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Theorem fourierd 38198
Description: Fourier series convergence for periodic, piecewise smooth functions. The series converges to the average value of the left and the right limit of the function. Thus, if the function is continuous at a given point, the series converges exactly to the function value, see fouriercnp 38202. Notice that for a piecewise smooth function, the left and right limits always exist, see fourier2 38203 for an alternative form of the theorem that makes this fact explicit. When the first derivative is continuous, a simpler version of the theorem can be stated, see fouriercn 38208. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierd.f  |-  ( ph  ->  F : RR --> RR )
fourierd.t  |-  T  =  ( 2  x.  pi )
fourierd.per  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierd.g  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
fourierd.dmdv  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
fourierd.dvcn  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
fourierd.rlim  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
fourierd.llim  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
fourierd.x  |-  ( ph  ->  X  e.  RR )
fourierd.l  |-  ( ph  ->  L  e.  ( ( F  |`  ( -oo (,) X ) ) lim CC  X ) )
fourierd.r  |-  ( ph  ->  R  e.  ( ( F  |`  ( X (,) +oo ) ) lim CC  X ) )
fourierd.a  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
fourierd.b  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
Assertion
Ref Expression
fourierd  |-  ( ph  ->  ( ( ( A `
 0 )  / 
2 )  +  sum_ n  e.  NN  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) )  =  ( ( L  +  R
)  /  2 ) )
Distinct variable groups:    n, F, x    x, G    x, T    x, X, n    ph, x
Allowed substitution hints:    ph( n)    A( x, n)    B( x, n)    R( x, n)    T( n)    G( n)    L( x, n)

Proof of Theorem fourierd
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fourierd.f . . 3  |-  ( ph  ->  F : RR --> RR )
2 fourierd.t . . 3  |-  T  =  ( 2  x.  pi )
3 fourierd.per . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
4 fourierd.g . . 3  |-  G  =  ( ( RR  _D  F )  |`  ( -u pi (,) pi ) )
5 fourierd.dmdv . . 3  |-  ( ph  ->  ( ( -u pi (,) pi )  \  dom  G )  e.  Fin )
6 fourierd.dvcn . . 3  |-  ( ph  ->  G  e.  ( dom 
G -cn-> CC ) )
7 fourierd.rlim . . 3  |-  ( (
ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  G ) )  ->  (
( G  |`  (
x (,) +oo )
) lim CC  x )  =/=  (/) )
8 fourierd.llim . . 3  |-  ( (
ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  G ) )  ->  (
( G  |`  ( -oo (,) x ) ) lim
CC  x )  =/=  (/) )
9 fourierd.x . . 3  |-  ( ph  ->  X  e.  RR )
10 fourierd.l . . 3  |-  ( ph  ->  L  e.  ( ( F  |`  ( -oo (,) X ) ) lim CC  X ) )
11 fourierd.r . . 3  |-  ( ph  ->  R  e.  ( ( F  |`  ( X (,) +oo ) ) lim CC  X ) )
12 fourierd.a . . 3  |-  A  =  ( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
13 fourierd.b . . 3  |-  B  =  ( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
14 nfcv 2612 . . . 4  |-  F/_ k
( ( ( A `
 n )  x.  ( cos `  (
n  x.  X ) ) )  +  ( ( B `  n
)  x.  ( sin `  ( n  x.  X
) ) ) )
15 nfmpt1 4485 . . . . . . . 8  |-  F/_ n
( n  e.  NN0  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( cos `  (
n  x.  x ) ) )  _d x  /  pi ) )
1612, 15nfcxfr 2610 . . . . . . 7  |-  F/_ n A
17 nfcv 2612 . . . . . . 7  |-  F/_ n
k
1816, 17nffv 5886 . . . . . 6  |-  F/_ n
( A `  k
)
19 nfcv 2612 . . . . . 6  |-  F/_ n  x.
