»°ËÞ½I—¹‚Ì‚¨’m‚点
ô‚ݍž‚Ý
¥l‹C‹LŽ–×ݷݸÞ
øêŠT—v
ô‚ݍž‚݁i‚½‚½‚Ý‚±‚ݤ‰p: convolutionj‚Ƃ͊֐” g ‚𕽍sˆÚ“®‚µ‚È‚ª‚çŠÖ” f ‚ɏd‚Ë‘«‚µ‡‚킹‚é“ñ€‰‰ŽZ‚Å‚ ‚顏ô‚ݍž‚ݐϕª¤‡¬Ï¤dôÏ•ª¤‚ ‚é‚¢‚͉pŒê‚É•í‚¢ºÝÎÞØ­°¼®Ý‚Æ‚àŒÄ‚΂ê‚é¡

—ðŽj[•ÒW]

ô‚ݍž‚ݐϕª‚ª—p‚¢‚ç‚ꂽÅ‰Šú‚Ì—á‚̈ê‚Â‚Í d'Alembert (1754) Recherches sur diff?rents points importants du syst?me du monde ‚É‚¨‚¯‚éòװ‚̒藝‚Ì“±o‚É‚ ‚é[1]¡‚Ü‚½

f ( u ) g ( x u ) d u {\displaystyle \int f(u)\cdot g(x-u){\mathit {du}}} ‚ÌŒ`‚ÌŽ®‚Í Lacroixi‰pŒê”Łj Treatise on differences and series[’Žß 1] ‚Ì505•Å‚Å—p‚¢‚ç‚ê[2]¤‚»‚Ì‚·‚®Œã‚ÉLaplace, Fourier, Poisson‚ç‚ÌŒ¤‹†‚ɏô‚ݍž‚݉‰ŽZ‚ªŒ»‚ê‚Ä‚¢‚é¡–¼ÌŽ©‘Ì‚ªL‚­—p‚¢‚ç‚ê‚é‚悤‚É‚È‚é‚É‚Í1950”N‘ã‚ ‚é‚¢‚Í1960”N‘ã‚ð‘Ò‚½‚È‚¯‚ê‚΂Ȃç‚È‚¢¡‚»‚ê‚ɐ旧‚Á‚Ä‚ÍÄ޲Œê: faltungi¢ô‚ݍž‚Ý£j¤composition producti¢‡¬Ï£j¤ superposition integrali¢d‚ˍ‡‚킹Ï•ª£j‚È‚Ç‚Æ‚àŒÄ‚΂꤂ ‚é‚¢‚Ͷ°Ù¿Ý‚̐ϕª (Carson's integral)[3]
‚Æ‚àŒ¾‚Á‚½¡Œ»‘ã“I‚È’è‹`‚ª‚æ‚èŒÃ‚¢—p—á‚É“éõ‚ނ킯‚Å‚à‚È‚¢‚ª¤‚»‚ê‚Å‚à‘‚­‚Í1903”N‚²‚ë‚ɂ͏oŒ»‚µ‚Ä‚¢‚é[4][5]¡

‡¬Ï‚Ì“Á•Ê‚̏ꍇ‚Æ‚µ‚Ẳ‰ŽZ

0 t φ ( s ) ψ ( t s ) d s ( 0 t < ) {\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s){\mathit {ds}}\quad (0\leq t<\infty )} ‚Í Volterra (1913)
"Le?ons sur les fonctions de lignes" ‚É‚ ‚é[6]

ˆêŽŸŒ³[•ÒW]

’è‹`[•ÒW]

ŠÖ” f, g ‚̏ô‚ݍž‚Ý‚Í f ? g ‚Ə‘‚«¤ˆÈ‰º‚̂悤‚É’è‹`‚³‚ê‚éF

( f g ) ( t ) = f ( τ ) g ( t τ ) d τ {\displaystyle (f*g)(t)=\int f(\tau )g(t-\tau )\,d\tau } Ï•ª”͈͂͊֐”‚Ì’è‹`ˆæ‚Ɉˑ¶‚·‚顒ʏí‚Í‹æŠÔ (−‡, +‡) ‚Å’è‹`‚³‚ê‚éŠÖ”‚ðˆµ‚¤‚±‚Æ‚ª‘½‚¢‚̂ŤÏ•ª”ÍˆÍ‚Í −‡ ‚©‚ç +‡ ‚ÅŒvŽZ‚³‚ê‚邱‚Æ‚ª‘½‚¢¡ˆê•û f, g ‚ª—LŒÀ‹æŠÔ‚Å‚µ‚©’è‹`‚³‚ê‚È‚¢ê‡‚ɂͤg(t − τ) ‚ª’è‹`ˆæ“à‚É“ü‚é‚悤‚É f, g ‚ðŽüŠúŠÖ”‚ÆŒ©‚È‚µ‚ÄŒvŽZ‚³‚ê‚é¡‚±‚ÌŽüŠúŠÖ”‚ÆŒ©‚È‚µ‚ďô‚ݍž‚Ý‚ð‚·‚é‚±‚Æ‚ðzŠÂô‚ݍž‚݁i‚¶‚ã‚ñ‚©‚ñ‚½‚½‚Ý‚±‚ݤ‰p: cyclic convolutionj‚ƌĂԡ

