lesson 3 2-norm and fourier series · •we want to use the fact that the fourier series is...
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Lesson 32-norm and Fourier series
� 6IGEPP�JVSQ�PEWX�PIGXYVI�[I�HMWGYWWIH PIEWX�WUYEVIW JSV�KIRIVEP�MRRIV�TVSHYGXW �x, y��
� -R�XLMW�PIGXYVI� [I�[MPP�ZMI[�XLI�*SYVMIV�FEWMW
. . . , � k�, . . . , � �, 1, �, . . . , k�, . . .
EW�ZIGXSVW�MR�E�ZIGXSV�WTEGI
� ;I�[MPP�½RH�XLEX�XLI]�EVI�SVXLSKSREP�JSV�FSXL�E�GSRXMRYSYW�ERH�E HMWGVIXI MRRIVTVSHYGX
� 8LMW�GSRRIGXMSR�[MPP��IZIRXYEPP] �PIEH�XS�JEWX�ERH�EGGYVEXI�GSQTYXEXMSR�SJ�*SYVMIVWIVMIW
� 0IXA =
�a1 | · · · | an
�
FI�E UYEWMQEXVM\ MR�E��KIRIVEP �ZIGXSV�WTEGI V
� 0EWX�PIGXYVI�[I�WLS[IH�XLEX�MJ�XLI�GSPYQRW�SJ A EVI PMRIEVP]�MRHITIRHIRX� XLIR�XLIYRMUYI�GSQTPI\�ZIGXSV c � Cn XLEX QMRMQM^IW
�Ac � b�
JSV�E�KMZIR b MW�TVIGMWIP]
c = K�1
�
���a1, b�
����an, b�
�
��
[LIVI�XLI�+VEQ�QEXVM\ K � Cn�n MW�HI½RIH�F]
K =
�
���a1, a1� · · · �a1, an�
���� � �
����a1, an� · · · �an, an�
�
��
� -J�[I�GER�IZEPYEXI�XLI�MRRIV�TVSHYGX�RYQIVMGEPP]� [I�GER�GSRWXVYGX K I\TPMGMXP]
3RGI MW�WIX�YT� [I�[MPP�I\TPSVI�XLMW�ETTVSEGL�EW�E RYQIVMGEP�EPKS�VMXLQ ERH�WII�XLEX�MX�MW�MRREGGYVEXI
8S�EGLMIZI�EGGYVEG]� [I�[MPP�HIZIPST�XLI +VEQ��7GLQMHX�EPKSVMXLQ
� 0IXA =
�a1 | · · · | an
�
FI�E UYEWMQEXVM\ MR�E��KIRIVEP �ZIGXSV�WTEGI V
� 0EWX�PIGXYVI�[I�WLS[IH�XLEX�MJ�XLI�GSPYQRW�SJ A EVI PMRIEVP]�MRHITIRHIRX� XLIR�XLIYRMUYI�GSQTPI\�ZIGXSV c � Cn XLEX QMRMQM^IW
�Ac � b�
JSV�E�KMZIR b MW�TVIGMWIP]
c = K�1
�
���a1, b�
����an, b�
�
��
[LIVI�XLI�+VEQ�QEXVM\ K � Cn�n MW�HI½RIH�F]
K =
�
���a1, a1� · · · �a1, an�
���� � �
����a1, an� · · · �an, an�
�
��
� -J�[I�GER�IZEPYEXI�XLI�MRRIV�TVSHYGX�RYQIVMGEPP]� [I�GER�GSRWXVYGX K I\TPMGMXP]
3RGI MW�WIX�YT� [I�[MPP�I\TPSVI�XLMW�ETTVSEGL�EW�E RYQIVMGEP�EPKS�VMXLQ ERH�WII�XLEX�MX�MW�MRREGGYVEXI
8S�EGLMIZI�EGGYVEG]� [I�[MPP�HIZIPST�XLI +VEQ��7GLQMHX�EPKSVMXLQ
� 0IXA =
�a1 | · · · | an
�
FI�E UYEWMQEXVM\ MR�E��KIRIVEP �ZIGXSV�WTEGI V
� 0EWX�PIGXYVI�[I�WLS[IH�XLEX�MJ�XLI�GSPYQRW�SJ A EVI PMRIEVP]�MRHITIRHIRX� XLIR�XLIYRMUYI�GSQTPI\�ZIGXSV c � Cn XLEX QMRMQM^IW
�Ac � b�
JSV�E�KMZIR b MW�TVIGMWIP]
c = K�1
�
���a1, b�
����an, b�
�
��
[LIVI�XLI�+VEQ�QEXVM\ K � Cn�n MW�HI½RIH�F]
K =
�
���a1, a1� · · · �a1, an�
���� � �
����a1, an� · · · �an, an�
�
��
� -J�[I�GER�IZEPYEXI�XLI�MRRIV�TVSHYGX�RYQIVMGEPP]� [I�GER�GSRWXVYGX K I\TPMGMXP]
3RGI MW�WIX�YT� [I�[MPP�I\TPSVI�XLMW�ETTVSEGL�EW�E RYQIVMGEP�EPKS�VMXLQ ERH�WII�XLEX�MX�MW�MRREGGYVEXI
8S�EGLMIZI�EGGYVEG]� [I�[MPP�HIZIPST�XLI +VEQ��7GLQMHX�EPKSVMXLQ
Orthonormal vectors and calculating least squares approximations
� % WIX�SJ�RSR^IVS�ZIGXSVW v1, . . . , vn EVI�GEPPIH SVXLSKSREP MJ
�vi, vj� = 0 [LIRIZIV i �= k.
� 8LI]�EVI�GEPPIH SVXLSRSVQEP MJ�XLI]�EVI�SVXLSKSREP�ERH�EPP�ZIGXSVW�EVI�SJ�YRMX�RSVQ�
1 = �vi� � SV�IUYMZEPIRXP]� �vi, vi� = 1.
� 8LI�+VEQ�QEXVM\�SJ�SVXLSRSVQEP�ZIGXSVW�MW�XLI�MHIRXMX]�
K =
�
���v1, v1� · · · �v1, vn�
���� � �
����vn, v1� · · · �vn, vn�
�
��
=
�
��1
� � �1
�
�� = I
� % WIX�SJ�RSR^IVS�ZIGXSVW v1, . . . , vn EVI�GEPPIH SVXLSKSREP MJ
�vi, vj� = 0 [LIRIZIV i �= k.
� 8LI]�EVI�GEPPIH SVXLSRSVQEP MJ�XLI]�EVI�SVXLSKSREP�ERH�EPP�ZIGXSVW�EVI�SJ�YRMX�RSVQ�
1 = �vi� � SV�IUYMZEPIRXP]� �vi, vi� = 1.
