{"topic":{"id":6869,"title":"建一个理科主题水楼！","slug":"topic","userId":1707,"categoryId":4,"createdAt":"2023-10-29 06:37:10.594279","bumpedAt":"2026-06-05 11:40:40.472164","views":2248,"postsCount":40,"highestPostNumber":43,"likeCount":94,"closed":false,"pinned":false,"pinnedGlobally":false,"pinnedAt":null,"pinnedUntil":null,"isBanner":false,"excerpt":"各种理科主题的科普、讨论、闲聊、灌水、钓鱼、发疯等等都行！","deletedAt":null,"isAnonymous":false,"isPrivate":false,"isMembersOnly":false,"category":{"id":4,"name":"谈笑风生","slug":"general","color":"25AAE2"},"author":{"id":1707,"username":"wbzd","name":"我不知道","avatarUrl":"/uploads/image/29/1f/291f099d72b6e1ab6e873cd1b0fe4d2f133a9e713c066ea299ae3040b0e2ef27.avif"},"realAuthor":{"id":1707,"username":"wbzd","name":"我不知道","avatarUrl":"/uploads/image/29/1f/291f099d72b6e1ab6e873cd1b0fe4d2f133a9e713c066ea299ae3040b0e2ef27.avif"},"tags":[]},"posts":[{"id":15268,"postNumber":1,"raw":"各种理科主题的科普、讨论、闲聊、灌水、钓鱼、发疯等等都行！","cooked":"

各种理科主题的科普、讨论、闲聊、灌水、钓鱼、发疯等等都行！

\n","createdAt":"2023-10-29 06:37:10.680273","updatedAt":"2023-10-29 06:37:10.680273","replyToPostNumber":null,"replyCount":0,"likeCount":4,"wordCount":29,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":4,"users":[{"id":1798,"username":"sooooog","name":"爱你越久我越背痛"},{"id":1817,"username":"anon1678451","name":null},{"id":1863,"username":"交大椒猫","name":"兴庆宫第一皮革兔"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1707,"username":"wbzd","name":"我不知道","avatarUrl":"/uploads/image/29/1f/291f099d72b6e1ab6e873cd1b0fe4d2f133a9e713c066ea299ae3040b0e2ef27.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15269,"postNumber":2,"raw":"先来个益智小问题\n\n给定一个圆和一条直径，但不给圆心，能否仅用无刻度直尺作图找出其圆心？","cooked":"

先来个益智小问题

给定一个圆和一条直径，但不给圆心，能否仅用无刻度直尺作图找出其圆心？

\n","createdAt":"2023-10-29 06:37:52.646042","updatedAt":"2023-10-29 06:37:52.646042","replyToPostNumber":null,"replyCount":0,"likeCount":2,"wordCount":42,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":2,"users":[{"id":1817,"username":"anon1678451","name":null},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1707,"username":"wbzd","name":"我不知道","avatarUrl":"/uploads/image/29/1f/291f099d72b6e1ab6e873cd1b0fe4d2f133a9e713c066ea299ae3040b0e2ef27.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15270,"postNumber":3,"raw":"东边钓不到，来西边钓是吧","cooked":"

东边钓不到，来西边钓是吧

\n","createdAt":"2023-10-29 06:54:10.230435","updatedAt":"2023-10-29 06:54:10.230435","replyToPostNumber":null,"replyCount":1,"likeCount":8,"wordCount":12,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"🐈️","count":1,"users":[{"id":30,"username":"一只玉米人","name":"玉米🌽"}],"hasReacted":false},{"reaction":"🤣️","count":1,"users":[{"id":5905,"username":"PipaQinse233","name":"PipaQinse233"}],"hasReacted":false},{"reaction":"❤️","count":8,"users":[{"id":30,"username":"一只玉米人","name":"玉米🌽"},{"id":1707,"username":"wbzd","name":"我不知道"},{"id":1817,"username":"anon1678451","name":null},{"id":1835,"username":"Dusk","name":"夕大画家"},{"id":1850,"username":"Reina","name":"星野露比"},{"id":5083,"username":"Puinn","name":null},{"id":5905,"username":"PipaQinse233","name":"PipaQinse233"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1701,"username":"sincerely","name":"等风也等你","avatarUrl":"/uploads/image/42/f9/42f93baa6a22349c4a71d805c55c5ab69434d8790f4c2a29249c0a1143b952c2.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15271,"postNumber":4,"raw":"你说的对但是教我拓扑","cooked":"

