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<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/626455c64f8811599e455ce2/bg1.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls0 ws0">图像变换</div><div class="t m0 x2 h3 y3 ff1 fs0 fc1 sc1 ls0 ws0">图像变换</div><div class="t m0 x3 h4 y4 ff2 fs0 fc0 sc2 ls0 ws0">(</div><div class="t m0 x4 h4 y3 ff2 fs0 fc1 sc2 ls0 ws0">(</div><div class="t m0 x5 h3 y2 ff1 fs0 fc0 sc0 ls0 ws0">二</div><div class="t m0 x6 h3 y3 ff1 fs0 fc1 sc1 ls0 ws0">二</div><div class="t m0 x7 h4 y4 ff2 fs0 fc0 sc2 ls0 ws0">)</div><div class="t m0 x8 h4 y3 ff2 fs0 fc1 sc2 ls0 ws0">)</div><div class="t m0 x9 h5 y5 ff1 fs1 fc1 sc1 ls0 ws0">【目录】</div><div class="t m0 xa h5 y6 ff1 fs1 fc2 sc3 ls0 ws0">【目录】</div><div class="t m0 xa h6 y7 ff1 fs2 fc1 sc2 ls0 ws0">图像变换<span class="ff3 fs3">(</span>二<span class="ff3 fs3">).................................................................................................................................1</span></div><div class="t m0 xb h7 y8 ff1 fs3 fc1 sc2 ls0 ws0">一、正交变换<span class="ff3">...........................................................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>1</span></div><div class="t m0 xc h7 y9 ff3 fs3 fc1 sc2 ls0 ws0">1<span class="ff1">、一维函数的正交性</span>.....................................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>1</div><div class="t m0 xc h7 ya ff3 fs3 fc1 sc2 ls0 ws0">2<span class="ff1">、二维函数的正交性</span>.....................................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>3</div><div class="t m0 xb h7 yb ff1 fs3 fc1 sc2 ls0 ws0">二、空间周期<span class="ff3">...........................................................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>5</span></div><div class="t m0 xb h7 yc ff1 fs3 fc1 sc2 ls0 ws0">三、傅立叶变换应用<span class="ff3">...............................................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>6</span></div><div class="t m0 xc h7 yd ff3 fs3 fc1 sc2 ls0 ws0">1<span class="ff1">、图像频谱显示</span>.............................................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>6</div><div class="t m0 xc h7 ye ff3 fs3 fc1 sc2 ls0 ws0">2<span class="ff1">、特殊函数的傅立叶变换</span>...........................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>11</div><div class="t m0 xc h7 yf ff3 fs3 fc1 sc2 ls0 ws0">2<span class="ff1">、图像滤波</span>...................................................................................................<span class="_"> </span>.<span class="_ _0"></span>...<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>....<span class="_ _0"></span>17</div><div class="t m0 x9 h5 y10 ff1 fs1 fc1 sc1 ls0 ws0">【正文】</div><div class="t m0 xa h5 y11 ff1 fs1 fc2 sc3 ls0 ws0">【正文】</div><div class="t m0 x9 h8 y12 ff1 fs4 fc0 sc0 ls0 ws0">一、正交变换</div><div class="t m0 xa h8 y13 ff1 fs4 fc1 sc1 ls0 ws0">一、正交变换</div><div class="t m0 xd h9 y14 ff2 fs5 fc0 sc2 ls0 ws0">1</div><div class="t m0 xe h9 y15 ff2 fs5 fc1 sc2 ls0 ws0">1</div><div class="t m0 xf ha y14 ff1 fs5 fc0 sc0 ls0 ws0">、一维函数的正交性</div><div class="t m0 x10 ha y15 ff1 fs5 fc1 sc1 ls0 ws0">、一维函数的正交性</div><div class="t m0 x9 h6 y16 ff1 fs2 fc1 sc1 ls0 ws0">【定义】</div><div class="t m0 xa h6 y17 ff1 fs2 fc3 sc4 ls0 ws0">【定义】</div><div class="t m0 x11 h6 y18 ff1 fs2 fc0 sc0 ls0 ws0">如果有</div><div class="t m0 x12 h6 y19 ff1 fs2 fc1 sc1 ls0 ws0">如果有</div><div class="t m0 x13 hb y1a ff2 fs2 fc0 sc2 ls0 ws0">N</div><div class="t m0 x13 hb y19 ff2 fs2 fc1 sc2 