� 8LI�+VEQ�QEXVM\�SJ�SVXLSRSVQEP�ZIGXSVW�MW�XLI�MHIRXMX]�
K =
�
���v1, v1� · · · �v1, vn�
���� � �
����vn, v1� · · · �vn, vn�
�
��
=
�
��1
� � �1
�
�� = I
The L2 inner product and norm� ;I�HI½RI�XLI 2 MRRIV�TVSHYGX��SR T �F]
�f, g� =1
2�
� �
��f̄(�)g(�) �
� %WWSGMEXIH�[MXL�XLMW�MRRIV�TVSHYGX�MW�XLI 2 RSVQ�
�f� =
�1
2�
� �
��|f(�)|2 �
� ;LEX�ZIGXSV� WTEGI�HSIW 2 RSVQ�ERH� MRRIV�TVSHYGX� ETTP]� XSS# 'SRZIRMIRXP]IRSYKL� XLI 2 WTEGI�
� 0IXAn =
� � n� | . . . | � � | 1 | � | . . . | n��
FI�E UYEWMQEXVM\ [LSWI�GSPYQRW�EVI�MR 2[T]
� ;LEX�W�XLI�PIEWX�WUYEVIW�ETTVS\MQEXMSR�SJ�E�JYRGXMSR f � 2[T] F] An#
� ;I�LEZI�
k�, k��
=1
2�
� �
��� = 1
ERH�JSV k �= j
�k�, j�
�=
1
2�
� �
��
(j�k)� � =(j�k)� � � (j�k)�
2� (j � k)= 0
� -R�SXLIV�[SVHW� XLI�VS[W�SJ An EVI�SVXLSKSREP�
• The Gram matrix is therefore
K =
�
��
� � n�, � n��
· · ·� � n�, n�
�
�n�, � n�
�· · ·
�n�, n�
�
�
�� =
�
��1
1
�
��
• The Gram matrix is therefore
• The least squares approximation of f is thus precisely the Fourier series:
K�1
�
��
� � n�, f�
�n�, f
�
�
�� =1
2�
�
��
� ��� f(�) n� �
� ��� f(�) � n� �
�
�� =
�
��f̂�n
f̂n
�
��
K =
�
��
� � n�, � n��
· · ·� � n�, n�
�
�n�, � n�
�· · ·
�n�, n�
�
�
�� =
�
��1
1
�
��
Normwise convergence of Fourier series
• We want to use the fact that the Fourier series is precisely the least squares approximation by complex exponentials to show that it converges in the 2-norm
• In other words:
• This will follow from completeness
�����f(�) �n�
k=�n
f̂keik�
�����2
� 0
� 8LI�GSQTPI\�I\TSRIRXMEPW
. . . , � k�, . . . , � �, 1, �, . . . , k�, . . .
JSVQ�E GSQTPIXI�SVXLSRSVQEP�W]WXIQ SJ 2[T]�
-R�SXLIV�[SVHW� JSV�IZIV] f � 2[T] ERH � > 0 XLIVI�I\MWXW�ER n ERH
�n(�) = s�n� n� + · · · sn
n� = Asn
WYGL�XLEX�f � �n�2 < �.
� 8LI�GSQTPI\�I\TSRIRXMEPW
. . . , � k�, . . . , � �, 1, �, . . . , k�, . . .
JSVQ�E GSQTPIXI�SVXLSRSVQEP�W]WXIQ SJ 2[T]�
-R�SXLIV�[SVHW� JSV�IZIV] f � 2[T] ERH � > 0 XLIVI�I\MWXW�ER n ERH
