endobj /Filter /FlateDecode 167 0 obj endobj 24 0 obj 195 0 obj /FontDescriptor 14 0 R << /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R endobj (Introduction) /D [185 0 R /XYZ 105.869 699.082 null] There are two types of Fourier expansions: † Fourier series: If a (reasonably well-behaved) function is periodic, then it can be 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 /BaseFont/MCADNU+CMR10 542.4 542.4 456.8 513.9 1027.8 513.9 513.9 513.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 endobj /LastChar 196 << stream (The zeta function and the Mellin transform of the theta function) endobj 314.8 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 314.8 314.8 t��YOMM N#�)7u��!�A�=�y=�7"W�#��}VL��Ii�<5c=��80�q���Y/iF��}�V}���e��Wn�9���`O&�5Z]� pf�#�DИ�(銾'h� 4��:��F� w#Ŏ�"rRD�$I��3��d�g�S���M�j�I��}3�`gj3��� G؂A�_��. endobj 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 >> << /S /GoTo /D (section.6.6) >> endobj 180 0 obj endobj 51 0 obj endobj endobj /Name/F2 �l��;�I��o����%�륺Mf�t�z����3C���! endobj 850.9 472.2 550.9 734.6 734.6 524.7 906.2 1011.1 787 262.3 524.7] (Theta functions) endobj 76 0 obj /FontDescriptor 23 0 R 168 0 obj �7�A���H�i.� 7]s8MnJ����ikNV��]�&�����h��a4. 188 0 obj (Distributional solutions of differential equations) 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 115 0 obj endobj endobj 513.9 770.7 456.8 513.9 742.3 799.4 513.9 927.8 1042 799.4 285.5 513.9] 100 0 obj << /S /GoTo /D (section.5.4) >> /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 << /S /GoTo /D (subsection.2.3.3) >> << endobj � �|�?�w�+C�����BZ8�2 ^U��tԴ$�����`��O_�EjuyX��Sjh`� << /S /GoTo /D (section.4.1) >> 92 0 obj endobj << 136 0 obj << /S /GoTo /D (section.3.6) >> 63 0 obj 87 0 obj 787 0 0 734.6 629.6 577.2 603.4 905.1 918.2 314.8 341.1 524.7 524.7 524.7 524.7 524.7 endobj (The wave equation in d=3) 155 0 obj (Three dimensions) /Filter /FlateDecode 35 0 obj /Widths[306.7 514.4 817.8 769.1 817.8 766.7 306.7 408.9 408.9 511.1 766.7 306.7 357.8 (The finite Fourier transform) endobj << /S /GoTo /D (chapter.6) >> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.3 856.5 799.4 713.6 685.2 770.7 742.3 799.4 123 0 obj 68 0 obj /LastChar 196 (The Fourier transform) << /S /GoTo /D (section.5.2) >> 43 0 obj 2 Fourier Transform 2.1 De nition The Fourier transform allows us to deal with non-periodic functions. 140 0 obj 285.5 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 285.5 285.5 << endobj 124 0 obj << /S /GoTo /D (section.2.3) >> endobj 186 0 obj 31 0 obj 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 << /S /GoTo /D (subsection.3.4.1) >> 108 0 obj /ProcSet [ /PDF /Text ] endobj /Name/F3 (The heat and Schr\366dinger equations in higher dimensions) (Distributions: examples) 111 0 obj Equally important, Fourier analysis is the tool with which many of the everyday phenomena - the 12 0 obj >> /BaseFont/RSIIJF+CMR9 12 0 obj endobj << /S /GoTo /D (chapter.4) >> /Font << /F8 197 0 R >> /FontDescriptor 17 0 R endobj << /S /GoTo /D (subsection.3.4.2) >> << �hw�q���//�~�w* /Parent 192 0 R << /S /GoTo /D (subsection.2.2.1) >> 83 0 obj endobj 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 95 0 obj (The functional equation for the zeta function) 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] 473.8 498.5 419.8 524.7 1049.4 524.7 524.7 524.7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 << /Subtype/Type1 endobj (Convolution of a function and a distribution) 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 << /S /GoTo /D (section.6.2) >> << endobj 770.7 628.1 285.5 513.9 285.5 513.9 285.5 285.5 513.9 571 456.8 571 457.2 314 513.9 Fourier Analysis Notes, Spring 2020 Peter Woit Department of Mathematics, Columbia University woit@math.columbia.edu September 3, 2020 55 0 obj Mathematically, Fourier analysis has spawned some of the most fundamental developments in our understanding of infinite series and function approxima-tion - developments which are, unfortunately, much beyond the scope of these notes. 163 0 obj endobj stream It can be derived in a rigorous fashion but here we will follow the time-honored approach 9 0 obj /Widths[314.8 527.8 839.5 786.1 839.5 787 314.8 419.8 419.8 524.7 787 314.8 367.3 /Length 283 96 0 obj 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 52 0 obj 127 0 obj 194 0 obj << /S /GoTo /D (chapter.3) >> 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 643.8 839.5 787 710.5 682.1 763 734.6 787 734.6 (Translations) (Fourier analysis on commutative groups) endobj /BaseFont/RXYAQQ+CMBX12 << 15 0 obj 39 0 obj 187 0 obj 26 0 obj 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 116 0 obj 179 0 obj 184 0 obj endobj << /S /GoTo /D (section.3.2) >> 48 0 obj 175 0 obj (The wave equation in d=1) endobj << 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 endobj 135 0 obj endobj 189 0 obj xڅ��j�0E���YJO���]�)Z endobj 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 172 0 obj endobj /Type/Font /Subtype/Type1 /FirstChar 33 << /S /GoTo /D (subsection.4.3.1) >> /Subtype/Type1 Fourier Transform series analysis, but it is clearly oscillatory and very well behaved for t>0 ( >0).