ls0 ws0">N</div><div class="t m0 x14 h6 y18 ff1 fs2 fc0 sc0 ls0 ws0">个函数：</div><div class="t m0 x15 h6 y19 ff1 fs2 fc1 sc1 ls0 ws0">个函数：</div><div class="t m0 x11 h6 y1b ff1 fs2 fc0 sc0 ls0 ws0">构<span class="_ _1"></span>成<span class="_ _1"></span>一<span class="_ _1"></span>个<span class="_ _1"></span>函<span class="_ _1"></span>数<span class="_ _2"></span>集</div><div class="t m0 x12 h6 y1c ff1 fs2 fc1 sc1 ls0 ws0">构<span class="_ _1"></span>成<span class="_ _2"></span>一<span class="_ _1"></span>个<span class="_ _1"></span>函<span class="_ _1"></span>数<span class="_ _1"></span>集</div><div class="t m0 x16 h6 y1b ff1 fs2 fc0 sc0 ls0 ws0">，<span class="_ _2"></span>这<span class="_ _1"></span>些<span class="_ _1"></span>函<span class="_ _1"></span>数<span class="_ _1"></span>在<span class="_ _1"></span>区<span class="_ _2"></span>间</div><div class="t m0 x17 h6 y1c ff1 fs2 fc1 sc1 ls0 ws0">，<span class="_ _2"></span>这<span class="_ _1"></span>些<span class="_ _1"></span>函<span class="_ _1"></span>数<span class="_ _1"></span>在<span class="_ _1"></span>区<span class="_ _2"></span>间</div><div class="t m0 x18 h6 y1b ff1 fs2 fc0 sc0 ls0 ws0">内<span class="_ _2"></span>满<span class="_ _1"></span>足<span class="_ _1"></span>下<span class="_ _1"></span>列<span class="_ _1"></span>特</div><div class="t m0 x19 h6 y1c ff1 fs2 fc1 sc1 ls0 ws0">内<span class="_ _2"></span>满<span class="_ _1"></span>足<span class="_ _1"></span>下<span class="_ _1"></span>列<span class="_ _1"></span>特</div><div class="t m0 x9 h6 y1d ff1 fs2 fc0 sc0 ls0 ws0">性：</div><div class="t m0 xa h6 y1e ff1 fs2 fc1 sc1 ls0 ws0">性：</div><div class="t m0 x11 h6 y1f ff1 fs2 fc0 sc0 ls0 ws0">则<span class="_ _0"></span>此<span class="_ _0"></span>函<span class="_ _2"></span>数<span class="_ _0"></span>集</div><div class="t m0 x12 h6 y20 ff1 fs2 fc1 sc1 ls0 ws0">则<span class="_ _0"></span>此<span class="_ _0"></span>函<span class="_ _2"></span>数<span class="_ _0"></span>集</div><div class="t m0 x1a h6 y1f ff1 fs2 fc0 sc0 ls0 ws0">称<span class="_ _0"></span>为<span class="_ _0"></span>正<span class="_ _2"></span>交<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>集<span class="_ _2"></span>。<span class="_ _0"></span>当</div><div class="t m0 x1b h6 y20 ff1 fs2 fc1 sc1 ls0 ws0">称<span class="_ _0"></span>为<span class="_ _0"></span>正<span class="_ _2"></span>交<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>集<span class="_ _2"></span>。<span class="_ _0"></span>当</div><div class="t m0 x1c h6 y1f ff1 fs2 fc0 sc0 ls0 ws0">时<span class="_ _0"></span>，<span class="_ _0"></span>则<span class="_ _0"></span>称<span class="_ _2"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>集</div><div class="t m0 x1d h6 y20 ff1 fs2 fc1 sc1 ls0 ws0">时<span class="_ _0"></span>，<span class="_ _0"></span>则<span class="_ _0"></span>称<span class="_ _2"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>集</div><div class="t m0 x9 h6 y21 ff1 fs2 fc0 sc0 ls0 ws0">为归一化正交函数集。</div><div class="t m0 xa h6 y22 ff1 fs2 fc1 sc1 ls0 ws0">为归一化正交函数集。