—£ŽU’l‚Å’è‹`‚³‚ꂽŠÖ”‚ɑ΂·‚éô‚ݍž‚݂ͤÏ•ª‚Ì‚©‚í‚è‚É‘˜a‚ðŽg‚Á‚Ä“¯—l‚É’è‹`‚³‚ê‚éF

( f g ) ( m ) = n f ( n ) g ( m n ) {\displaystyle (f*g)(m)=\sum _{n}{f(n)\,g(m-n)}} ‘˜a‚͈̔͂àŠÖ”‚Ì’è‹`ˆæ‚Ɉˑ¶‚µ¤ŠÖ”‚ª—LŒÀ‹æŠÔ‚Å‚µ‚©’è‹`‚³‚ê‚Ä‚¢‚È‚¢ê‡‚ÍŽüŠúŠÖ”‚Æ‚Ý‚È‚µ‚ďô‚ݍž‚݉‰ŽZ‚ªs‚í‚ê‚é¡‚Ü‚½¤—£ŽUŒn‚̏ꍇ¤’è‹`ˆæŠO‚Ì’l‚ð 0 ‚Æ’è‹`‚µ’¼‚µ‚½ŠÖ”‚ł̏ô‚ݍž‚Ý‚ª‚æ‚­s‚í‚ê‚é¡‚±‚ê‚ðüŒ`ô‚ݍž‚݁i‚¹‚ñ‚¯‚¢‚½‚½‚Ý‚±‚ݤ‰p: linear convolutionj‚ƌĂԡüŒ`ô‚ݍž‚Ý‚Í’¼üô‚ݍž‚݁i‚¿‚å‚­‚¹‚ñ‚½‚½‚Ý‚±‚݁j‚Æ‚àŒÄ‚΂ê‚é¡‚È‚¨—£ŽUŒn‚̏ꍇ‚͐ϕª‚ðŽg‚킸‚É‘˜a‚ðŽg‚¤‚̂Ťô‚ݍž‚ݐϕª¥dôÏ•ª‚Ƃ͌Ă΂¸¤ô‚ݍž‚ݘa¥dô˜a‚ƌĂԡ

«Ž¿[•ÒW]

Ï•ª‰‰ŽZ‚É—R—ˆ‚·‚鐫Ž¿‚Æ‚µ‚Ĉȉº‚̐«Ž¿‚ª‚ ‚é¡

ù Œ‹‡—¥
ù •ª”z—¥
ù ½¶×°”{
‚½‚¾‚µ¤a ‚Í”CˆÓ‚ÌŽÀ”‚Ü‚½‚Í•¡‘f”¡
ù ”÷•ª
‚½‚¾‚µ¤D ‚Í”÷•ª‰‰ŽZŽqi—£ŽUŒn‚̏ꍇ‚Í Df(n) = f(n + 1) − f(n)j¡ ‚½‚¾‚µ ‚͊֐” f ‚ÌÌ°Ø´•ÏŠ·‚Å‚ ‚é¡‚±‚̒藝‚Í×Ìß×½•ÏŠ·¥Z•ÏŠ·‚âÒØݕϊ·‚Æ‚¢‚Á‚½•ÏŠ·‚ɑ΂µ‚Ä‚à“K—p‚Å‚«‚é¡

Ì°Ø´•ÏŠ·‚ðŽg‚Á‚ďô‚ݍž‚݉‰ŽZ‚ð’Pƒ‚ÈŠ|‚¯ŽZ‚É•ÏŠ·‚·‚邱‚Æ‚ªo—ˆ‚é¡—£ŽUŒn‚ł̊֐”‚̏ꍇ¤’è‹`’Ê‚è‚̏ô‚ݍž‚ÝŒvŽZ‚ð‚µ‚È‚¢‚ŤŠÖ” f, g ‚̍‚‘¬Ì°Ø´•ÏŠ· (FFT) ‚ðŠ|‚¯ŽZ‚µ‚½Œ‹‰Ê‚ð‹t‚‘¬Ì°Ø´•ÏŠ· (IFFT) ‚ð‚·‚é‚±‚ƂŤ‚‘¬‚ɏô‚ݍž‚Ý‚ÌŒvŽZˆ—‚ð‚·‚é‚Ì‚ªˆê”Ê“I‚Å‚ ‚é¡

‰ž—p[•ÒW]

Šm—¦‘ª“x‚É‚¨‚¯‚éô‚ݍž‚Ý[•ÒW]


W‡ŠÖ”‚̈êŽí‚Å‚ ‚éŠm—¦‘ª“x‚̏ô‚ݍž‚Ý‚ÍŽŸ‚̂悤‚É•\Œ»‚³‚ê‚é¡Šm—¦‘ª“x ƒÊ1, ƒÊ2 ‚É‚¨‚¢‚Ä”CˆÓ‚ÌÎÞÚُW‡ B ‚ɑ΂µ¤

( μ 1 μ 2 ) ( B ) = 1 B ( x + y )   μ 1 ( d x ) μ 2 ( d y ) {\displaystyle (\mu _{1}*\mu _{2})(B)=\int 1_{B}(x+y)\ \mu _{1}(dx)\mu _{2}(dy)} ‚Æ•\Œ»‚³‚ê‚éD‚±‚±‚Å1B‚ÍB‚Ì’è‹`ŠÖ”‚Å‚ ‚éD‚±‚ê‚Í ƒÊ1, ƒÊ2 ‚ðW‡ŠÖ”‚Æ‚µ‚Ä‘¨‚¦‚Ĥ•Ï”•ÏŠ·‚·‚邱‚Æ‚Å‹‚Ü‚é¡‚±‚ê‚É‚æ‚褃Ê1, ƒÊ2 ‚𕪕z‚ÉŽ‚Šm—¦•Ï” X, Y ‚É‚¨‚¢‚Ä‚»‚̘a X + Y ‚Ì•ª•z‚ªô‚ݍž‚Ý‚É‚ ‚½‚邱‚Æ‚ª•ª‚©‚é¡

‘½€Ž®‚ÌŠ|‚¯ŽZ[•ÒW]