你说的对但是教我拓扑

\n","createdAt":"2023-10-29 06:54:55.716735","updatedAt":"2023-10-29 06:54:55.716735","replyToPostNumber":3,"replyCount":1,"likeCount":1,"wordCount":10,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":1,"users":[{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1707,"username":"wbzd","name":"我不知道","avatarUrl":"/uploads/image/29/1f/291f099d72b6e1ab6e873cd1b0fe4d2f133a9e713c066ea299ae3040b0e2ef27.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15272,"postNumber":5,"raw":"拓扑学不了一点 :pouting_cat:","cooked":"

拓扑学不了一点 😾

\n","createdAt":"2023-10-29 06:55:57.358755","updatedAt":"2023-10-29 06:55:57.358755","replyToPostNumber":4,"replyCount":0,"likeCount":3,"wordCount":8,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":3,"users":[{"id":1707,"username":"wbzd","name":"我不知道"},{"id":1817,"username":"anon1678451","name":null},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1701,"username":"sincerely","name":"等风也等你","avatarUrl":"/uploads/image/42/f9/42f93baa6a22349c4a71d805c55c5ab69434d8790f4c2a29249c0a1143b952c2.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15273,"postNumber":6,"raw":"来个有点难度的，证明:\n$$\n\\int_0^{+\\infty}\\frac{\\mathrm dx}{\\sqrt{x^4+25x^2+160}}=\\int_0^{+\\infty}\\frac{\\mathrm dx}{\\sqrt{x^4-95x^2+2560}}\n$$","cooked":"

来个有点难度的，证明:
\n

\n\\int_0^{+\\infty}\\frac{\\mathrm dx}{\\sqrt{x^4+25x^2+160}}=\\int_0^{+\\infty}\\frac{\\mathrm dx}{\\sqrt{x^4-95x^2+2560}}\n

\n","createdAt":"2023-10-29 06:56:39.228862","updatedAt":"2023-10-29 06:56:39.228862","replyToPostNumber":null,"replyCount":2,"likeCount":5,"wordCount":16,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"😅️","count":1,"users":[{"id":1288,"username":"LittleWangInShanghai","name":"小王在上海"}],"hasReacted":false},{"reaction":"❤️","count":5,"users":[{"id":1716,"username":"Mk_XVII_CQC","name":"CQC"},{"id":1746,"username":"Vasavi_Shakti","name":"赞美太阳菜菜酱"},{"id":1817,"username":"anon1678451","name":null},{"id":5905,"username":"PipaQinse233","name":"PipaQinse233"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1707,"username":"wbzd","name":"我不知道","avatarUrl":"/uploads/image/29/1f/291f099d72b6e1ab6e873cd1b0fe4d2f133a9e713c066ea299ae3040b0e2ef27.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15280,"postNumber":8,"raw":"I is fish","cooked":"

I is fish

\n","createdAt":"2023-10-29 11:25:53.014339","updatedAt":"2023-10-29 11:25:53.014339","replyToPostNumber":6,"replyCount":0,"likeCount":3,"wordCount":3,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":3,"users":[{"id":1707,"username":"wbzd","name":"我不知道"},{"id":1817,"username":"anon1678451","name":null},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1724,"username":"Philia","name":"耳玖","avatarUrl":"/uploads/image/c5/ed/c5edfff4ff47377a09cf6d694ecda658436fce7b6d02c364489ecac6551955e1.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15283,"postNumber":9,"raw":":whale:！\n好久不见","cooked":"

🐋！
\n好久不见

\n","createdAt":"2023-10-29 11:29:51.315277","updatedAt":"2023-10-29 11:29:51.315277","replyToPostNumber":7,"replyCount":0,"likeCount":4,"wordCount":5,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":4,"users":[{"id":1707,"username":"wbzd","name":"我不知道"},{"id":1746,"username":"Vasavi_Shakti","name":"赞美太阳菜菜酱"},{"id":1817,"username":"anon1678451","name":null},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1744,"username":"qianbaidu","name":"千百度","avatarUrl":"/uploads/image/1d/c8/1dc82b9526a1201b0554ef1fc6e81a507b2982d96c8741683f59d1c76d940b57.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15284,"postNumber":10,"raw":"前排顶帖，最优化学不了一点😾","cooked":"