</div><div class="t m0 x11 h6 y23 ff1 fs2 fc0 sc0 ls0 ws0">任<span class="_ _0"></span>一函<span class="_ _0"></span>数</div><div class="t m0 x12 h6 y24 ff1 fs2 fc1 sc1 ls0 ws0">任<span class="_ _0"></span>一函<span class="_ _0"></span>数</div><div class="t m0 x1e h6 y23 ff1 fs2 fc0 sc0 ls0 ws0">在<span class="_ _0"></span>区间</div><div class="t m0 x1f h6 y24 ff1 fs2 fc1 sc1 ls0 ws0">在<span class="_ _0"></span>区间</div><div class="t m0 x20 h6 y23 ff1 fs2 fc0 sc0 ls0 ws0">内<span class="_ _0"></span>都可<span class="_ _0"></span>以<span class="_ _0"></span>用<span class="_ _0"></span>正<span class="_ _0"></span>交<span class="_ _0"></span>函<span class="_ _0"></span>数集<span class="_ _0"></span>各<span class="_ _0"></span>分<span class="_ _0"></span>量<span class="_ _0"></span>的<span class="_ _0"></span>线性</div><div class="t m0 x21 h6 y24 ff1 fs2 fc1 sc1 ls0 ws0">内<span class="_ _0"></span>都可<span class="_ _0"></span>以<span class="_ _0"></span>用<span class="_ _0"></span>正<span class="_ _0"></span>交<span class="_ _0"></span>函<span class="_ _0"></span>数集<span class="_ _0"></span>各<span class="_ _0"></span>分<span class="_ _0"></span>量<span class="_ _0"></span>的<span class="_ _0"></span>线性</div><div class="t m0 x9 h6 y25 ff1 fs2 fc0 sc0 ls0 ws0">组合近似：</div><div class="t m0 xa h6 y26 ff1 fs2 fc1 sc1 ls0 ws0">组合近似：</div><div class="t m0 x5 h6 y27 ff1 fs2 fc0 sc0 ls0 ws0">。</div><div class="t m0 x22 h6 y28 ff1 fs2 fc1 sc1 ls0 ws0">。</div><div class="t m0 x11 h6 y29 ff1 fs2 fc0 sc0 ls0 ws0">为<span class="_ _0"></span>获<span class="_ _0"></span>得<span class="_ _0"></span>最<span class="_ _0"></span>佳<span class="_ _0"></span>的<span class="_ _0"></span>近<span class="_ _0"></span>似<span class="_ _0"></span>，<span class="_ _0"></span>即<span class="_ _0"></span>求<span class="_ _0"></span>得<span class="_ _0"></span>系<span class="_ _0"></span>数</div><div class="t m0 x12 h6 y2a ff1 fs2 fc1 sc1 ls0 ws0">为<span class="_ _0"></span>获<span class="_ _0"></span>得<span class="_ _0"></span>最<span class="_ _0"></span>佳<span class="_ _0"></span>的<span class="_ _0"></span>近<span class="_ _0"></span>似<span class="_ _0"></span>，<span class="_ _0"></span>即<span class="_ _0"></span>求<span class="_ _0"></span>得<span class="_ _0"></span>系<span class="_ _0"></span>数</div><div class="t m0 x23 h6 y29 ff1 fs2 fc0 sc0 ls0 ws0">，<span class="_ _0"></span>可<span class="_ _0"></span>用<span class="_ _0"></span>均<span class="_ _0"></span>方<span class="_ _0"></span>误<span class="_ _0"></span>差</div><div class="t m0 x24 h6 y2a ff1 fs2 fc1 sc1 ls0 ws0">，<span class="_ _0"></span>可<span class="_ _0"></span>用<span class="_ _0"></span>均<span class="_ _0"></span>方<span class="_ _0"></span>误<span class="_ _0"></span>差</div><div class="t m0 x25 h6 y29 ff1 fs2 fc0 sc0 ls0 ws0">最<span class="_ _0"></span>小<span class="_ _0"></span>的<span class="_ _0"></span>条<span class="_ _0"></span>件<span class="_ _0"></span>求</div><div class="t m0 x26 h6 y2a ff1 fs2 fc1 sc1 ls0 ws0">最<span class="_ _0"></span>小<span class="_ _0"></span>的<span class="_ _0"></span>条<span class="_ _0"></span>件<span class="_ _0"></span>求</div><div class="t m0 x9 h6 y2b ff1 fs2 fc0 sc0 ls0 ws0">出。</div><div class="t m0 xa h6 y2c ff1 fs2 fc1 sc1 ls0 ws0">出。</div><div class="t m0 x9 h6 y2d ff1 fs2 fc1 sc1 ls0 ws0">【推导】</div><div class="t m0 xa h6 y2e ff1 fs2 fc3 sc4 ls0 ws0">【推导】</div><div class="t m0 x27 hc y2f ff4 fs1 fc1 sc2 ls0 ws0">12-1</div></div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div>
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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/626455c64f8811599e455ce2/bg2.