‘½€Ž®‚ÌŠ|‚¯ŽZ‚ÌŒ‹‰Ê‚ÌŒW”—ñ‚ͤŒ³‚Ì‘½€Ž®‚ÌŒW”—ñ‚̐üŒ`ô‚ݍž‚Ý‚É‚È‚é¡ŽÀÛ

( i = 0 m a i x i ) ( j = 0 l b j x j ) = k = 0 m + l ( i + j = k a i b j ) x k = k = 0 m + l ( i = 0 k a i b k i ) x k {\displaystyle \left(\sum _{i=0}^{m}a_{i}x^{i}\right)\left(\sum _{j=0}^{l}b_{j}x^{j}\right)=\sum _{k=0}^{m+l}\left(\sum _{i+j=k}a_{i}b_{j}\right)x^{k}=\sum _{k=0}^{m+l}\left(\sum _{i=0}^{k}a_{i}b_{k-i}\right)x^{k}} ‚Å‚ ‚褊|‚¯ŽZ‚ÌŒ‹‰Ê‚ÌŒW”‚ª a*b ‚Æ‚È‚é¡

üŒ`¼½ÃÑ[•ÒW]


“d‹C‰ñ˜H‚Æ‚¢‚Á‚½ŒÃ“T“I‚ÈŽž•s•Ïi¼Ìĕs•ÏjüŒ`¼½Ãтͤ”CˆÓ‚Ì“ü—Í x(t) ‚ɑ΂·‚éo—Í y(t) ‚ª x(t) ‚ƲÝÊßÙ½‰ž“š h(t) ‚̏ô‚ݍž‚Ý‚Å‹Lq‚Å‚«‚éF

y ( t ) = h ( t ) x ( t ) {\displaystyle y(t)=h(t)*x(t)} ‚±‚±‚Å“Á‚ɤ“ü—Í x(t) ‚ªÃÞÙÀŠÖ” δ(t) ‚Ì‚Æ‚«o—Í‚Í h(t) ‚»‚Ì‚à‚Ì‚É‚È‚é¡

‚±‚±‚ŏ㎮‚Ì—¼•Ó‚ðÌ°Ø´•ÏŠ·‚à‚µ‚­‚Í×Ìß×½•ÏŠ·i—£ŽUŒn‚̏ꍇ‚ÍZ•ÏŠ·j‚·‚é‚Ƥ#ô‚ݍž‚ݒ藝‚æ‚艺Ž®‚̂悤‚É‚È‚é¡

Y = H X {\displaystyle Y=HX} ‚±‚±‚Ť

H = Y X {\displaystyle H={\frac {Y}{X}}} ‚ð“`’BŠÖ”‚Æ‚¢‚¢¤‚±‚ÌŽ®‚͌ÓT§Œä˜_‚ÌŠî‘b‚Æ‚È‚Á‚Ä‚¢‚é¡

‰¹‹¿Šw[•ÒW]


´º°‚ÍŒ³‚̉¹”g‚Ƥ‰¹‚𔽎˂·‚邳‚Ü‚´‚Ü‚È•¨‘̂Ɉö‚é“Á«i²ÝÊßÙ½‰ž“šj‚Ƃ̏ô‚ݍž‚Ý‚Å‹Lq‚³‚ê‚顶׵¹‚â¼Ý¾»²»Þ°‚É“‹Ú‚³‚ê‚Ä‚¢‚é´º°‹@”\‚ͤ‚±‚̏ô‚ݍž‚Ý‚ÌŒø‰Ê‚ð“d‹C‰ñ˜H‚à‚µ‚­‚ͺÝËß­°À‚żЭڰĂ·‚邱‚Æ‚ÅŽÀŒ»‚µ‚Ä‚¢‚é¡

ŒõŠw‚¨‚æ‚щ摜ˆ—[•ÒW]


ŽB‘œŽž‚ÌÌÞڂȂǂ̑½‚­‚Ì‚Ô‚ê (blur) ‚͏ô‚ݍž‚Ý‚Å‹Lq‚Å‚«‚顗Ⴆ‚ΤËßÝĂª‚Ú‚¯‚½ŽÊ^‚ͤËßÝĂª‚ ‚Á‚½‰¼‘z“I‚ȉ摜‚Ƥi‚è‚Ì“Á«‚ðŽ¦‚·‰~‚Ƃ̏ô‚ݍž‚Ý‚Å‚ ‚é¡‚Ü‚½”íŽÊ‘Ì“™‚Ì“®‚«‚É‚æ‚éÌÞڂऐÎ~‚µ‚½‰¼‘z“I‚ȉ摜‚Æ“®‚«‚Ì“Á«‚Ƃ̏ô‚ݍž‚Ý‚Å‚ ‚褸Þ×̨¯¸¿Ìijª±‚ÌÓ°¼®ÝÌÞ×°‚Í‚±‚̏ô‚ݍž‚݉‰ŽZ‚ðŒvŽZ‚É‚æ‚è¼Ð­Ú°Ä‚·‚邱‚Æ‚ÅŽÀŒ»‚µ‚Ä‚¢‚é¡

‰æ‘œ”FŽ¯‚É‚¨‚¢‚ĂईقȂ齹°Ù‚̉摜‚ð”FŽ¯‚·‚é‚É‚ ‚½‚褏ô‚ݍž‚Ý‚Å‚Ô‚ê‚ð‚‚­‚Á‚Ä‚©‚礉摜ˆ—‚·‚邱‚Æ‚ª‚ ‚é¡

“ŒvŠw[•ÒW]


X, Y ‚ª‚»‚ꂼ‚ê“Æ—§‚ȘA‘±Œ^Šm—¦•Ï”‚Æ‚·‚é‚Ƥ˜a‚Ì ‚ÌŠm—¦–§“xŠÖ”‚͏ô‚ݍž‚Ý‚É‚æ‚Á‚Ä—^‚¦‚ç‚ê‚é¡X, Y ‚ÌŠm—¦–§“xŠÖ”‚ð‚»‚ê‚¼‚ê ‚Æ•\‹L‚·‚é‚ƤS ‚Ì–§“xŠÖ”‚͈ȉº‚ÌŽ®‚Å—^‚¦‚ç‚ê‚é.