前排顶帖，最优化学不了一点😾

\n","createdAt":"2023-10-29 11:30:44.500571","updatedAt":"2023-10-29 11:30:44.500571","replyToPostNumber":null,"replyCount":0,"likeCount":2,"wordCount":14,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"😅️","count":1,"users":[{"id":1288,"username":"LittleWangInShanghai","name":"小王在上海"}],"hasReacted":false},{"reaction":"❤️","count":2,"users":[{"id":1707,"username":"wbzd","name":"我不知道"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1709,"username":"One_Yuan","name":"一块钱","avatarUrl":"/uploads/image/48/38/483867d97065f3344ba0f2600c3c35f7f5933cb85fbabccc13640bb78763f32a.avif","admin":false,"moderator":false,"signatureRaw":"SJTUer","signatureCooked":"

SJTUer

\n"}},{"id":15350,"postNumber":12,"raw":"解答：\n\n令\n$$\nz=\\frac{x^5+40x^3+320x}{x^4+20x^2+80}\n$$\n则有\n$$\n\\frac{\\mathrm dz}{\\sqrt{z^4-95z^2+2560}}=\\frac{\\mathrm dx}{\\sqrt{x^4+25x^2+160}}\n$$\n然后积一下就好了\n\n:nose_steam:","cooked":"

解答：

令
\n

\nz=\\frac{x^5+40x^3+320x}{x^4+20x^2+80}\n

\n则有
\n

\n\\frac{\\mathrm dz}{\\sqrt{z^4-95z^2+2560}}=\\frac{\\mathrm dx}{\\sqrt{x^4+25x^2+160}}\n

\n然后积一下就好了

😤

\n","createdAt":"2023-10-30 02:32:25.314409","updatedAt":"2023-10-30 02:32:25.314409","replyToPostNumber":6,"replyCount":0,"likeCount":3,"wordCount":23,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":3,"users":[{"id":1047,"username":"anonymous_coward_old","name":"Anonymous Coward Old"},{"id":1817,"username":"anon1678451","name":null},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1707,"username":"wbzd","name":"我不知道","avatarUrl":"/uploads/image/29/1f/291f099d72b6e1ab6e873cd1b0fe4d2f133a9e713c066ea299ae3040b0e2ef27.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":15373,"postNumber":13,"raw":"这下出口转内销了","cooked":"

这下出口转内销了

\n","createdAt":"2023-10-30 04:07:28.288592","updatedAt":"2023-10-30 04:07:28.288592","replyToPostNumber":null,"replyCount":0,"likeCount":4,"wordCount":8,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":4,"users":[{"id":1411,"username":"lwe","name":null},{"id":1707,"username":"wbzd","name":"我不知道"},{"id":1819,"username":"Thevenin","name":"交大金猫"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1817,"username":"anon1678451","name":null,"avatarUrl":"/uploads/image/c6/9a/c69a738e32dc09506ebfea8035a962d53e98afc1828783d42c3dac2eaa627df5.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":20804,"postNumber":14,"raw":"非常好理科水楼","cooked":"

非常好理科水楼

\n","createdAt":"2023-12-30 06:39:35.936793","updatedAt":"2023-12-30 06:39:35.936793","replyToPostNumber":null,"replyCount":0,"likeCount":2,"wordCount":7,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":2,"users":[{"id":1817,"username":"anon1678451","name":null},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1811,"username":"Time_Limit_Exceed","name":null,"avatarUrl":"/uploads/image/d8/9f/d89f1d48f18bbbe6e6743852b213ef5226500b8782ee99f80b02495c9fdd631b.avif","admin":false,"moderator":false,"signatureRaw":"再挑灯火读文章 :bulb: :book: :face_with_monocle:","signatureCooked":"

再挑灯火读文章 💡 📖 🧐

\n"}},{"id":20805,"postNumber":15,"raw":"我不知道！！！","cooked":"

我不知道！！！

\n","createdAt":"2023-12-30 06:44:51.345536","updatedAt":"2023-12-30 06:44:51.345536","replyToPostNumber":null,"replyCount":0,"likeCount":2,"wordCount":5,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":2,"users":[{"id":1817,"username":"anon1678451","name":null},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1811,"username":"Time_Limit_Exceed","name":null,"avatarUrl":"/uploads/image/d8/9f/d89f1d48f18bbbe6e6743852b213ef5226500b8782ee99f80b02495c9fdd631b.avif","admin":false,"moderator":false,"signatureRaw":"再挑灯火读文章 :bulb: :book: :face_with_monocle:","signatureCooked":"