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x11 h6 y30 ff1 fs2 fc0 sc0 ls0 ws0">要使</div><div class="t m0 x12 h6 y31 ff1 fs2 fc1 sc1 ls0 ws0">要使</div><div class="t m0 x28 h6 y30 ff1 fs2 fc0 sc0 ls0 ws0">最小，对于每一个系数</div><div class="t m0 x29 h6 y31 ff1 fs2 fc1 sc1 ls0 ws0">最小，对于每一个系数</div><div class="t m0 x2a h6 y30 ff1 fs2 fc0 sc0 ls0 ws0">，应满足</div><div class="t m0 x2b h6 y31 ff1 fs2 fc1 sc1 ls0 ws0">，应满足</div><div class="t m0 x2c h6 y30 ff1 fs2 fc0 sc0 ls0 ws0">，即：</div><div class="t m0 x2d h6 y31 ff1 fs2 fc1 sc1 ls0 ws0">，即：</div><div class="t m0 x9 h6 y32 ff1 fs2 fc0 sc0 ls0 ws0">整理一下，有：</div><div class="t m0 xa h6 y33 ff1 fs2 fc1 sc1 ls0 ws0">整理一下，有：</div><div class="t m0 x9 h6 y34 ff1 fs2 fc0 sc0 ls0 ws0">若函数集是归一化的，</div><div class="t m0 xa h6 y35 ff1 fs2 fc1 sc1 ls0 ws0">若函数集是归一化的，</div><div class="t m0 x17 h6 y34 ff1 fs2 fc0 sc0 ls0 ws0">，则有：</div><div class="t m0 x17 h6 y35 ff1 fs2 fc1 sc1 ls0 ws0">，则有：</div><div class="t m0 x9 h6 y36 ff1 fs2 fc1 sc1 ls0 ws0">【意义】</div><div class="t m0 xa h6 y37 ff1 fs2 fc3 sc4 ls0 ws0">【意义】</div><div class="t m0 x11 h6 y38 ff1 fs2 fc0 sc0 ls0 ws0">对<span class="_ _0"></span>于</div><div class="t m0 x12 h6 y39 ff1 fs2 fc1 sc1 ls0 ws0">对<span class="_ _0"></span>于</div><div class="t m0 x2e h6 y38 ff1 fs2 fc0 sc0 ls0 ws0">的<span class="_ _0"></span>正<span class="_ _2"></span>交<span class="_ _0"></span>函<span class="_ _2"></span>数<span class="_ _0"></span>集</div><div class="t m0 x2f h6 y39 ff1 fs2 fc1 sc1 ls0 ws0">的<span class="_ _0"></span>正<span class="_ _2"></span>交<span class="_ _0"></span>函<span class="_ _2"></span>数<span class="_ _0"></span>集</div><div class="t m0 x30 h6 y38 ff1 fs2 fc0 sc0 ls0 ws0">，<span class="_ _0"></span>当<span class="_ _2"></span>下<span class="_ _0"></span>述<span class="_ _2"></span>两<span class="_ _0"></span>点<span class="_ _0"></span>成<span class="_ _2"></span>立<span class="_ _0"></span>时<span class="_ _2"></span>，<span class="_ _0"></span>称<span class="_ _0"></span>为</div><div class="t m0 x31 h6 y39 ff1 fs2 fc1 sc1 ls0 ws0">，<span class="_ _0"></span>当<span class="_ _2"></span>下<span class="_ _0"></span>述<span class="_ _2"></span>两<span class="_ _0"></span>点<span class="_ _0"></span>成<span class="_ _2"></span>立<span class="_ _0"></span>时<span class="_ _2"></span>，<span class="_ _0"></span>称<span class="_ _0"></span>为</div><div class="t m0 x9 h6 y3a ff1 fs2 fc0 sc0 ls0 ws0">完备的正交函数集：</div><div class="t m0 xa h6 y3b ff1 fs2 fc1 sc1 ls0 ws0">完备的正交函数集：</div><div class="t m0 x12 hb y3c ff2 fs2 fc0 sc2 ls0 ws0">(1) </div><div class="t m0 x12 hb y3d ff2 fs2 fc1 sc2 ls0 ws0">(1) </div><div class="t m0 x32 h6 y3e ff1 fs2 fc0 sc0 ls0 ws0">不存在这样的函数</div><div class="t m0 x33 h6 y3d ff1 fs2 fc1 sc1 ls0 ws0">不存在这样的函数</div><div class="t m0 x34 h6 y3e ff1 fs2 fc0 sc0 ls0 ws0">，且：</div><div class="t m0 x35 h6 y3d ff1 fs2 fc1 sc1 ls0 ws0">，且：</div><div class="t m0 x11 h6 y3f ff1 fs2 fc0 sc0 ls0 ws0">显然，</div><div class="t m0 x12 h6 y40 ff1 fs2 fc1 sc1 ls0 ws0">显然，</div><div class="t m0 x36 h6 y3f ff1 fs2 fc0 sc0 ls0 ws0">如果能<span class="_ _0"></span>使上式成立<span class="_ _0"></span>，说明</div><div class="t m0 x37 h6 y40 ff1 fs2 fc1 sc1 ls0 ws0">如果能<span class="_ _0"></span>使上式成立<span class="_ _0"></span>，说明</div><div class="t m0 x38 h6 y3f ff1 fs2 fc0 sc0 ls0 ws0">与函数<span class="_ _0"></span>集</div><div class="t m0 x39 h6 y40 ff1 fs2 fc1 sc1 ls0 ws0">与函数<span class="_ _0"></span>集</div><div class="t m0 x3a h6 y3f ff1 fs2 fc0 sc0 ls0 ws0">中的每<span class="_ _0"></span>一个</div><div class="t m0 x3b h6 y40 ff1 fs2 fc1 sc1 ls0 ws0">中的每<span class="_ _0"></span>一个</div><div class="t m0 x9 h6 y41 ff1 fs2 fc0 sc0 ls0 ws0">成员<span class="_ _0"></span>都是<span class="_ _0"></span>正<span class="_ _0"></span>交的<span class="_ _0"></span>，因<span class="_ _0"></span>为</div><div class="t m0 xa h6 y42 ff1 fs2 fc1 sc1 ls0 ws0">成员<span class="_ _0"></span>都是<span class="_ _0"></span>正<span class="_ _0"></span>交的<span class="_ _0"></span>，因<span class="_ _0"></span>为</div><div class="t m0 x3c h6 y41 ff1 fs2 fc0 sc0 ls0 ws0">就应<span class="_ _0"></span>该属<span class="_ _0"></span>于<span class="_ _0"></span>此函<span class="_ _0"></span>数集<span class="_ _0"></span>。