�n(�) = s�n� n� + · · · sn
n� = Asn
WYGL�XLEX�f � �n�2 < �.
�
� %WWYQI�XLEX f MW�GSRXMRYSYW� f � C[T]
� 6IGEPP� *SYVMIV�WIVMIW HS�RSX RIGIWWEVMP]�GSRZIVKI�TSMRX[MWI�JSV�GSRXMRYSYW�JYRG�XMSRW� ,S[IZIV� [I�GER�HI½RI�E� VIPEXIH�GSQFMREXMSR�SJ�GSQTPI\�I\TSRIRXMEPW[LMGL�GSRZIVKIW YRMJSVQP] JSV�EPP�GSRXMRYSYW�JYRGXMSRW�
� ;I�GSRWXVYGX�WYGL�ER �n I\TPMGMXP]�EW�E�WSVX�SJ�EZIVEKI�SZIV�XLI�*SYVMIV�WIVMIW�
�n(�) =n�
k=�n
n + 1 � |k|n + 1
f̂kk� =
1
2�
� �
��f(t)Kn(� � t) t
[LIVI
Kn(�) =n�
k=�n
n + 1 � |k|n + 1
k�
� % WMQTPI�MRXIKVEXMSR�ZEVMEFPI�XVERWJSVQEXMSR�WLS[W�XLEX
�n(�) =1
2�
� �
��f(t)Kn(� � t) t =
1
2�
� �
��f(� � t)Kn(t) t
�
� %WWYQI�XLEX f MW�GSRXMRYSYW� f � C[T]
� 6IGEPP� *SYVMIV�WIVMIW HS�RSX RIGIWWEVMP]�GSRZIVKI�TSMRX[MWI�JSV�GSRXMRYSYW�JYRG�XMSRW� ,S[IZIV� [I�GER�HI½RI�E� VIPEXIH�GSQFMREXMSR�SJ�GSQTPI\�I\TSRIRXMEPW[LMGL�GSRZIVKIW YRMJSVQP] JSV�EPP�GSRXMRYSYW�JYRGXMSRW�
� ;I�GSRWXVYGX�WYGL�ER �n I\TPMGMXP]�EW�E�WSVX�SJ�EZIVEKI�SZIV�XLI�*SYVMIV�WIVMIW�
�n(�) =n�
k=�n
n + 1 � |k|n + 1
f̂kk� =
1
2�
� �
��f(t)Kn(� � t) t
[LIVI
Kn(�) =n�
k=�n
n + 1 � |k|n + 1
k�
� % WMQTPI�MRXIKVEXMSR�ZEVMEFPI�XVERWJSVQEXMSR�WLS[W�XLEX
�n(�) =1
2�
� �
��f(t)Kn(� � t) t =
1
2�
� �
��f(� � t)Kn(t) t
�
� %WWYQI�XLEX f MW�GSRXMRYSYW� f � C[T]
� 6IGEPP� *SYVMIV�WIVMIW HS�RSX RIGIWWEVMP]�GSRZIVKI�TSMRX[MWI�JSV�GSRXMRYSYW�JYRG�XMSRW� ,S[IZIV� [I�GER�HI½RI�E� VIPEXIH�GSQFMREXMSR�SJ�GSQTPI\�I\TSRIRXMEPW[LMGL�GSRZIVKIW YRMJSVQP] JSV�EPP�GSRXMRYSYW�JYRGXMSRW�
� ;I�GSRWXVYGX�WYGL�ER �n I\TPMGMXP]�EW�E�WSVX�SJ�EZIVEKI�SZIV�XLI�*SYVMIV�WIVMIW�
�n(�) =n�
k=�n
n + 1 � |k|n + 1
f̂kk� =
1
2�
� �
��f(t)Kn(� � t) t
[LIVI
Kn(�) =n�
k=�n
n + 1 � |k|n + 1
k�
� % WMQTPI�MRXIKVEXMSR�ZEVMEFPI�XVERWJSVQEXMSR�WLS[W�XLEX
�n(�) =1
2�
� �
��f(t)Kn(� � t) t =
1
2�
� �
��f(� � t)Kn(t) t
-3 -2 -1 1 2 3
5
10
15
20
� Kn(�) MW TSWMXMZI ERH PSGEPM^IH RIEV�^IVS�
n = 5, 10, 20
-3 -2 -1 1 2 3
5
10
15
20
� Kn(�) MW TSWMXMZI ERH PSGEPM^IH RIEV�^IVS�
n = 5, 10, 20