再挑灯火读文章 💡 📖 🧐

\n"}},{"id":22825,"postNumber":16,"raw":"存在不能被任何图灵机 (Turing machine) 计算的函数 $ UC:\\{0,1\\}^*\\to\\{0,1\\} $.\n\n函数 $ UC $ 如下定义：\n $$\n \\forall \\alpha\\in\\{0,1\\}^*, \nUC(\\alpha)=\n\\begin{cases}\n0,M_\\alpha(\\alpha)=1;\\\\\n1,\\text{otherwise}.\n\\end{cases}\n$$\n证明：假设函数 $ UC $ 可被图灵机计算，则存在图灵机 $ M $ 使得 $ \nM(\\alpha)\\equiv UC(\\alpha), \\forall \\alpha\\in\\{0,1\\}^* $\n所以有 $ M(\\lfloor M\\rfloor)=UC(\\lfloor M\\rfloor) $.这与定义矛盾。[]\n注： $ \\{0,1\\}^* $ 为 0 和 1 组成的字符串， $ M_\\alpha $ 为以二进制字符串 $ \\alpha $ 表示的图灵机（数据和指令等价）， $ \\lfloor M\\rfloor $ 为图灵机 $ M $ 的二进制字符串表示\n这个函数的构造方式有点像理发师悖论\n来源：COMPUTATIONAL COMPLEXITY A Modern Approach","cooked":"

存在不能被任何图灵机 (Turing machine) 计算的函数 $ UC:{0,1}^*\\to{0,1} $.

函数 $ UC $ 如下定义：
\n

\n\\forall \\alpha\\in\\{0,1\\}^*, \nUC(\\alpha)=\n\\begin{cases}\n0,M_\\alpha(\\alpha)=1;\\\\\n1,\\text{otherwise}.\n\\end{cases}\n

\n证明：假设函数 $ UC $ 可被图灵机计算，则存在图灵机 $ M $ 使得 $
\nM(\\alpha)\\equiv UC(\\alpha), \\forall \\alpha\\in{0,1}^* $
\n所以有 $ M(\\lfloor M\\rfloor)=UC(\\lfloor M\\rfloor) $.这与定义矛盾。
\n注： $ {0,1}^* $ 为 0 和 1 组成的字符串， $ M_\\alpha $ 为以二进制字符串 $ \\alpha $ 表示的图灵机（数据和指令等价）， $ \\lfloor M\\rfloor $ 为图灵机 $ M $ 的二进制字符串表示
\n这个函数的构造方式有点像理发师悖论
\n来源：COMPUTATIONAL COMPLEXITY A Modern Approach

\n","createdAt":"2024-01-29 11:36:14.539367","updatedAt":"2024-01-29 11:37:51.782405","replyToPostNumber":null,"replyCount":1,"likeCount":5,"wordCount":178,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":5,"users":[{"id":172,"username":"HCCH","name":"HCCH"},{"id":1411,"username":"lwe","name":null},{"id":1811,"username":"Time_Limit_Exceed","name":null},{"id":1819,"username":"Thevenin","name":"交大金猫"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1817,"username":"anon1678451","name":null,"avatarUrl":"/uploads/image/c6/9a/c69a738e32dc09506ebfea8035a962d53e98afc1828783d42c3dac2eaa627df5.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":22829,"postNumber":17,"raw":"怎么没人水，那我多水几个","cooked":"