如<span class="_ _0"></span>果<span class="_ _0"></span>函数<span class="_ _0"></span>集</div><div class="t m0 x3d h6 y42 ff1 fs2 fc1 sc1 ls0 ws0">就应<span class="_ _0"></span>该属<span class="_ _0"></span>于<span class="_ _0"></span>此函<span class="_ _0"></span>数集<span class="_ _0"></span>。如<span class="_ _0"></span>果<span class="_ _0"></span>函数<span class="_ _0"></span>集</div><div class="t m0 x3e h6 y41 ff1 fs2 fc0 sc0 ls0 ws0">不含</div><div class="t m0 x3f h6 y42 ff1 fs2 fc1 sc1 ls0 ws0">不含</div><div class="t m0 x40 h6 y43 ff1 fs2 fc0 sc0 ls0 ws0">，则函数集就不完备；</div><div class="t m0 x41 h6 y44 ff1 fs2 fc1 sc1 ls0 ws0">，则函数集就不完备；</div><div class="t m0 x12 hb y45 ff2 fs2 fc0 sc2 ls0 ws0">(2)<span class="_ _0"></span> </div><div class="t m0 x12 hb y46 ff2 fs2 fc1 sc2 ls0 ws0">(2)<span class="_ _0"></span> </div><div class="t m0 x42 h6 y47 ff1 fs2 fc0 sc0 ls0 ws0">如<span class="_ _0"></span>果<span class="_ _0"></span>函<span class="_ _0"></span>数</div><div class="t m0 x32 h6 y46 ff1 fs2 fc1 sc1 ls0 ws0">如<span class="_ _0"></span>果<span class="_ _0"></span>函<span class="_ _0"></span>数</div><div class="t m0 x1b h6 y47 ff1 fs2 fc0 sc0 ls0 ws0">在<span class="_ _0"></span>区<span class="_ _0"></span>间</div><div class="t m0 x43 h6 y46 ff1 fs2 fc1 sc1 ls0 ws0">在<span class="_ _0"></span>区<span class="_ _0"></span>间</div><div class="t m0 x44 h6 y47 ff1 fs2 fc0 sc0 ls0 ws0">上<span class="_ _0"></span>可<span class="_ _0"></span>用<span class="_ _0"></span>正<span class="_ _0"></span>交<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>集</div><div class="t m0 x45 h6 y46 ff1 fs2 fc1 sc1 ls0 ws0">上<span class="_ _0"></span>可<span class="_ _0"></span>用<span class="_ _0"></span>正<span class="_ _0"></span>交<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>集</div><div class="t m0 x46 h6 y47 ff1 fs2 fc0 sc0 ls0 ws0">近<span class="_ _0"></span>似<span class="_ _0"></span>表</div><div class="t m0 x47 h6 y46 ff1 fs2 fc1 sc1 ls0 ws0">近<span class="_ _0"></span>似<span class="_ _0"></span>表</div><div class="t m0 x9 h6 y48 ff1 fs2 fc0 sc0 ls0 ws0">示成：</div><div class="t m0 xa h6 y49 ff1 fs2 fc1 sc1 ls0 ws0">示成：</div><div class="t m0 x9 h6 y4a ff1 fs2 fc0 sc0 ls0 ws0">均方误差：</div><div class="t m0 xa h6 y4b ff1 fs2 fc1 sc1 ls0 ws0">均方误差：</div><div class="t m0 x27 hc y2f ff4 fs1 fc1 sc2 ls0 ws0">12-2</div></div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div>
<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/626455c64f8811599e455ce2/bg3.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x9 h6 y4c ff1 fs2 fc0 sc0 ls0 ws0">则当</div><div class="t m0 xa h6 y4d ff1 fs2 fc1 sc1 ls0 ws0">则当</div><div class="t m0 x48 h6 y4c ff1 fs2 fc0 sc0 ls0 ws0">时，必需：</div><div class="t m0 x49 h6 y4d ff1 fs2 fc1 sc1 ls0 ws0">时，必需：</div><div class="t m0 x35 h6 y4c ff1 fs2 fc0 sc0 ls0 ws0">，函数集才完备。此时意味着：</div><div class="t m0 x4a h6 y4d ff1 fs2 fc1 sc1 ls0 ws0">，函数集才完备。