‚ŽŸŒ³[•ÒW]

Rd ã‚Ì•¡‘f”’l”Ÿ” f, g ‚̏ô‚ݍž‚݂ͤ‚»‚ꎩg‚ª Rd ã‚Ì•¡‘f”’l”Ÿ”‚Æ‚µ‚Ä

( f g ) ( x ) = R d f ( y ) g ( x y ) d y = R d f ( x y ) g ( y ) d y {\displaystyle (f*g)(x)=\int _{\mathbf {R} ^{d}}f(y)g(x-y)\,dy=\int _{\mathbf {R} ^{d}}f(x-y)g(y)\,dy} ‚Å’è‹`‚³‚ê‚é‚à‚Ì‚Å‚ ‚邪¤‰E•Ó‚̐ϕª‚ª‘¶Ý‚µ‚Ä‚±‚ꂪ’è‹`‰Â”\‚Æ‚È‚é‚ɂͤf, g ‚ª–³ŒÀ‰“‚É‚¨‚¢‚ď\•ª‹}‘¬‚ÉŒ¸­‚·‚éi‰pŒê”Łj•K—v‚ª‚ ‚é¡‚Æ‚Í‚¢‚¦¤—Ⴆ‚Î g ‚ª–³ŒÀ‰“‚É‚¨‚¢‚Ä”š”­‚·‚é‚Æ‚µ‚Ăं»‚̉e‹¿‚Í f ‚ª\•ª‚É‹}Œ¸­‚È‚ç‚ΗeˆÕ‚É‘Å‚¿Á‚·‚±‚Æ‚ª‚Å‚«‚é‚©‚礂±‚̐ϕª‚Ì‘¶ÝðŒ‚͍ž‚Ý“ü‚Á‚½‚à‚Ì‚àl‚¦“¾‚é¡‚±‚Ì–â‘è‚ð¸Ø±‚·‚锟”‚ÌðŒ‚Æ‚µ‚Ä‚æ‚­—p‚¢‚ç‚ê‚éê‡‚ðˆÈ‰º‚É‹“‚°‚é¡

ºÝÊ߸đä•t‚«”Ÿ”[•ÒW]

f ‚Æ g ‚ª‚Æ‚à‚ɺÝÊ߸đä˜A‘±”Ÿ”‚È‚ç‚Î, ‚»‚ê‚ç‚̏ô‚ݍž‚Ý‚Í‘¶Ý‚µ‚Ä, ‚â‚Í‚èºÝÊ߸đä˜A‘±”Ÿ”‚Æ‚È‚é[7]. ‚æ‚èˆê”Ê‚É, ˆê•û‚ªºÝÊ߸đä, ‘¼•û‚ª‹ÇŠ‰ÂÏ•ª”Ÿ”‚È‚ç‚Î, ô‚ݍž‚Ý f ? g ‚ª’è‹`‚³‚ê‚ĘA‘±‚Å‚ ‚é.

R ã‚Å‚Í—¼ŽÒ‚ª‹ÇŠŽ©æ‰ÂÏ•ª‚̏ꍇ, ‚ ‚é‚¢‚Í—¼ŽÒ‚ª‚Æ‚à‚É”¼–³ŒÀ‹æŠÔ [a, +∞) (‚ ‚é‚¢‚Í‚Æ‚à‚É (-∞, a]) ‚É‘ä‚ðŽ‚Âê‡‚Å‚àô‚ݍž‚Ý‚ª’è‚Ü‚é.

‰ÂÏ•ª”Ÿ”[•ÒW]

f ‚Æ g ‚ª‚Æ‚à‚ÉL1(Rd)‚É‘®‚·‚éÙÍÞ°¸Þ‰ÂÏ•ª”Ÿ”‚È‚ç‚Î, ‚»‚ê‚ç‚̏ô‚ݍž‚Ý f ? g ‚ª‘¶Ý‚µ‚Ä‚â‚Í‚è‰ÂÏ•ª‚Å‚ ‚é[8]. ‚±‚ê‚ÍÄÈ؂̒藝‚Ì‹AŒ‹‚Å‚ ‚é. ‚±‚Ì‚±‚Æ‚Í ?1 ‚É‘®‚·‚鐔—ñ‚Ì—£ŽUô‚ݍž‚݂⤂æ‚èˆê”Ê‚ÌŒQã‚Ì L1 ‚̏ô‚ݍž‚Ý‚Å‚à¬—§‚·‚é.

“¯—l‚É‚µ‚Ä, f ¸ L1(Rd) ‚Æ g ¸ Lp(Rd) ‚ª 1 ? p ? ‡ ‚Ì‚Æ‚«, f ? g ¸ Lp(Rd) ‚©‚Â

f g p f 1 g p {\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}} ‚ð–ž‚½‚·. “Á‚É p = 1 ‚Ì‚Æ‚«, ‚±‚ê‚É‚æ‚è L1 ‚͏ô‚ݍž‚Ý‚ðÏ‚Æ‚µ‚ÄÊÞůʑ㐔‚𐬂·. (‚Ü‚½, “™†¬—§‚Í f ‚Æ g ‚ª‚Æ‚à‚É–w‚ÇŽŠ‚鏊”ñ•‰‚Ì‚Æ‚«‚Å‚ ‚é.)