怎么没人水，那我多水几个

\n","createdAt":"2024-01-29 11:42:55.339068","updatedAt":"2024-01-29 11:42:55.339068","replyToPostNumber":null,"replyCount":0,"likeCount":2,"wordCount":12,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":2,"users":[{"id":1819,"username":"Thevenin","name":"交大金猫"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1817,"username":"anon1678451","name":null,"avatarUrl":"/uploads/image/c6/9a/c69a738e32dc09506ebfea8035a962d53e98afc1828783d42c3dac2eaa627df5.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":22846,"postNumber":18,"raw":"相信大家都在信号与系统中学过 Parseval 方程：\n$$\n(f,f)=\\sum_{i=1}^\\infty{\\left|c_i\\right|^2},\\forall f\\in\\mathscr{H}.\n$$\n其中 $ \\mathscr{H} $ 为全体定义在 $ [a,b] $ 上复值平方可积的函数构成的线性空间。\n课本的证明是最容易想到、最直观的证明：\n$$\n(f,f)=\\Bigl(\\sum_{i=1}^\\infty{c_if_i},\\sum_{i=1}^\\infty{c_if_i}\\Bigr)=\\sum_{i=1}^\\infty{\\left|c_i\\right|^2}.\n$$\n该证明看似简洁，其实并不正确。正交归一函数集 $ \\{f_i\\} $ 的完备性不能保证逐点收敛，所得函数和原函数可能相差一个广义零函数，因此第一个等号不成立。从完备性直接得到的是由积分定义的平均收敛。正确的证明如下：\n$$\n\\begin{align*}\n&\\int_a^b{\\Bigl|f(x)-\\sum_{i=1}^{n}{c_if_i(x)}\\Bigr|^2\\rho(x)\\mathrm{d}x}\\\\\n=&(f,f)-\\sum_{i=1}^{n}{c_i^*(f_i,f)}-\\sum_{i=1}^{n}{c_i(f,f_i)}+\\sum_{i=1}^{n}{|c_i|^2}\\\\\n=&(f,f)-\\sum_{i=1}^{n}{|c_i|^2}.\n\\end{align*}\n$$\n由基函数 $ \\{f_i\\} $ 的完备性得，\n$$\n\\lim_{n\\to\\infty}\\int_a^b{\\Bigl|f(x)-\\sum_{i=1}^{n}{c_if_i(x)}\\Bigr|^2\\rho(x)\\mathrm{d}x}=0,\\\\\ni.e.(f,f)=\\sum_{i=1}^\\infty{\\left|c_i\\right|^2}.\n$$\n\n该方程还有更一般的形式：\n$$\n(f,g)=\\sum_{i=1}^\\infty{(f,f_i)(f_i,g)},\\forall f,g\\in\\mathscr{H}.\n$$\n证明如下：\n$$\n\\begin{align*}\nf(x)&=\\sum_{i=1}^\\infty{c_if_i(x)}=\\sum_{i=1}^\\infty{\\Bigl[\\int_a^b{f_i(x)f_i^*(x')\\rho(x')\\mathrm{d}x'}\\Bigr]f_i(x)}\n\\\\&=\\int_a^b{\\Bigl[\\sum_{i=1}^\\infty{f_i(x)f_i^*(x')\\rho(x')}\\Bigr]f(x')\\mathrm{d}x'}\n\\\\&=\\int_a^b{\\delta(x'-x)\\mathrm{d}x'}\\\\\n&\\Rightarrow\\sum_{i=1}^\\infty{f_i(x)f_i^*(x')\\rho(x')}=\\delta(x-x'),a\\leqslant x,x'\\leqslant b.\\\\\n(f,g)&=\\int_a^b{f^*(x)g(x)\\rho(x)\\mathrm{d}x}\\\\\n&=\\int_a^b{\\Bigl[\\int_a^b{f^*(x)g(x')\\rho(x)\\delta(x-x')\\mathrm{d}x'}\\Bigr]\\mathrm{d}x}\\\\\n&=\\sum_{i=1}^\\infty{\\int_a^b{\\Bigl[\\int_a^b{f^*(x)f_i(x)f_i^*(x')\\rho(x')g(x')\\rho(x)\\mathrm{d}x'}\\Bigr]\\mathrm{d}x}}\\\\\n&=\\sum_{i=1}^\\infty{\\int_a^b{f^*(x)f_i(x)\\rho(x)\\mathrm{d}x}\\int_a^b{f_i^*(x')g(x')\\rho(x')\\mathrm{d}x'}}\\\\\n&=\\sum_{i=1}^\\infty{(f,f_i)(f_i,g)}.\n\\end{align*}\n$$","cooked":"

相信大家都在信号与系统中学过 Parseval 方程：
\n

\n(f,f)=\\sum_{i=1}^\\infty{\\left|c_i\\right|^2},\\forall f\\in\\mathscr{H}.\n

\n其中 $ \\mathscr{H} $ 为全体定义在 $ [a,b] $ 上复值平方可积的函数构成的线性空间。
\n课本的证明是最容易想到、最直观的证明：
\n

\n(f,f)=\\Bigl(\\sum_{i=1}^\\infty{c_if_i},\\sum_{i=1}^\\infty{c_if_i}\\Bigr)=\\sum_{i=1}^\\infty{\\left|c_i\\right|^2}.\n