此时意味着：</div><div class="t m0 x9 h6 y4e ff1 fs2 fc1 sc1 ls0 ws0">【说明】</div><div class="t m0 xa h6 y4f ff1 fs2 fc3 sc4 ls0 ws0">【说明】</div><div class="t m0 x12 hb y50 ff2 fs2 fc0 sc2 ls0 ws0">(1)<span class="_ _0"></span> </div><div class="t m0 x12 hb y51 ff2 fs2 fc1 sc2 ls0 ws0">(1)<span class="_ _0"></span> </div><div class="t m0 x42 h6 y52 ff1 fs2 fc0 sc0 ls0 ws0">任<span class="_ _0"></span>一<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>，<span class="_ _0"></span>若<span class="_ _2"></span>能<span class="_ _0"></span>量<span class="_ _0"></span>有<span class="_ _0"></span>限</div><div class="t m0 x32 h6 y51 ff1 fs2 fc1 sc1 ls0 ws0">任<span class="_ _0"></span>一<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>，<span class="_ _0"></span>若<span class="_ _2"></span>能<span class="_ _0"></span>量<span class="_ _0"></span>有<span class="_ _0"></span>限</div><div class="t m0 x4b hb y50 ff2 fs2 fc0 sc2 ls0 ws0">,</div><div class="t m0 x4b hb y51 ff2 fs2 fc1 sc2 ls0 ws0">,</div><div class="t m0 x4c h6 y52 ff1 fs2 fc0 sc0 ls0 ws0">即</div><div class="t m0 x4d h6 y51 ff1 fs2 fc1 sc1 ls0 ws0">即</div><div class="t m0 x4e h6 y52 ff1 fs2 fc0 sc0 ls0 ws0">，<span class="_ _0"></span>总<span class="_ _0"></span>可<span class="_ _0"></span>以<span class="_ _0"></span>用<span class="_ _0"></span>有<span class="_ _2"></span>限<span class="_ _0"></span>项<span class="_ _0"></span>级<span class="_ _0"></span>数</div><div class="t m0 x4f h6 y51 ff1 fs2 fc1 sc1 ls0 ws0">，<span class="_ _0"></span>总<span class="_ _0"></span>可<span class="_ _0"></span>以<span class="_ _0"></span>用<span class="_ _0"></span>有<span class="_ _2"></span>限<span class="_ _0"></span>项<span class="_ _0"></span>级<span class="_ _0"></span>数</div><div class="t m0 x9 h6 y53 ff1 fs2 fc0 sc0 ls0 ws0">来逼近。</div><div class="t m0 xa h6 y54 ff1 fs2 fc1 sc1 ls0 ws0">来逼近。</div><div class="t m0 x12 hb y55 ff2 fs2 fc0 sc2 ls0 ws0">(2)<span class="_ _0"></span> </div><div class="t m0 x12 hb y56 ff2 fs2 fc1 sc2 ls0 ws0">(2)<span class="_ _0"></span> </div><div class="t m0 x42 h6 y57 ff1 fs2 fc0 sc0 ls0 ws0">因<span class="_ _0"></span>为<span class="_ _0"></span>奇<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>的<span class="_ _0"></span>累<span class="_ _0"></span>加<span class="_ _0"></span>仍<span class="_ _0"></span>为<span class="_ _0"></span>奇函<span class="_ _2"></span>数，<span class="_ _2"></span>偶函<span class="_ _0"></span>数<span class="_ _0"></span>的<span class="_ _0"></span>累<span class="_ _0"></span>加<span class="_ _0"></span>仍<span class="_ _0"></span>为<span class="_ _0"></span>偶<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>，<span class="_ _0"></span>所<span class="_ _0"></span>以</div><div class="t m0 x32 h6 y56 ff1 fs2 fc1 sc1 ls0 ws0">因<span class="_ _0"></span>为<span class="_ _0"></span>奇<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>的<span class="_ _0"></span>累<span class="_ _0"></span>加<span class="_ _0"></span>仍<span class="_ _0"></span>为<span class="_ _0"></span>奇函<span class="_ _2"></span>数，<span class="_ _2"></span>偶函<span class="_ _0"></span>数<span class="_ _0"></span>的<span class="_ _0"></span>累<span class="_ _0"></span>加<span class="_ _0"></span>仍<span class="_ _0"></span>为<span class="_ _0"></span>偶<span class="_ _0"></span>函<span class="_ _0"></span>数<span class="_ _0"></span>，<span class="_ _0"></span>所<span class="_ _0"></span>以</div><div class="t m0 x9 h6 y58 ff1 fs2 fc0 sc0 ls0 ws0">一个函数集要表示任何函数，则函数集中必需有奇函数和偶函数。</div><div class="t m0 xa h6 y59 ff1 fs2 fc1 sc1 ls0 ws0">一个函数集要表示任何函数，则函数集中必需有奇函数和偶函数。</div><div class="t m0 x9 h6 y5a ff1 fs2 fc1 sc1 ls0 ws0">【举例】</div><div class="t m0 xa h6 y5b ff1 fs2 fc3 sc4 ls0 ws0">【举例】</div><div class="t m0 x11 h6 y5c ff1 fs2 fc0 sc0 ls0 ws0">下面考察三角函数的正交性。三角函数的表达式为：</div><div class="t m0 x12 h6 y5d ff1 fs2 fc1 sc1 ls0 ws0">下面考察三角函数的正交性。