� -XW�RSVQ�MW ½\IH� 12�
� ��� Kn(�) � = 1
• We thus get
|sn(�) � f(�)| =
����1
2�
� �
��f(� � t)Kn(t) dt � 1
2�
� �
��f(�)Kn(t) dt
����
=1
2�
����� �
��(f(� � t) � f(�))Kn(t) dt
����
� 1
2�
�����
� �
��(f(� � t) � f(�))Kn(t) dt
����� +1
2�
�����
�� �
��+
� �
�
�(f(� � t) � f(�))Kn(t) dt
�����
• We thus get
|sn(�) � f(�)| =
����1
2�
� �
��f(� � t)Kn(t) dt � 1
2�
� �
��f(�)Kn(t) dt
����
=1
2�
����� �
��(f(� � t) � f(�))Kn(t) dt
����
� 1
2�
�����
� �
��(f(� � t) � f(�))Kn(t) dt
����� +1
2�
�����
�� �
��+
� �
�
�(f(� � t) � f(�))Kn(t) dt
�����
• We thus get
|sn(�) � f(�)| =
����1
2�
� �
��f(� � t)Kn(t) dt � 1
2�
� �
��f(�)Kn(t) dt
����
=1
2�
����� �
��(f(� � t) � f(�))Kn(t) dt
����
� 1
2�
�����
� �
��(f(� � t) � f(�))Kn(t) dt
����� +1
2�
�����
�� �
��+
� �
�
�(f(� � t) � f(�))Kn(t) dt
�����
• We thus get
|sn(�) � f(�)| =
����1
2�
� �
��f(� � t)Kn(t) dt � 1
2�
� �
��f(�)Kn(t) dt
����
=1
2�
����� �
��(f(� � t) � f(�))Kn(t) dt
����
� 1
2�
�����
� �
��(f(� � t) � f(�))Kn(t) dt
����� +1
2�
�����
�� �
��+
� �
�
�(f(� � t) � f(�))Kn(t) dt
�����
Small because f is continuous Small because Kn is isolated
• We thus get
|sn(�) � f(�)| =
����1
2�
� �
��f(� � t)Kn(t) dt � 1
2�
� �
��f(�)Kn(t) dt
����
=1
2�
����� �
��(f(� � t) � f(�))Kn(t) dt
����
� 1
2�
�����
� �
��(f(� � t) � f(�))Kn(t) dt
����� +1
2�
�����
�� �
��+
� �
�
�(f(� � t) � f(�))Kn(t) dt
�����
Small because f is continuous Small because Kn is isolated
• Uniform convergence implies convergence in 2-norm
� ;LEX�EFSYX�KIRIVEP 2[T] JYRGXMSRW#
� %R] 2[T] JYRGXMSR�GER�FI�ETTVS\MQEXIH�MR�RSVQ�F]�E GSRXMRYSYW�JYRGXMSR
� 8LYW� [I�½VWX� ETTVS\MQEXI f � 2[T] F]� E� GSRXMRYSYW� JYRGXMSR f̃ � [LMGL�[IETTVS\MQEXI�F] sn(�)
� 8LI�*SYVMIV�WIVMIW�GSRZIVKIW�MR�RSVQ�JSV�EPP f � 2[T]�
�
� ;I�LEZI�WLS[R�XLEX�XLIVI�I\MWXW�ER sn(�) = As WYGL�XLEX
�f � As�2 < �
� 8LI�*SYVMIV�WIVMIW�MW�XLI�YRMUYI�ZIGXSV c XLEX�QMRMQM^IW�XLI�RSVQ� XLIVIJSVI�
�f � Ac�2 � �f � As�2 < �
� 8LI�*SYVMIV�WIVMIW�GSRZIVKIW�MR�RSVQ�JSV�EPP f � 2[T]�
�
� ;I�LEZI�WLS[R�XLEX�XLIVI�I\MWXW�ER sn(�) = As WYGL�XLEX
�f � As�2 < �
� 8LI�*SYVMIV�WIVMIW�MW�XLI�YRMUYI�ZIGXSV c XLEX�QMRMQM^IW�XLI�RSVQ� XLIVIJSVI�
�f � Ac�2 � �f � As�2 < �
� 8LI�*SYVMIV�WIVMIW�GSRZIVKIW�TSMRX[MWI�JSV�EPP f � C2[T]�
�
� ;I�LEZI�WLS[R�MR�XLI�½VWX�PIGXYVI�XLEX�XLI�*SYVMIV�WIVMIW�GSRZIVKIW�YRMJSVQP]�XSE�GSRXMRYSYW�JYRGXMSR f̃
� 8LI�TVIZMSYW�XLISVIQ�WLS[W�XLEX���f � f̃
���2
= 0
� 7MRGI f ERH f̃ EVI�FSXL�GSRXMRYSYW� XLMW�MQTPMIW�XLEX f = f̃
� 8LI�*SYVMIV�WIVMIW�GSRZIVKIW�TSMRX[MWI�JSV�EPP f � C2[T]�
�
� ;I�LEZI�WLS[R�MR�XLI�½VWX�PIGXYVI�XLEX�XLI�*SYVMIV�WIVMIW�GSRZIVKIW�YRMJSVQP]�XSE�GSRXMRYSYW�JYRGXMSR f̃
� 8LI�TVIZMSYW�XLISVIQ�WLS[W�XLEX���f � f̃
���2
= 0
� 7MRGI f ERH f̃ EVI�FSXL�GSRXMRYSYW� XLMW�MQTPMIW�XLEX f = f̃
Discrete inner product
-3 -2 -1 0 1 2 3
� ;I�LEZI�WLS[R�XLEX�XLI�GSQTPI\�I\TSRIRXMEPW�EVI�SVXLSKSREP�[MXL�VIWTIGX�XS�XLIMRRIV�TVSHYGX
�f, g� =1
2�
� �
��f̄(�)g(�) �
� % VIQEVOEFPI�JEGX�MW�XLEX�XLI]�EVI�EPWS�SVXLSKSREP�[MXL�VIWTIGX�XS�XLI�JSPPS[MRKHMWGVIXI�WIQM�MRRIV�TVSHYGX�
�f, g�m =1
m
m�
j=1
f̄(�j)g(�j) = f(�)�g(�)
[LIVI � = (�1, . . . , �m) EVI�IZIRP]�WTEGIH�TSMRXW�
� =
���,
�1
m� 1
��, . . . ,
�1 � 1
m
��
�=
� 8LMW�SFWIVZEXMSR�JSVQW�XLI�FEWMW�SJ RYQIVMEP�*SYVMIV�WIVMIW
� -R�TEVXMGYPEV� MX�[MPP�JSVQ�XLI�FEWMW�SJ�XLI JEWX�*SYVMIV�XVERWJSVQ� EVKYEFP]�XLI�QSWXMQTSVXERX�RYQIVMGEP�EPKSVMXLQ�MR�I\MWXIRGI
Evenly spaced points on the unit circle
�
ei✓
z
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
-3 -2 -1 0 1 2 3
Some identities (shown for even m):
z
m�
j=1
ei�j =m�
j=1
zj = 0-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Some identities (shown for even m):
z
m�
j=1
ei2�j =m�
j=1
z2j = 0
-1.0 -0.5 0.5 1.0
-0.5
0.5
�
m�
j=1
k�j = (�)km JSV k = . . . , �2m, �m, 0, m, 2m. . .
m�
j=1
k�j = 0 JSV�EPP�SXLIV�MRXIKIV k
� 2SXI�XLEXzj = � 2� (j�1)/m = ��j�1
JSV � = 11/m = 2� /m
� 8LIVIJSVI�m�
j=1
zkj = (�)k
m�1�
j=0
�kj
� -J k = �m MW�E�QYPXMTPI�SJ m� XLIR�[I�LEZI
m�1�
j=0
�kj =m�1�
j=0
(�m)�j =m�1�
j=0
1�j = m
� 2SXI�XLEXzj = � 2� (j�1)/m = ��j�1
JSV � = 11/m = 2� /m
� 8LIVIJSVI�m�
j=1
zkj = (�)k
m�1�
j=0
�kj
� -J k = �m MW�E�QYPXMTPI�SJ m� XLIR�[I�LEZI
m�1�
j=0
�kj =m�1�
j=0
(�m)�j =m�1�
j=0
1�j = m
� 2SXI�XLEXzj = � 2� (j�1)/m = ��j�1
JSV � = 11/m = 2� /m
� 8LIVIJSVI�m�
j=1
zkj = (�)k
m�1�
j=0
�kj
� -J k = �m MW�E�QYPXMTPI�SJ m� XLIR�[I�LEZI
m�1�
j=0
�kj =m�1�
j=0
(�m)�j =m�1�
j=0
1�j = m
� 2SXI�XLEXzj = � 2� (j�1)/m = ��j�1
JSV � = 11/m = 2� /m
� 8LIVIJSVI�m�
j=1
zkj = (�)k
m�1�
j=0
�kj
� -J k = �m MW�E�QYPXMTPI�SJ m� XLIR�[I�LEZI
m�1�
j=0
�kj =m�1�
j=0
(�m)�j =m�1�
j=0
1�j = m
� *SV k RSX�E�QYPXMTPI�SJ m� VIGEPP�XLI�KISQIXVMG�WYQ�JSVQYPE�
m�1�
j=0
zj =zm � 1
z � 1
� 8LYWm�1�
j=0
�kj =m�1�
j=0
(�k)j =�km � 1
�k � 1
� &IGEYWI k MW�RSX�E�QYPXMTPI�SJ m� XLI�HIRSQMREXSV�MW�RSR^IVS
� &IGEYWI �km = (�m)k = 1k � XLI�RYQIVEXSV�MW�^IVS
� 2SXI�XLEXzj = � 2� (j�1)/m = ��j�1
JSV � = 11/m = 2� /m
� 8LIVIJSVI�m�
j=1
zkj = (�)k
m�1�
j=0
�kj
� -J k = �m MW�E�QYPXMTPI�SJ m� XLIR�[I�LEZI
m�1�
j=0
�kj =m�1�
j=0
(�m)�j =m�1�
j=0
1�j = m
� *SV k RSX�E�QYPXMTPI�SJ m� VIGEPP�XLI�KISQIXVMG�WYQ�JSVQYPE�
m�1�
j=0
zj =zm � 1
z � 1
� 8LYWm�1�
j=0
�kj =m�1�
j=0
(�k)j =�km � 1
�k � 1
� &IGEYWI k MW�RSX�E�QYPXMTPI�SJ m� XLI�HIRSQMREXSV�MW�RSR^IVS
� &IGEYWI �km = (�m)k = 1k � XLI�RYQIVEXSV�MW�^IVS
� 2SXI�XLEXzj = � 2� (j�1)/m = ��j�1
JSV � = 11/m = 2� /m
� 8LIVIJSVI�m�
j=1
zkj = (�)k
m�1�
j=0
�kj
� -J k = �m MW�E�QYPXMTPI�SJ m� XLIR�[I�LEZI
m�1�
j=0
�kj =m�1�
j=0
(�m)�j =m�1�
j=0
1�j = m
� *SV k RSX�E�QYPXMTPI�SJ m� VIGEPP�XLI�KISQIXVMG�WYQ�JSVQYPE�
m�1�
j=0
zj =zm � 1
z � 1
� 8LYWm�1�
j=0
�kj =m�1�
j=0
(�k)j =�km � 1
�k � 1
� &IGEYWI k MW�RSX�E�QYPXMTPI�SJ m� XLI�HIRSQMREXSV�MW�RSR^IVS
� &IGEYWI �km = (�m)k = 1k � XLI�RYQIVEXSV�MW�^IVS
��
k�, j��
m= (�1)j�k JSV j � k = . . . , �2m, �m, 0, m, 2m, . . .