\n该证明看似简洁，其实并不正确。正交归一函数集 $ {f_i} $ 的完备性不能保证逐点收敛，所得函数和原函数可能相差一个广义零函数，因此第一个等号不成立。从完备性直接得到的是由积分定义的平均收敛。正确的证明如下：
\n

\n\\begin{align*}\n&\\int_a^b{\\Bigl|f(x)-\\sum_{i=1}^{n}{c_if_i(x)}\\Bigr|^2\\rho(x)\\mathrm{d}x}\\\\\n=&(f,f)-\\sum_{i=1}^{n}{c_i^*(f_i,f)}-\\sum_{i=1}^{n}{c_i(f,f_i)}+\\sum_{i=1}^{n}{|c_i|^2}\\\\\n=&(f,f)-\\sum_{i=1}^{n}{|c_i|^2}.\n\\end{align*}\n

\n由基函数 $ {f_i} $ 的完备性得，
\n

\n\\lim_{n\\to\\infty}\\int_a^b{\\Bigl|f(x)-\\sum_{i=1}^{n}{c_if_i(x)}\\Bigr|^2\\rho(x)\\mathrm{d}x}=0,\\\\\ni.e.(f,f)=\\sum_{i=1}^\\infty{\\left|c_i\\right|^2}.\n

该方程还有更一般的形式：
\n

\n(f,g)=\\sum_{i=1}^\\infty{(f,f_i)(f_i,g)},\\forall f,g\\in\\mathscr{H}.\n

\n证明如下：
\n

\n\\begin{align*}\nf(x)&=\\sum_{i=1}^\\infty{c_if_i(x)}=\\sum_{i=1}^\\infty{\\Bigl[\\int_a^b{f_i(x)f_i^*(x')\\rho(x')\\mathrm{d}x'}\\Bigr]f_i(x)}\n\\\\&=\\int_a^b{\\Bigl[\\sum_{i=1}^\\infty{f_i(x)f_i^*(x')\\rho(x')}\\Bigr]f(x')\\mathrm{d}x'}\n\\\\&=\\int_a^b{\\delta(x'-x)\\mathrm{d}x'}\\\\\n&\\Rightarrow\\sum_{i=1}^\\infty{f_i(x)f_i^*(x')\\rho(x')}=\\delta(x-x'),a\\leqslant x,x'\\leqslant b.\\\\\n(f,g)&=\\int_a^b{f^*(x)g(x)\\rho(x)\\mathrm{d}x}\\\\\n&=\\int_a^b{\\Bigl[\\int_a^b{f^*(x)g(x')\\rho(x)\\delta(x-x')\\mathrm{d}x'}\\Bigr]\\mathrm{d}x}\\\\\n&=\\sum_{i=1}^\\infty{\\int_a^b{\\Bigl[\\int_a^b{f^*(x)f_i(x)f_i^*(x')\\rho(x')g(x')\\rho(x)\\mathrm{d}x'}\\Bigr]\\mathrm{d}x}}\\\\\n&=\\sum_{i=1}^\\infty{\\int_a^b{f^*(x)f_i(x)\\rho(x)\\mathrm{d}x}\\int_a^b{f_i^*(x')g(x')\\rho(x')\\mathrm{d}x'}}\\\\\n&=\\sum_{i=1}^\\infty{(f,f_i)(f_i,g)}.\n\\end{align*}\n

\n","createdAt":"2024-01-29 14:54:32.024756","updatedAt":"2024-01-29 14:56:31.644771","replyToPostNumber":null,"replyCount":1,"likeCount":3,"wordCount":234,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"😍️","count":1,"users":[{"id":5905,"username":"PipaQinse233","name":"PipaQinse233"}],"hasReacted":false},{"reaction":"❤️","count":3,"users":[{"id":1819,"username":"Thevenin","name":"交大金猫"},{"id":5905,"username":"PipaQinse233","name":"PipaQinse233"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1817,"username":"anon1678451","name":null,"avatarUrl":"/uploads/image/c6/9a/c69a738e32dc09506ebfea8035a962d53e98afc1828783d42c3dac2eaa627df5.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":22852,"postNumber":19,"raw":"超，你什么时候改行搞数学了\n这里的 $\\rho$ 是什么","cooked":"

