fourierd - Mathbox for Glauco Siliprandi

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Theorem fourierd 42597

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 42601. Notice that for a piecewise smooth function, the left and right limits always exist, see fourier2 42602 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 42607. (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 2977	. . . 4 ⊢ Ⅎ𝑘(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))
15		nfmpt1 5150	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
16	12, 15	nfcxfr 2975	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
17		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑛𝑘
18	16, 17	nffv 6666	. . . . . 6 ⊢ Ⅎ𝑛(𝐴‘𝑘)
19		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑛 ·
20		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑛(cos‘(𝑘 · 𝑋))
21	18, 19, 20	nfov 7172	. . . . 5 ⊢ Ⅎ𝑛((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))
22		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑛 +
23		nfmpt1 5150	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
24	13, 23	nfcxfr 2975	. . . . . . 7 ⊢ Ⅎ𝑛𝐵
25	24, 17	nffv 6666	. . . . . 6 ⊢ Ⅎ𝑛(𝐵‘𝑘)
26		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑛(sin‘(𝑘 · 𝑋))
27	25, 19, 26	nfov 7172	. . . . 5 ⊢ Ⅎ𝑛((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))
28	21, 22, 27	nfov 7172	. . . 4 ⊢ Ⅎ𝑛(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
29		fveq2 6656	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘))
30		oveq1 7149	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
31	30	fveq2d 6660	. . . . . 6 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
32	29, 31	oveq12d 7160	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))))
33		fveq2 6656	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
34	30	fveq2d 6660	. . . . . 6 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
35	33, 34	oveq12d 7160	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
36	32, 35	oveq12d 7160	. . . 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 5153	. . 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 42596	. 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 498	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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3921 ∅c0 4279 class class class wbr 5052 ↦ cmpt 5132 dom cdm 5541 ↾ cres 5543 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 Fincfn 8495 ℂcc 10521 ℝcr 10522 0cc0 10523 1c1 10524 + caddc 10526 · cmul 10528 +∞cpnf 10658 -∞cmnf 10659 − cmin 10856 -cneg 10857 / cdiv 11283 ℕcn 11624 2c2 11679 ℕ₀cn0 11884 (,)cioo 12725 (,]cioc 12726 [,)cico 12727 seqcseq 13359 ⇝ cli 14826 Σcsu 15027 sincsin 15402 cosccos 15403 πcpi 15405 –cn→ccncf 23467 ∫citg 24202 lim_ℂ climc 24445 D cdv 24446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cc 9843 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-symdif 4207 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-disj 5018 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-ofr 7396 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-omul 8093 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-fi 8861 df-sup 8892 df-inf 8893 df-oi 8960 df-dju 9316 df-card 9354 df-acn 9357 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-xnn0 11955 df-z 11969 df-dec 12086 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ioo 12729 df-ioc 12730 df-ico 12731 df-icc 12732 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14411 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-limsup 14813 df-clim 14830 df-rlim 14831 df-sum 15028 df-ef 15406 df-sin 15408 df-cos 15409 df-pi 15411 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-hom 16572 df-cco 16573 df-rest 16679 df-topn 16680 df-0g 16698 df-gsum 16699 df-topgen 16700 df-pt 16701 df-prds 16704 df-xrs 16758 df-qtop 16763 df-imas 16764 df-xps 16766 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-mulg 18208 df-cntz 18430 df-cmn 18891 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-fbas 20525 df-fg 20526 df-cnfld 20529 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-lp 21727 df-perf 21728 df-cn 21818 df-cnp 21819 df-t1 21905 df-haus 21906 df-cmp 21978 df-tx 22153 df-hmeo 22346 df-fil 22437 df-fm 22529 df-flim 22530 df-flf 22531 df-xms 22913 df-ms 22914 df-tms 22915 df-cncf 23469 df-ovol 24048 df-vol 24049 df-mbf 24203 df-itg1 24204 df-itg2 24205 df-ibl 24206 df-itg 24207 df-0p 24254 df-ditg 24430 df-limc 24449 df-dv 24450

This theorem is referenced by: fourier 42600 fouriercnp 42601 fourier2 42602

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