‚æ‚èˆê”Ê‚É, ô‚ݍž‚݂ɑ΂·‚éÔݸނ̕s“™Ž®‚É‚æ‚è, ô‚ݍž‚ݐς͓K“–‚È Lp-‹óŠÔã‚̘A‘±‘oüŒ^‰‰ŽZ‚ƂȂ邱‚Æ‚ª]‚¤. ‹ï‘Ì“I‚ɏ‘‚¯‚Î, 1 ? p,q,r ? ‡ ‚ª

1 p + 1 q = 1 r + 1 {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1} ‚È‚éŠÖŒW‚ð–ž‘«‚·‚é‚Æ‚µ‚Ä,

f g r f p g q ( f L p ( R d ) , g L q ( R d ) ) {\displaystyle \lVert f*g\rVert _{r}\leq \lVert f\rVert _{p}\,\lVert g\rVert _{q}\quad (f\in L^{p}(\mathbb {R} ^{d}),\,g\in L^{q}(\mathbb {R} ^{d}))} ‚Æ‚È‚é‚©‚ç, ô‚ݍž‚ÝÏ‚Í Lp × Lq ¨ Lr ‚È‚é˜A‘±‘oüŒ^ŽÊ‘œ‚ð’è‚ß‚Ä‚¢‚é.

ô‚ݍž‚݂ɑ΂·‚éÔݸނ̕s“™Ž®‚Í, zŠÂô‚ݍž‚Ý‚â—£ŽUô‚ݍž‚Ý‚È‚Ç‚Ù‚©‚Ì•¶–¬‚Å‚à¬—§‚·‚é. ‚Ü‚½, R ã‚ł͐æ‚ÉŒf‚°‚½•s“™Ž®‚Í‚æ‚茵‚µ‚­•]‰¿‚Å‚«‚é: æ‚Æ“¯—l‚ÌŠÖŒW‚ðŽ‚Â 1 < p, q, r < ∞ ‚ɑ΂µ, ’萔 Bp,q < 1 ‚ª‘¶Ý‚µ‚Ä

f g r B p , q f p g q ( f L p ( R ) , g L q ( R ) ) . {\displaystyle \lVert f*g\rVert _{r}\leq B_{p,q}\lVert f\rVert _{p}\,\lVert g\rVert _{q}\quad (f\in L^{p}(\mathbb {R} ),\,g\in L^{q}(\mathbb {R} )).} Bp,q ‚̍œK’l‚Í Beckner (1975) ‚É‚ ‚é[9]. ‚æ‚è‹­‚¢•]‰¿‚Æ‚µ‚Ä 1 < p, q, r < ∞ ‚ɑ΂µ

f g r C p , q f p g q , w {\displaystyle \lVert f*g\rVert _{r}\leq C_{p,q}\lVert f\rVert _{p}\,\lVert g\rVert _{q,w}} ‚à“¾‚ç‚ê‚é. ‚½‚¾‚µ, ‖ g ‖q,w ‚ÍŽã Lp-ÉÙтł ‚é. 1 < p, q, r < ∞ ‚ɑ΂µŽã‚¢”Å‚ÌÔݸޕs“™Ž®

f g r , w C p , q f p , w g r , w {\displaystyle \|f*g\|_{r,w}\leq C_{p,q}\|f\|_{p,w}\|g\|_{r,w}} ‚ðl‚¦‚ê‚Î, ô‚ݍž‚݂͘A‘±‘oüŒ^ŽÊ‘œ ‚Æ‚àŒ©‚ç‚ê‚é[10].

‹}Œ¸­”Ÿ”[•ÒW]

ºÝÊ߸đä•t‚«‚â‰ÂÏ•ª‚È”Ÿ”‚Æ“¯—l‚ɤ”Ÿ”‚ª–³ŒÀ‰“‚ŏ\•ª‹}‘¬‚ÉŒ¸­i‰pŒê”Łj‚·‚ê‚Ώô‚ݍž‚Ý‚ª‚Å‚«‚Ä, ‚»‚ê‚ç‚̏ô‚ݍž‚Ý‚à‚Ü‚½‹}‘¬‚ÉŒ¸­‚·‚邱‚Ƃ͏d—v‚Ȑ«Ž¿‚Å‚ ‚é. ‚Æ‚­‚É f ‚Æ g ‚ª‹}Œ¸­”Ÿ”‚È‚ç‚Î, ‚»‚ê‚ç‚̏ô‚ݍž‚Ý f ? g ‚à‚Ü‚½‹}Œ¸­”Ÿ”‚Æ‚È‚é. ‚±‚Ì‚±‚Æ‚ð, ô‚ݍž‚Ý‚ª”÷•ª‚ƉŠ·‚Å‚ ‚é‚Æ‚¢‚¤Ž–ŽÀ‚Æ‘g‚ݍ‡‚킹‚ê‚Î, ¼­³Þ§Ù”Ÿ”‚̸׽‚ªô‚ݍž‚݂ŕ‚¶‚Ä‚¢‚邱‚Æ‚ª“±‚©‚ê‚é[11].

•ª•z[•ÒW]

“K“–‚ÈðŒ‚Ì‰º‚Å, ”Ÿ”‚Æ•ª•z‚ ‚é‚¢‚Í•ª•z“¯Žm‚̏ô‚ݍž‚Ý‚ª’è‹`‚Å‚«‚é. f ‚ªºÝÊ߸đä•t‚«”Ÿ”‚Å G ‚ª•ª•z‚È‚ç‚Î f ? G ‚Í, ”Ÿ”‚̏ô‚ݍž‚Ý‚ÌŽ®‚𕪕z”Å‚É‚µ‚½

R d f ( x y ) d G ( y ) {\displaystyle \int _{\mathbf {R} ^{d}}f(x-y)dG(y)} ‚Å’è‹`‚³‚ê‚銊‚ç‚©‚È”Ÿ”‚Å‚ ‚é (G ‚ª–§“x”Ÿ” g ‚ðŽ‚Ä‚Î’Êí‚Ì”Ÿ”‚̏ô‚ݍž‚݂ɏ‘‚«’¼‚¹‚é). ‚æ‚èˆê”Ê‚É, ŽŽŒ±”Ÿ” ƒÓ ‚ɑ΂µ‚ÄŒ‹‡—¥

f ( g φ ) = ( f g ) φ {\displaystyle f*(g*\varphi )=(f*g)*\varphi } ‚ª¬‚è—§‚‚悤‚ȈêˆÓ“I‚È•û–@‚ŏô‚ݍž‚Ý‚Ì’è‹`‚ðŠg’£‚·‚邱‚Æ‚ª‚Å‚«‚Ä, ‚»‚ê‚Í f ‚ª•ª•z, g ‚ªºÝÊ߸đä•t‚«•ª•z‚Ì‚Æ‚«‚É‚Í—LŒø‚Å‚ ‚é[12].