超，你什么时候改行搞数学了
\n这里的 $\\rho$ 是什么

\n","createdAt":"2024-01-29 15:40:02.829573","updatedAt":"2024-01-29 15:45:07.462282","replyToPostNumber":18,"replyCount":1,"likeCount":1,"wordCount":20,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":1,"users":[{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":2,"publicVersion":2,"deletedAt":null,"isAnonymous":false,"user":{"id":1411,"username":"lwe","name":null,"avatarUrl":null,"admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":22853,"postNumber":20,"raw":"我也来写点","cooked":"

我也来写点

\n","createdAt":"2024-01-29 15:46:44.651935","updatedAt":"2024-01-29 15:46:44.651935","replyToPostNumber":null,"replyCount":0,"likeCount":1,"wordCount":5,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":1,"users":[{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1411,"username":"lwe","name":null,"avatarUrl":null,"admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":22854,"postNumber":21,"raw":" $ \\rho(x) $ 是定义在 $ [a,b] $ 上不恒为零的非负实函数，在线性空间 $ \\mathscr{H} $ 的内积定义中用作权函数。 $ \\rho(x)\\equiv 1 $ 是一种常见的特殊情况。","cooked":"

$ \\rho(x) $ 是定义在 $ [a,b] $ 上不恒为零的非负实函数，在线性空间 $ \\mathscr{H} $ 的内积定义中用作权函数。 $ \\rho(x)\\equiv 1 $ 是一种常见的特殊情况。

\n","createdAt":"2024-01-29 15:49:19.769417","updatedAt":"2024-01-29 15:50:26.597946","replyToPostNumber":19,"replyCount":0,"likeCount":2,"wordCount":57,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":2,"users":[{"id":1819,"username":"Thevenin","name":"交大金猫"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":1,"publicVersion":1,"deletedAt":null,"isAnonymous":false,"user":{"id":1817,"username":"anon1678451","name":null,"avatarUrl":"/uploads/image/c6/9a/c69a738e32dc09506ebfea8035a962d53e98afc1828783d42c3dac2eaa627df5.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}},{"id":23054,"postNumber":22,"raw":"Fourier 变换前后形式不变的信号——Gauss 型函数\n$$\ng_{\\alpha}\\left( t \\right) =\\frac{1}{2\\sqrt{\\pi \\alpha}}\\exp \\left\\{ -\\frac{t^2}{4\\alpha} \\right\\} \n$$\n不难发现\n$$\n\\begin{align}\n\\mathscr{F} \\left\\{ g_{\\alpha}\\left( t \\right) \\right\\} &=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{+\\infty}{\\frac{1}{2\\sqrt{\\pi \\alpha}}\\exp \\left\\{ -\\frac{t^2}{4\\alpha} \\right\\} \\exp \\left\\{ -\\mathrm{i}\\omega t \\right\\} \\mathrm{d}t}\n\\\\\n&=\\frac{1}{\\sqrt{2}\\pi}\\int_{-\\infty}^{+\\infty}{\\exp \\left\\{ -\\left( \\frac{t}{2\\sqrt{\\alpha}}+\\sqrt{\\alpha}\\mathrm{i}\\omega \\right) ^2-\\alpha \\omega ^2 \\right\\} \\mathrm{d}\\frac{t}{2\\sqrt{\\alpha}}}\n\\\\\n&=\\frac{1}{\\sqrt{2\\pi}}\\exp \\left\\{ -\\alpha \\omega ^2 \\right\\} \n\\end{align}\n$$\n以此为窗进行的短时 Fourier 变换称为 Gabor 变换：\n$$\n\\begin{align}\n\\mathscr{F} \\left\\{ f\\left( t \\right) g_{\\alpha}\\left( t-b \\right) \\right\\} &=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{+\\infty}{f\\left( t \\right) g_{\\alpha}\\left( t-b \\right) \\mathrm{e}^{-\\mathrm{i}\\omega t}\\mathrm{d}t}\n\\\\\n&=\\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty}{F\\left( \\xi \\right) \\mathrm{e}^{-\\mathrm{i}\\left( \\omega -\\xi \\right) b}\\mathrm{e}^{-\\alpha \\left( \\omega -\\xi \\right) ^2}\\mathrm{d}\\xi}\n\\\\\n&=\\frac{1}{2\\sqrt{\\pi \\alpha}}\\mathrm{e}^{-\\mathrm{i}\\omega b}\\int_{-\\infty}^{+\\infty}{F\\left( \\xi \\right) g_{\\frac{1}{4\\alpha}}\\left( \\xi -\\omega \\right) \\mathrm{e}^{\\mathrm{i}\\xi b}\\mathrm{d}\\xi}\n\\end{align}\n$$\n时域乘积在频域表现为卷积。 $ g_{\\alpha}\\left( t \\right) $ 是时间窗函数， $ g_{\\frac{1}{4\\alpha}}(\\xi) $ 则是相应的频率窗函数。时间窗宽度和频率窗宽度成反比。","cooked":"

