiyahoi metri dar on khosiyati funksiyaho bo yorii cheni (nig. Majmu) majmuhoe omukhta meshavad, ki dar onho in khosiyatho joy dorand. Dar Nazariyai funksiyahoi metri integroni va differevsironii funksiyaho (nig. Integral, Dif­ferensial, Hosila) dar shakli tom omukhta meshavavd, mafhumi nazdikshavii paydarpaihoi funksionaliro bo tarzhoi gunogun tamim mekunand, sokhti umumitarin funksiyahoi kanishdor tadqiq karda meshavad va gayra Sinfi asosii Nazariyai funksiyahoi metri funk­siyahoi chenshavanda ba hisob meravand. 2) Nazariyai funksiyahoi tavsifi (deskriptivi), ki dar on obekti asosii tadqiqot amali ba khududguzari meboshad. 3) Nazariyai funksiyahoi konstruktivi, ki masalahoi bo yorii funksiyahoi tahlili ifoda namudani funksiyahoi ikhtiyoriro tadqiq mekunad (nig. Nazdikkuni va interpolironii funksiyaho).

Dar borai nazariyai funksiyahoi tagyiryobandaash komvleksi nig Funktsiyahoi tahlili.

Ad.: Kolmogorov A. N., Fomin S. V., Elementi teorii funktsiy i funk­tsionalnogo analiza, 4 izd., M., 1974.