‘ª“x[•ÒW]

“ñ‚‚̗LŠE•Ï“®ÎÞÚّª“x ƒÊ ‚Æ ƒË ‚̏ô‚ݍž‚Ý‚Æ‚Í,

R d f ( x ) d λ ( x ) = R d R d f ( x + y ) d μ ( x ) d ν ( y ) {\displaystyle \int _{\mathbf {R} ^{d}}f(x)d\lambda (x)=\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x+y)\,d\mu (x)\,d\nu (y)} ‚Å’è‹`‚³‚ê‚鑪“x ƒÉ ‚ðŒ¾‚¤[13]. ‚±‚ê‚Í ƒÊ ‚Æ ƒË ‚𕪕z‚ÆŒ©‚é‚Æ‚«, ‘Oß‚É‚¢‚¤•ª•z‚̏ô‚ݍž‚݂Ɉê’v‚·‚é. ‚Ü‚½ ƒÊ ‚Æ ƒË ‚ªÙÍÞ°¸Þ‘ª“x‚ÉŠÖ‚µ‚Đâ‘ΘA‘±‚Å‚ ‚é‚Æ‚«, ‚»‚ê‚ç‚Ì–§“x”Ÿ”‚Ì L1-”Ÿ”‚Æ‚µ‚Ă̏ô‚ݍž‚Ý‚Æ‚àˆê’v‚·‚é.

‘ª“x‚̏ô‚ݍž‚Ý‚Í, ‘ª“x‚Ì‘S•Ï“®i‰pŒê”Łj‚ðÉÙтƂµ‚Ä

μ ν μ ν {\displaystyle \lVert \mu *\nu \rVert \leq \lVert \mu \rVert \,\lVert \nu \rVert } ‚ð–ž‚½‚·‚Æ‚¢‚¤ˆÓ–¡‚Å‚ÌÔݸނ̕s“™Ž®‚ª¬—§‚·‚é. —LŠE•Ï“®‘ª“x‚Ì‹óŠÔ‚ÍÊÞůʋóŠÔ‚Å‚ ‚é‚©‚ç, ‘ª“x‚̏ô‚ݍž‚Ý‚Í”Ÿ”‰ðÍŠw‚Ì•W€“I‚È (•ª•z‚̏ô‚ݍž‚݂ɑ΂µ‚Ä‚Í“K—p‚Å‚«‚È‚¢) •û–@‚ňµ‚¤‚±‚Æ‚ª‚Å‚«‚é.

ŒQã‚̏ô‚ݍž‚Ý[•ÒW]

“K“–‚È‘ª“x ƒÉ ‚ð”õ‚¦‚½ŒQ G ‚Æ‚»‚̏ã‚ÌŽÀ‚Ü‚½‚Í•¡‘f”’lÙÍÞ°¸Þ‰ÂÏ•ª”Ÿ” f ‚Æ g ‚ª—^‚¦‚ç‚ê‚ê‚Î, ‚»‚ê‚ç‚̏ô‚ݍž‚Ý‚ð

( f g ) ( x ) = G f ( y ) g ( y 1 x ) d λ ( y ) {\displaystyle (f*g)(x)=\int _{G}f(y)g(y^{-1}x)\,d\lambda (y)} ‚Å’è‹`‚·‚邱‚Æ‚ª‚Å‚«‚é. ‚µ‚©‚µˆê”ʂɂ͉Š·«‚ª¬‚è—§‚½‚È‚¢‚±‚Æ‚É’ˆÓ‚·‚ׂ«‚Å‚ ‚é.

‹ÇŠºÝÊ߸ČQã‚Ì•s•ÏÏ•ª‚̏ꍇ[•ÒW]

“TŒ^“I‚ȏꍇ‚Æ‚µ‚Ä, G ‚ª‹ÇŠºÝÊ߸Äʳ½ÄÞÙ̈ʑŠŒQ‚Å ƒÉ ‚ª¶Ê°Ù‘ª“xi¶•s•Ï‘ª“xj‚̏ꍇ‚Å‚ ‚é. ‰E•s•Ï‘ª“x ƒÏ ‚ɑ΂µ‚Ä‚à“¯—l‚̐ϕª

f ( x y 1 ) g ( y ) d ρ ( y ) {\displaystyle \int f(xy^{-1})g(y)\,d\rho (y)} ‚ðl‚¦‚邱‚Æ‚ª‚Å‚«‚邪, G ‚ª’P–Í‚Å‚È‚¢‚È‚ç‚Η¼ŽÒ‚͈ê’v‚µ‚È‚¢. ‘OŽÒ‚Ì’è‹`‚Å‚Í, ŒÅ’肵‚½”Ÿ” g ‚É‚æ‚éô‚ݍž‚Ý‚ªŒQ‚ւ̍¶ˆÚ“®ì—p‚ƉŠ·:

L h ( f g ) = ( L h f ) g {\displaystyle L_{h}(f*g)=(L_{h}f)*g} ‚ƂȂ邱‚Æ‚©‚ç‚æ‚­‘I‚΂ê‚é. ‚³‚ç‚É‚±‚Ì’è‹`‚Å‚ÍŒã‚ŏq‚ׂ鑪“x‚̏ô‚ݍž‚Ý‚Ì’è‹`‚Æ–µ‚‚µ‚È‚¢. ˆê•û, ¶•s•Ï‚Å‚Í‚È‚­‰E•s•Ï‘ª“x‚ðŽæ‚è, ŒãŽÒ‚Ì’è‹`‚ð—p‚¢‚ê‚ΉEˆÚ“®ì—p‚ƉŠ·‚É‚È‚é.