Fourier 变换前后形式不变的信号——Gauss 型函数
\n

\ng_{\\alpha}\\left( t \\right) =\\frac{1}{2\\sqrt{\\pi \\alpha}}\\exp \\left\\{ -\\frac{t^2}{4\\alpha} \\right\\}\n

\n不难发现
\n

\n\\begin{align}\n\\mathscr{F} \\left\\{ g_{\\alpha}\\left( t \\right) \\right\\} &=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{+\\infty}{\\frac{1}{2\\sqrt{\\pi \\alpha}}\\exp \\left\\{ -\\frac{t^2}{4\\alpha} \\right\\} \\exp \\left\\{ -\\mathrm{i}\\omega t \\right\\} \\mathrm{d}t}\n\\\\\n&=\\frac{1}{\\sqrt{2}\\pi}\\int_{-\\infty}^{+\\infty}{\\exp \\left\\{ -\\left( \\frac{t}{2\\sqrt{\\alpha}}+\\sqrt{\\alpha}\\mathrm{i}\\omega \\right) ^2-\\alpha \\omega ^2 \\right\\} \\mathrm{d}\\frac{t}{2\\sqrt{\\alpha}}}\n\\\\\n&=\\frac{1}{\\sqrt{2\\pi}}\\exp \\left\\{ -\\alpha \\omega ^2 \\right\\} \n\\end{align}\n

\n以此为窗进行的短时 Fourier 变换称为 Gabor 变换：
\n

\n\\begin{align}\n\\mathscr{F} \\left\\{ f\\left( t \\right) g_{\\alpha}\\left( t-b \\right) \\right\\} &=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{+\\infty}{f\\left( t \\right) g_{\\alpha}\\left( t-b \\right) \\mathrm{e}^{-\\mathrm{i}\\omega t}\\mathrm{d}t}\n\\\\\n&=\\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty}{F\\left( \\xi \\right) \\mathrm{e}^{-\\mathrm{i}\\left( \\omega -\\xi \\right) b}\\mathrm{e}^{-\\alpha \\left( \\omega -\\xi \\right) ^2}\\mathrm{d}\\xi}\n\\\\\n&=\\frac{1}{2\\sqrt{\\pi \\alpha}}\\mathrm{e}^{-\\mathrm{i}\\omega b}\\int_{-\\infty}^{+\\infty}{F\\left( \\xi \\right) g_{\\frac{1}{4\\alpha}}\\left( \\xi -\\omega \\right) \\mathrm{e}^{\\mathrm{i}\\xi b}\\mathrm{d}\\xi}\n\\end{align}\n

\n时域乘积在频域表现为卷积。 $ g_{\\alpha}\\left( t \\right) $ 是时间窗函数， $ g_{\\frac{1}{4\\alpha}}(\\xi) $ 则是相应的频率窗函数。时间窗宽度和频率窗宽度成反比。

\n","createdAt":"2024-01-31 14:34:38.987217","updatedAt":"2024-01-31 15:18:29.959781","replyToPostNumber":null,"replyCount":0,"likeCount":2,"wordCount":188,"liked":false,"bookmarked":false,"polls":[],"reactions":[{"reaction":"❤️","count":2,"users":[{"id":1819,"username":"Thevenin","name":"交大金猫"},{"id":7352,"username":"Love_FDU","name":"🌲➕🖐️"}],"hasReacted":false}],"wiki":false,"version":2,"publicVersion":2,"deletedAt":null,"isAnonymous":false,"user":{"id":1817,"username":"anon1678451","name":null,"avatarUrl":"/uploads/image/c6/9a/c69a738e32dc09506ebfea8035a962d53e98afc1828783d42c3dac2eaa627df5.avif","admin":false,"moderator":false,"signatureRaw":null,"signatureCooked":null}}],"hasMore":true,"lastReadPostNumber":8,"hasUrlPostNumber":true}