三角函数的表达式为：</div><div class="t m0 x50 h6 y5e ff1 fs2 fc0 sc0 ls0 ws0">，</div><div class="t m0 x51 h6 y5f ff1 fs2 fc1 sc1 ls0 ws0">，</div><div class="t m0 x9 h6 y60 ff1 fs2 fc0 sc0 ls0 ws0">其函数集为：</div><div class="t m0 xa h6 y61 ff1 fs2 fc1 sc1 ls0 ws0">其函数集为：</div><div class="t m0 x9 h6 y62 ff1 fs2 fc0 sc0 ls0 ws0">由于有：</div><div class="t m0 xa h6 y63 ff1 fs2 fc1 sc1 ls0 ws0">由于有：</div><div class="t m0 x9 h6 y64 ff1 fs2 fc0 sc0 ls0 ws0">所以三角级数的函数集为正交函数集。</div><div class="t m0 xa h6 y65 ff1 fs2 fc1 sc1 ls0 ws0">所以三角级数的函数集为正交函数集。</div><div class="t m0 xd h9 y66 ff2 fs5 fc0 sc2 ls0 ws0">2</div><div class="t m0 xe h9 y67 ff2 fs5 fc1 sc2 ls0 ws0">2</div><div class="t m0 xf ha y66 ff1 fs5 fc0 sc0 ls0 ws0">、二维函数的正交性</div><div class="t m0 x10 ha y67 ff1 fs5 fc1 sc1 ls0 ws0">、二维函数的正交性</div><div class="t m0 x9 h6 y68 ff1 fs2 fc1 sc1 ls0 ws0">【定义】</div><div class="t m0 xa h6 y69 ff1 fs2 fc3 sc4 ls0 ws0">【定义】</div><div class="t m0 x11 h6 y6a ff1 fs2 fc0 sc0 ls0 ws0">若</div><div class="t m0 x12 h6 y6b ff1 fs2 fc1 sc1 ls0 ws0">若</div><div class="t m0 x52 hb y6c ff2 fs2 fc0 sc2 ls0 ws0">M</div><div class="t m0 x52 hb y6b ff2 fs2 fc1 sc2 ls0 ws0">M</div><div class="t m0 x53 h6 y6a ff1 fs2 fc0 sc0 ls0 ws0">阶实数矩阵满足</div><div class="t m0 x54 h6 y6b ff1 fs2 fc1 sc1 ls0 ws0">阶实数矩阵满足</div><div class="t m0 x21 h6 y6a ff1 fs2 fc0 sc0 ls0 ws0">，则</div><div class="t m0 x55 h6 y6b ff1 fs2 fc1 sc1 ls0 ws0">，则</div><div class="t m0 x5 h6 y6a ff1 fs2 fc0 sc0 ls0 ws0">称为正交矩阵；</div><div class="t m0 x22 h6 y6b ff1 fs2 fc1 sc1 ls0 ws0">称为正交矩阵；</div><div class="t m0 x11 h6 y6d ff1 fs2 fc0 sc0 ls0 ws0">若</div><div class="t m0 x12 h6 y6e ff1 fs2 fc1 sc1 ls0 ws0">若</div><div class="t m0 x52 hb y6f ff2 fs2 fc0 sc2 ls0 ws0">M</div><div class="t m0 x52 hb y6e ff2 fs2 fc1 sc2 ls0 ws0">M</div><div class="t m0 x53 h6 y6d ff1 fs2 fc0 sc0 ls0 ws0">阶复数矩阵满足</div><div class="t m0 x54 h6 y6e ff1 fs2 fc1 sc1 ls0 ws0">阶复数矩阵满足</div><div class="t m0 x56 h6 y6d ff1 fs2 fc0 sc0 ls0 ws0">，则</div><div class="t m0 x57 h6 y6e ff1 fs2 fc1 sc1 ls0 ws0">，则</div><div class="t m0 x58 h6 y6d ff1 fs2 fc0 sc0 ls0 ws0">称为酉矩阵。</div><div class="t m0 x59 h6 y6e ff1 fs2 fc1 sc1 ls0 ws0">称为酉矩阵。</div><div class="t m0 x9 h6 y70 ff1 fs2 fc1 sc1 ls0 ws0">【性质】正交归一</div><div class="t m0 xa h6 y71 ff1 fs2 fc3 sc4 ls0 ws0">【性质】正交归一</div><div class="t m0 x11 h6 y72 ff1 fs2 fc0 sc0 ls0 ws0">若</div><div class="t m0 x12 h6 y73 ff1 fs2 fc1 sc1 ls0 ws0">若</div><div class="t m0 x53 h6 y72 ff1 fs2 fc0 sc0 ls0 ws0">为酉<span class="_ _0"></span>矩阵<span class="_ _0"></span>或<span class="_ _0"></span>正交<span class="_ _0"></span>阵，<span class="_ _0"></span>则在<span class="_ _0"></span>矩<span class="_ _0"></span>阵中<span class="_ _0"></span>各行<span class="_ _0"></span>或各<span class="_ _0"></span>列向<span class="_ _0"></span>量<span class="_ _0"></span>的模<span class="_ _0"></span>为</div><div class="t m0 x54 h6 y73 ff1 fs2 fc1 sc1 ls0 ws0">为酉<span class="_ _0"></span>矩阵<span class="_ _0"></span>或<span class="_ _0"></span>正交<span class="_ _0"></span>阵，<span class="_ _0"></span>则在<span class="_ _0"></span>矩<span class="_ _0"></span>阵中<span class="_ _0"></span>各行<span class="_ _0"></span>或各<span class="_ _0"></span>列向<span class="_ _0"></span>量<span class="_ _0"></span>的模<span class="_ _0"></span>为</div><div class="t m0 x5a hb y74 ff2 fs2 fc0 sc2 ls0 ws0">1</div><div class="t m0 x5a hb y73 ff2 fs2 fc1 sc2 ls0 ws0">1</div><div class="t m0 x46 h6 y72 ff1 fs2 fc0 sc0 ls0 ws0">，任<span class="_ _0"></span>意</div><div class="t m0 x47 h6 y73 ff1 fs2 fc1 sc1 ls0 ws0">，任<span class="_ _0"></span>意</div><div class="t m0 x9 h6 y75 ff1 fs2 fc0 sc0 ls0 ws0">不同行或列向量之间正交。</div><div class="t m0 xa h6 y76 ff1 fs2 fc1 sc1 ls0 ws0">不同行或列向量之间正交。</div><div class="t m0 x11 h6 y77 ff1 fs2 fc0 sc0 ls0 ws0">矩阵</div><div class="t m0 x12 h6 y78 ff1 fs2 fc1 sc1 ls0 ws0">矩阵</div><div class="t m0 x29 h6 y77 ff1 fs2 fc0 sc0 ls0 ws0">可表示成：</div><div class="t m0 x29 h6 y78 ff1 fs2 fc1 sc1 ls0 ws0">可表示成：</div><div class="t m0 x27 hc y2f ff4 fs1 fc1 sc2 ls0 ws0">12-3</div></div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div>