‚æ‚­’m‚ç‚ꂽ—á‚Ì“±o[•ÒW]

‹ÇŠºÝÊ߸ı°ÍÞٌQã‚Å, ‚ ‚éŽí‚̏ô‚ݍž‚ݒ藝i‰pŒê”Łj (ô‚ݍž‚Ý‚ÌÌ°Ø´•ÏŠ·‚ÍÌ°Ø´•ÏŠ·‚Ì“_‚²‚Ƃ̐ςɈê’v‚·‚é) ‚ª¬—§‚·‚é.

‰~ŽüŒQ T ‚ÉÙÍÞ°¸Þ‘ª“x‚ðl‚¦‚½‚à‚Ì‚Í‚æ‚­’m‚ç‚ꂽzŠÂô‚ݍž‚݂̏ꍇ‚Ì—á‚ð—^‚¦‚é: g ¸ L1(T) ‚ðŒÅ’肵‚Ä, ËÙÍÞÙċóŠÔ L2(T) ‚ɍì—p‚·‚é‚æ‚­’m‚ç‚ꂽì—p‘f:

T f ( x ) = 1 2 π T f ( y ) g ( x y ) d y {\displaystyle T{f}(x)={\frac {1}{2\pi }}\int _{\mathbf {T} }{f}(y)g(x-y)\,dy} ‚ª‚Æ‚ê‚é. ì—p‘f T ‚ͺÝÊ߸čì—p‘f‚Å‚ ‚é. ’¼ÚŒvŽZ‚É‚æ‚è, ‚»‚̐”ºì—p‘f T* ‚Í g(−y) ‚É‚æ‚éô‚ݍž‚Ý‚Å‚ ‚邱‚Æ‚ªŽ¦‚¹‚é. ã‚ÅŒf‚°‚½‰ÂŠ·«‚É‚æ‚è, T ‚͐³‹Kì—p‘f (T*T = TT* ‚Å‚ ‚é. ‚Ü‚½ T ‚Í•½sˆÚ“®ì—p‘f‚Æ‚à‰ÂŠ·‚Å‚ ‚é. ‚»‚̂悤‚ȏô‚ݍž‚ݍì—p‘f‚Æ•½sˆÚ“®ì—p‘f‘S‘̂̐¬‚·ì—p‘f‘°‚ð S ‚Æ‚·‚ê‚Î, S ‚͐³‹Kì—p‘f‚©‚ç‚È‚é‰ÂŠ·‘°‚Å‚ ‚é. ËÙÍÞÙċóŠÔã‚̽Í߸Ä٘_‚ɏ]‚¦‚Î, S ‚𓯎ž‘Ίp‰»‚·‚鐳‹K’¼ŒðŠî’ê {hk} ‚ª‘¶Ý‚µ‚Ä, ‚±‚ꂪ‰~Žüã‚̏ô‚ݍž‚Ý‚ð“Á’¥•t‚¯‚é. ‹ï‘Ì“I‚É‚Í

h k ( x ) = e i k x ( k Z ) {\displaystyle h_{k}(x)=e^{ikx}\quad (k\in \mathbb {Z} )} ‚ª‚¿‚傤‚Ç T ‚ÌŽw•W‚Ì‘S‘̂̐¬‚·W‡‚Ɉê’v‚·‚é. ‚±‚ÌŠî’ê‚É‘®‚·‚éŠeô‚ݍž‚ݍì—p‘f‚ªºÝÊ߸ďæŽZì—p‘f‚Å‚ ‚邱‚Æ‚ª, ã‚ŏq‚ׂ½zŠÂô‚ݍž‚݂ɑ΂·‚éô‚ݍž‚ݒ藝‚Æ‚µ‚Ă݂邱‚Æ‚ª‚Å‚«‚é.

—£ŽUô‚ݍž‚Ý‚Ì—á‚͈ʐ” n ‚Ì—LŒÀ„‰ñŒQ‚ð‚Æ‚é. ‚±‚̏ꍇ‚̏ô‚ݍž‚ݍì—p‘f‚͏„‰ñs—ñ‚É‚æ‚Á‚Ä•\Œ»‚³‚ê, —£ŽUÌ°Ø´•ÏŠ·‚É‚æ‚Á‚đΊp‰»‚·‚邱‚Æ‚ª‚Å‚«‚é.

“¯—l‚ÌŒ‹‰Ê‚ª (±°ÍÞقƂ͌À‚ç‚È‚¢) ºÝÊ߸ČQ G ‚ɑ΂µ‚Ä‚à’m‚ç‚ê‚Ä‚¢‚é: —LŒÀŽŸŒ³ÕÆÀؕ\Œ»‚̍s—ñ—v‘f‚Ì‘S‘Ì‚ª L2(G) ‚̐³‹K’¼ŒðŠî’ê‚𐬂µ (Ëß°À°–ܲق̒藝i‰pŒê”Łj), “K“–‚ȈӖ¡‚ł̏ô‚ݍž‚ݒ藝‚ª (Ì°Ø´•ÏŠ·‚ÉŠî‚­’²˜a‰ðÍ‚Ì‘¼‚Ì‘½‚­‚Ì‘¤–Ê‚Æ‚Æ‚à‚É) ˆø‚«‘±‚«–ž‘«‚³‚ê‚é.

