fourierd - Mathbox for Glauco Siliprandi

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Theorem fourierd 42864

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 42868. Notice that for a piecewise smooth function, the left and right limits always exist, see fourier2 42869 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 42874. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierd.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierd.t	⊢ 𝑇 = (2 · π)
fourierd.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierd.g	⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
fourierd.dmdv	⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
fourierd.dvcn	⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
fourierd.rlim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourierd.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourierd.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierd.l	⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierd.r	⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierd.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierd.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))

**Assertion**
Ref	Expression
fourierd	⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))

Distinct variable groups: 𝑛,𝐹,𝑥 𝑥,𝐺 𝑥,𝑇 𝑛,𝑋,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(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.

Step	Hyp	Ref	Expression
1		fourierd.f	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fourierd.t	. . 3 ⊢ 𝑇 = (2 · π)
3		fourierd.per	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
4		fourierd.g	. . 3 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
5		fourierd.dmdv	. . 3 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
6		fourierd.dvcn	. . 3 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
7		fourierd.rlim	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
8		fourierd.llim	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
9		fourierd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
10		fourierd.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
11		fourierd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
12		fourierd.a	. . 3 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
13		fourierd.b	. . 3 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
14		nfcv 2955	. . . 4 ⊢ Ⅎ𝑘(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))
15		nfmpt1 5128	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
16	12, 15	nfcxfr 2953	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
17		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑛𝑘
18	16, 17	nffv 6655	. . . . . 6 ⊢ Ⅎ𝑛(𝐴‘𝑘)
19		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑛 ·
20		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑛(cos‘(𝑘 · 𝑋))
21	18, 19, 20	nfov 7165	. . . . 5 ⊢ Ⅎ𝑛((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))
22		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑛 +
23		nfmpt1 5128	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
24	13, 23	nfcxfr 2953	. . . . . . 7 ⊢ Ⅎ𝑛𝐵
25	24, 17	nffv 6655	. . . . . 6 ⊢ Ⅎ𝑛(𝐵‘𝑘)
26		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑛(sin‘(𝑘 · 𝑋))
27	25, 19, 26	nfov 7165	. . . . 5 ⊢ Ⅎ𝑛((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))
28	21, 22, 27	nfov 7165	. . . 4 ⊢ Ⅎ𝑛(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
29		fveq2 6645	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘))
30		oveq1 7142	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
31	30	fveq2d 6649	. . . . . 6 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
32	29, 31	oveq12d 7153	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))))
33		fveq2 6645	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
34	30	fveq2d 6649	. . . . . 6 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
35	33, 34	oveq12d 7153	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
36	32, 35	oveq12d 7153	. . . 4 ⊢ (𝑛 = 𝑘 → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
37	14, 28, 36	cbvmpt 5131	. . 3 ⊢ (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑘 ∈ ℕ ↦ (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
38	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37	fourierdlem115 42863	. 2 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))
39	38	simprd 499	1 ⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 +∞cpnf 10661 -∞cmnf 10662 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 (,)cioo 12726 (,]cioc 12727 [,)cico 12728 seqcseq 13364 ⇝ cli 14833 Σcsu 15034 sincsin 15409 cosccos 15410 πcpi 15412 –cn→ccncf 23481 ∫citg 24222 lim_ℂ climc 24465 D cdv 24466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-symdif 4169 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-t1 21919 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-ibl 24226 df-itg 24227 df-0p 24274 df-ditg 24450 df-limc 24469 df-dv 24470

This theorem is referenced by: fourier 42867 fouriercnp 42868 fourier2 42869

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