<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/626455c64f8811599e455ce2/bg4.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x9 h6 y79 ff1 fs2 fc0 sc0 ls0 ws0">其中：</div><div class="t m0 xa h6 y7a ff1 fs2 fc1 sc1 ls0 ws0">其中：</div><div class="t m0 x5b h6 y7b ff1 fs2 fc0 sc0 ls0 ws0">，</div><div class="t m0 x2e h6 y7c ff1 fs2 fc1 sc1 ls0 ws0">，</div><div class="t m0 x9 h6 y7d ff1 fs2 fc0 sc0 ls0 ws0">当</div><div class="t m0 xa h6 y7e ff1 fs2 fc1 sc1 ls0 ws0">当</div><div class="t m0 x5c h6 y7d ff1 fs2 fc0 sc0 ls0 ws0">为酉矩阵时，根据定义有：</div><div class="t m0 x5d h6 y7e ff1 fs2 fc1 sc1 ls0 ws0">为酉矩阵时，根据定义有：</div><div class="t m0 x9 h6 y7f ff1 fs2 fc0 sc0 ls0 ws0">故得：</div><div class="t m0 xa h6 y80 ff1 fs2 fc1 sc1 ls0 ws0">故得：</div><div class="t m0 x9 h6 y81 ff1 fs2 fc0 sc0 ls0 ws0">同样：</div><div class="t m0 xa h6 y82 ff1 fs2 fc1 sc1 ls0 ws0">同样：</div><div class="t m0 x9 h6 y83 ff1 fs2 fc0 sc0 ls0 ws0">可得：</div><div class="t m0 xa h6 y84 ff1 fs2 fc1 sc1 ls0 ws0">可得：</div><div class="t m0 x11 h6 y85 ff1 fs2 fc0 sc0 ls0 ws0">这个性质说明酉阵是一个正交归一矩阵。</div><div class="t m0 x12 h6 y86 ff1 fs2 fc1 sc1 ls0 ws0">这个性质说明酉阵是一个正交归一矩阵。</div><div class="t m0 x9 h6 y87 ff1 fs2 fc1 sc1 ls0 ws0">【性质】若</div><div class="t m0 xa h6 y88 ff1 fs2 fc3 sc4 ls0 ws0">【性质】若</div><div class="t m0 x5e h6 y87 ff1 fs2 fc1 sc1 ls0 ws0">为酉阵，则</div><div class="t m0 x5f h6 y88 ff1 fs2 fc3 sc4 ls0 ws0">为酉阵，则</div><div class="t m0 x60 h6 y87 ff1 fs2 fc1 sc1 ls0 ws0">和</div><div class="t m0 x61 h6 y88 ff1 fs2 fc3 sc4 ls0 ws0">和</div><div class="t m0 x20 h6 y87 ff1 fs2 fc1 sc1 ls0 ws0">也是酉阵。</div><div class="t m0 x21 h6 y88 ff1 fs2 fc3 sc4 ls0 ws0">也是酉阵。</div><div class="t m0 x11 h6 y89 ff1 fs2 fc0 sc0 ls0 ws0">因为，酉阵有：</div><div class="t m0 x12 h6 y8a ff1 fs2 fc1 sc1 ls0 ws0">因为，酉阵有：</div><div class="t m0 x62 h6 y89 ff1 fs2 fc0 sc0 ls0 ws0">，</div><div class="t m0 x63 h6 y8a ff1 fs2 fc1 sc1 ls0 ws0">，</div><div class="t m0 x9 h6 y8b ff1 fs2 fc0 sc0 ls0 ws0">所以：</div><div class="t m0 xa h6 y8c ff1 fs2 fc1 sc1 ls0 ws0">所以：</div><div class="t m0 x9 h6 y8d ff1 fs2 fc1 sc1 ls0 ws0">【性质】若</div><div class="t m0 xa h6 y8e ff1 fs2 fc3 sc4 ls0 ws0">【性质】若</div><div class="t m0 x5e h6 y8d ff1 fs2 fc1 sc1 ls0 ws0">是酉阵，则其行列式的模为</div><div class="t m0 x5f h6 y8e ff1 fs2 fc3 sc4 ls0 ws0">是酉阵，则其行列式的模为</div><div class="t m0 x64 hb y8f ff2 fs2 fc1 sc2 ls0 ws0">1</div><div class="t m0 x64 hb y8e ff2 fs2 fc3 sc2 ls0 ws0">1</div><div class="t m0 x11 h6 y90 ff1 fs2 fc0 sc0 ls0 ws0">由于</div><div class="t m0 x12 h6 y91 ff1 fs2 fc1 sc1 ls0 ws0">由于</div><div class="t m0 x29 h6 y90 ff1 fs2 fc0 sc0 ls0 ws0">为一个复数方阵，所以</div><div class="t m0 x29 h6 y91 ff1 fs2 fc1 sc1 ls0 ws0">为一个复数方阵，所以</div><div class="t m0 x65 h6 y90 ff1 fs2 fc0 sc0 ls0 ws0">也是复数，可令：</div><div class="t m0 x66 h6 y91 ff1 fs2 fc1 sc1 ls0 ws0">也是复数，可令：</div><div class="t m0 x9 h6 y92 ff1 fs2 fc0 sc0 ls0 ws0">则：</div><div class="t m0 xa h6 y93 ff1 fs2 fc1 sc1 ls0 ws0">则：</div><div class="t m0 x27 hc y2f ff4 fs1 fc1 sc2 ls0 ws0">12-4</div></div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div>