ŒQã‚Ì‘ª“x‚̏ô‚ݍž‚Ý[•ÒW]

ˆÊ‘ŠŒQ G ã‚Ì—LŒÀÎÞÚّª“x ƒÊ ‚Æ ƒË ‚ɑ΂µ, ‚»‚ê‚ç‚̏ô‚ݍž‚Ý ƒÊ ? ƒË ‚Í G ‚ÌŠe‰Â‘ª•”•ªW‡ E ‚ɑ΂µ‚Ä

( μ ν ) ( E ) = 1 E ( x y ) d μ ( x ) d ν ( y ) {\displaystyle (\mu *\nu )(E)=\iint 1_{E}(xy)\,d\mu (x)\,d\nu (y)} ‚Å’è‹`‚³‚ê, ‚â‚Í‚è—LŒÀ‘ª“x‚Æ‚È‚é. ŽÀÛ, ‘S•Ï“®‚ÉŠÖ‚·‚éÔݸނ̕s“™Ž®

μ ν μ ν {\displaystyle \lVert \mu *\nu \rVert \leq \lVert \mu \rVert \,\lVert \nu \rVert } ‚ª–ž‘«‚³‚ê‚é. G ‚ª‹ÇŠºÝÊ߸ČQ‚ō¶Ê°Ù‘ª“x ƒÉ ‚ðŽ‚¿, ƒÊ ‚Æ ƒË ‚ª ƒÉ ‚ÉŠÖ‚µ‚Đâ‘ΘA‘±‚ÅŠeX–§“x”Ÿ”‚ðŽ‚Âê‡‚ɂͤô‚ݍž‚Ý ƒÊ ? ƒË ‚à‚Ü‚½â‘ΘA‘±‚Å, ‚»‚Ì–§“x”Ÿ”‚ÍŠe‘ª“x‚Ì–§“x”Ÿ”‚Ì (’ʏí‚Ì”Ÿ”‚Æ‚µ‚Ä‚Ì) ô‚ݍž‚݂Ɉê’v‚·‚é.

l‚¦‚éˆÊ‘ŠŒQ‚ªŽÀ”‚̉Á–@ŒQ (R, +) ‚Ì‚Æ‚«, ‚»‚̏ã‚ÌŠm—¦‘ª“x ƒÊ ‚Æ ƒË ‚ð‚Æ‚ê‚Î, ‘ª“x‚̏ô‚ݍž‚Ý ƒÊ ? ƒË ‚Í, •ª•z ƒÊ ‚¨‚æ‚Ñ ƒË ‚ɏ]‚¤“Æ—§Šm—¦•Ï” X ‚¨‚æ‚Ñ Y ‚̘a X + Y ‚ÌŠm—¦•ª•z‚ɑΉž‚·‚é.

’Žß[•ÒW]

o“T[•ÒW]

R. N. Bracewell (2005), gEarly work on imaging theory in radio astronomyh, in W. T. Sullivan, The Early Years of Radio Astronomy: Reflections Fifty Years After Jansky's Discovery, Cambridge University Press, p. 172, , https://books.google.com/books?id=v2SqL0zCrwcC&pg=PA172 John Hilton Grace and Alfred Young (1903), The algebra of invariants, Cambridge University Press, p. 40, https://books.google.com/books?id=NIe4AAAAIAAJ&pg=PA40 Leonard Eugene Dickson (1914), Algebraic invariants, J. Wiley, p. 85, https://books.google.com/books?id=LRGoAAAAIAAJ&pg=PA85 Sitzungsberichte der S?chsischen Akademie der Wissenschaften zu Leipzig,Mathematisch-naturwissenschaftliche Klasse, volume 128, number 2, 6?7Beckner, William (1975), "Inequalities in Fourier analysis", Ann. of Math. (2) 102: 159–182. Independently, Brascamp, Herm J. and Lieb, Elliott H. (1976), "Best constants in Young's inequality, its converse, and its generalization to more than three functions", Advances in Math. 20: 151–173. See Brascamp?Lieb inequality

ŽQl•¶Œ£[•ÒW]

Dominguez-Torres, Alejandro (Nov 2, 2010). "Origin and history of convolution". 41 pgs. http://www.slideshare.net/Alexdfar/origin-adn-history-of-convolution. Cranfield, Bedford MK43 OAL, UK. Retrieved Mar 13, 2013.
H?rmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, , MR0717035 
Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press,  
Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt Brace Jovanovich, Publishers, pp. xv+361, , MR0493420 
Rudin, Walter (1962), Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York?London, , MR0152834 .

ŠÖ˜A€–Ú[•ÒW]

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ÌÞײÝÄÞ¥ÃÞºÝÎÞØ­°¼®Ý
²ÝÊßÙ½‰ž“š - “`’BŠÖ”
Ì°Ø´•ÏŠ· - ×Ìß×½•ÏŠ·
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Šî–{‰ð
Ž©ŒÈ‰ñ‹AˆÚ“®•½‹ÏÓÃÞÙ
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±Å۸ސM†ˆ—
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Ï˜a‰‰ŽZ

ŠO•”Øݸ[•ÒW]

Weisstein, Eric W. "Convolution". MathWorldi‰pŒêj.
convolution - PlanetMath.i‰pŒêj
Hazewinkel, Michiel, ed. (2001), "Convolution of functions", Encyclopaedia of Mathematics, Springer, ¡
Hazewinkel, Michiel, ed. (2001), "Convolution transform", Encyclopaedia of Mathematics, Springer, ¡
The Joy of Convolution Java Applet ‚ðŽg‚Á‚½Ž‹Šo“I‚ȏô‚ݍž‚Ý‚Ìà–¾
Examples of sampled impulse responses to be used in convolution reverbs (Fokke Van Saane)
Examples of impulse responses synthesized from oscillator spectra, to be used in convolution reverbs (Emmanuel Deruty)
BruteFIR; A software for applying long FIR filters to multi-channel digital audio, either offline or in realtime.
Freeverb3 Reverb Impulse Response Processor: DSP library with convolution engines.
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