�k�, j�
�m
= SXLIV[MWI
�
� *SPPS[W�JVSQ�TVIGIHMRK�XLISVIQ�
�k�, j�
�m
=1
m
m�
j=1
(j�k)�j
40
DFT
41
� ;I�LEZI�WLS[R�XLEX�XLI�GSQTPI\�I\TSRIRXMEPW
� ��, � (�+1)�, . . . , (��1)�, ��
EVI�SVXLSRSVQEP�[MXL�VIWTIGX�XS ��, ��m [LIRIZIV �m < � < � < m
� 8LYW�XLI�PIEWX�WUYEVIW�ETTVS\MQEXMSR�SJ�E�JYRGXMSR f MW
f ���
k=�
�k�, f
�m
k�
� ;LEX�WLSYPH � ERH � FI# *SV�VIEWSRW�[I�HMWGYWW�PEXIV� MX�QEOIW�WIRWI�XS�LEZII\EGXP] m GSIJ½GMIRXW� 3RI�GLSMGI�MW
� = 0, � = m � 1
[LMGL�QEOIW RS�WIRWI �;L]#
� % QSVI�WIRWMFPI�GLSMGI�JSV m SHH�MW
� = �m � 1
2ERH � =
m � 1
2,
*SV m IZIR� = �m
2ERH � =
m
2� 1
41
� ;I�LEZI�WLS[R�XLEX�XLI�GSQTPI\�I\TSRIRXMEPW
� ��, � (�+1)�, . . . , (��1)�, ��
EVI�SVXLSRSVQEP�[MXL�VIWTIGX�XS ��, ��m [LIRIZIV �m < � < � < m
� 8LYW�XLI�PIEWX�WUYEVIW�ETTVS\MQEXMSR�SJ�E�JYRGXMSR f MW
f ���
k=�
�k�, f
�m
k�
� ;LEX�WLSYPH � ERH � FI# *SV�VIEWSRW�[I�HMWGYWW�PEXIV� MX�QEOIW�WIRWI�XS�LEZII\EGXP] m GSIJ½GMIRXW� 3RI�GLSMGI�MW
� = 0, � = m � 1
[LMGL�QEOIW RS�WIRWI �;L]#
� % QSVI�WIRWMFPI�GLSMGI�JSV m SHH�MW
� = �m � 1
2ERH � =
m � 1
2,
*SV m IZIR� = �m
2ERH � =
m
2� 1
41
� ;I�LEZI�WLS[R�XLEX�XLI�GSQTPI\�I\TSRIRXMEPW
� ��, � (�+1)�, . . . , (��1)�, ��
EVI�SVXLSRSVQEP�[MXL�VIWTIGX�XS ��, ��m [LIRIZIV �m < � < � < m
� 8LYW�XLI�PIEWX�WUYEVIW�ETTVS\MQEXMSR�SJ�E�JYRGXMSR f MW
f ���
k=�
�k�, f
�m
k�
� ;LEX�WLSYPH � ERH � FI# *SV�VIEWSRW�[I�HMWGYWW�PEXIV� MX�QEOIW�WIRWI�XS�LEZII\EGXP] m GSIJ½GMIRXW� 3RI�GLSMGI�MW
� = 0, � = m � 1
[LMGL�QEOIW RS�WIRWI �;L]#
� % QSVI�WIRWMFPI�GLSMGI�JSV m SHH�MW
� = �m � 1
2ERH � =
m � 1
2,
*SV m IZIR� = �m
2ERH � =
m
2� 1
41
� ;I�LEZI�WLS[R�XLEX�XLI�GSQTPI\�I\TSRIRXMEPW
� ��, � (�+1)�, . . . , (��1)�, ��
EVI�SVXLSRSVQEP�[MXL�VIWTIGX�XS ��, ��m [LIRIZIV �m < � < � < m
� 8LYW�XLI�PIEWX�WUYEVIW�ETTVS\MQEXMSR�SJ�E�JYRGXMSR f MW
f ���
k=�
�k�, f
�m
k�
� ;LEX�WLSYPH � ERH � FI# *SV�VIEWSRW�[I�HMWGYWW�PEXIV� MX�QEOIW�WIRWI�XS�LEZII\EGXP] m GSIJ½GMIRXW� 3RI�GLSMGI�MW
� = 0, � = m � 1
[LMGL�QEOIW RS�WIRWI �;L]#
� % QSVI�WIRWMFPI�GLSMGI�JSV m SHH�MW
� = �m � 1
2ERH � =
m � 1
2,
*SV m IZIR� = �m
2ERH � =
m
2� 1