20 nfcv 2612 . . . . . 6  |-  F/_ n
( cos `  (
k  x.  X ) )
2118, 19, 20nfov 6334 . . . . 5  |-  F/_ n
( ( A `  k )  x.  ( cos `  ( k  x.  X ) ) )
22 nfcv 2612 . . . . 5  |-  F/_ n  +
23 nfmpt1 4485 . . . . . . . 8  |-  F/_ n
( n  e.  NN  |->  ( S. ( -u pi (,) pi ) ( ( F `  x )  x.  ( sin `  (
n  x.  x ) ) )  _d x  /  pi ) )
2413, 23nfcxfr 2610 . . . . . . 7  |-  F/_ n B
2524, 17nffv 5886 . . . . . 6  |-  F/_ n
( B `  k
)
26 nfcv 2612 . . . . . 6  |-  F/_ n
( sin `  (
k  x.  X ) )
2725, 19, 26nfov 6334 . . . . 5  |-  F/_ n
( ( B `  k )  x.  ( sin `  ( k  x.  X ) ) )
2821, 22, 27nfov 6334 . . . 4  |-  F/_ n
( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) )
29 fveq2 5879 . . . . . 6  |-  ( n  =  k  ->  ( A `  n )  =  ( A `  k ) )
30 oveq1 6315 . . . . . . 7  |-  ( n  =  k  ->  (
n  x.  X )  =  ( k  x.  X ) )
3130fveq2d 5883 . . . . . 6  |-  ( n  =  k  ->  ( cos `  ( n  x.  X ) )  =  ( cos `  (
k  x.  X ) ) )
3229, 31oveq12d 6326 . . . . 5  |-  ( n  =  k  ->  (
( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  =  ( ( A `  k )  x.  ( cos `  ( k  x.  X ) ) ) )
33 fveq2 5879 . . . . . 6  |-  ( n  =  k  ->  ( B `  n )  =  ( B `  k ) )
3430fveq2d 5883 . . . . . 6  |-  ( n  =  k  ->  ( sin `  ( n  x.  X ) )  =  ( sin `  (
k  x.  X ) ) )
3533, 34oveq12d 6326 . . . . 5  |-  ( n  =  k  ->  (
( B `  n
)  x.  ( sin `  ( n  x.  X
) ) )  =  ( ( B `  k )  x.  ( sin `  ( k  x.  X ) ) ) )
3632, 35oveq12d 6326 . . . 4  |-  ( n  =  k  ->  (
( ( A `  n )  x.  ( cos `  ( n  x.  X ) ) )  +  ( ( B `
 n )  x.  ( sin `  (
n  x.  X ) ) ) )  =  ( ( ( A `
 k )  x.  ( cos `  (
k  x.  X ) ) )  +  ( ( B `  k
)  x.  ( sin `  ( k  x.  X
) ) ) ) )
3714, 28, 36cbvmpt 4487 . . 3  |-  ( n  e.  NN  |->  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) )  =  ( k  e.  NN  |->  ( ( ( A `  k )  x.  ( cos `  ( k  x.  X ) ) )  +  ( ( B `
 k )  x.  ( sin `  (
k  x.  X ) ) ) ) )
381, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37fourierdlem115 38197 . 2  |-  ( ph  ->  (  seq 1 (  +  ,  ( n  e.  NN  |->  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) ) )  ~~>  ( ( ( L  +  R
)  /  2 )  -  ( ( A `
 0 )  / 
2 ) )  /\  ( ( ( A `
 0 )  / 
2 )  +  sum_ n  e.  NN  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) )  =  ( ( L  +  R
)  /  2 ) ) )
3938simprd 470 1  |-  ( ph  ->  ( ( ( A `
 0 )  / 
2 )  +  sum_ n  e.  NN  ( ( ( A `  n
)  x.  ( cos `  ( n  x.  X
) ) )  +  ( ( B `  n )  x.  ( sin `  ( n  x.  X ) ) ) ) )  =  ( ( L  +  R
)  /  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387   (/)c0 3722   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690   -oocmnf 9691    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   (,)cioo 11660   (,]cioc 11661   [,)cico 11662    seqcseq 12251    ~~> cli 13625   sum_csu 13829   sincsin 14193   cosccos 14194   picpi 14196   -cn->ccncf 21986   S.citg 22655   lim CC climc 22896    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-t1 20407  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-ditg 22881  df-limc 22900  df-dv 22901
This theorem is referenced by:  fourier  38201  fouriercnp  38202  fourier2  38203
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