fourierd - Mathbox for Glauco Siliprandi

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Theorem fourierd 41008

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 41012. Notice that for a piecewise smooth function, the left and right limits always exist, see fourier2 41013 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 41018. (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 2906	. . . 4 ⊢ Ⅎ𝑘(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))
15		nfmpt1 4905	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
16	12, 15	nfcxfr 2904	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
17		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑛𝑘
18	16, 17	nffv 6384	. . . . . 6 ⊢ Ⅎ𝑛(𝐴‘𝑘)
19		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑛 ·
20		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑛(cos‘(𝑘 · 𝑋))
21	18, 19, 20	nfov 6871	. . . . 5 ⊢ Ⅎ𝑛((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋)))
22		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑛 +
23		nfmpt1 4905	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
24	13, 23	nfcxfr 2904	. . . . . . 7 ⊢ Ⅎ𝑛𝐵
25	24, 17	nffv 6384	. . . . . 6 ⊢ Ⅎ𝑛(𝐵‘𝑘)
26		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑛(sin‘(𝑘 · 𝑋))
27	25, 19, 26	nfov 6871	. . . . 5 ⊢ Ⅎ𝑛((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))
28	21, 22, 27	nfov 6871	. . . 4 ⊢ Ⅎ𝑛(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
29		fveq2 6374	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘))
30		oveq1 6848	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
31	30	fveq2d 6378	. . . . . 6 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
32	29, 31	oveq12d 6859	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))))
33		fveq2 6374	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
34	30	fveq2d 6378	. . . . . 6 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
35	33, 34	oveq12d 6859	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
36	32, 35	oveq12d 6859	. . . 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 4907	. . 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 41007	. 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 489	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 384 = wceq 1652 ∈ wcel 2155 ≠ wne 2936 ∖ cdif 3728 ∅c0 4078 class class class wbr 4808 ↦ cmpt 4887 dom cdm 5276 ↾ cres 5278 ⟶wf 6063 ‘cfv 6067 (class class class)co 6841 Fincfn 8159 ℂcc 10186 ℝcr 10187 0cc0 10188 1c1 10189 + caddc 10191 · cmul 10193 +∞cpnf 10324 -∞cmnf 10325 − cmin 10519 -cneg 10520 / cdiv 10937 ℕcn 11273 2c2 11326 ℕ₀cn0 11537 (,)cioo 12376 (,]cioc 12377 [,)cico 12378 seqcseq 13007 ⇝ cli 14501 Σcsu 14702 sincsin 15077 cosccos 15078 πcpi 15080 –cn→ccncf 22957 ∫citg 23675 lim_ℂ climc 23916 D cdv 23917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2069 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2349 ax-ext 2742 ax-rep 4929 ax-sep 4940 ax-nul 4948 ax-pow 5000 ax-pr 5061 ax-un 7146 ax-inf2 8752 ax-cc 9509 ax-cnex 10244 ax-resscn 10245 ax-1cn 10246 ax-icn 10247 ax-addcl 10248 ax-addrcl 10249 ax-mulcl 10250 ax-mulrcl 10251 ax-mulcom 10252 ax-addass 10253 ax-mulass 10254 ax-distr 10255 ax-i2m1 10256 ax-1ne0 10257 ax-1rid 10258 ax-rnegex 10259 ax-rrecex 10260 ax-cnre 10261 ax-pre-lttri 10262 ax-pre-lttrn 10263 ax-pre-ltadd 10264 ax-pre-mulgt0 10265 ax-pre-sup 10266 ax-addf 10267 ax-mulf 10268

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2062 df-mo 2564 df-eu 2581 df-clab 2751 df-cleq 2757 df-clel 2760 df-nfc 2895 df-ne 2937 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3351 df-sbc 3596 df-csb 3691 df-dif 3734 df-un 3736 df-in 3738 df-ss 3745 df-pss 3747 df-symdif 4004 df-nul 4079 df-if 4243 df-pw 4316 df-sn 4334 df-pr 4336 df-tp 4338 df-op 4340 df-uni 4594 df-int 4633 df-iun 4677 df-iin 4678 df-disj 4777 df-br 4809 df-opab 4871 df-mpt 4888 df-tr 4911 df-id 5184 df-eprel 5189 df-po 5197 df-so 5198 df-fr 5235 df-se 5236 df-we 5237 df-xp 5282 df-rel 5283 df-cnv 5284 df-co 5285 df-dm 5286 df-rn 5287 df-res 5288 df-ima 5289 df-pred 5864 df-ord 5910 df-on 5911 df-lim 5912 df-suc 5913 df-iota 6030 df-fun 6069 df-fn 6070 df-f 6071 df-f1 6072 df-fo 6073 df-f1o 6074 df-fv 6075 df-isom 6076 df-riota 6802 df-ov 6844 df-oprab 6845 df-mpt2 6846 df-of 7094 df-ofr 7095 df-om 7263 df-1st 7365 df-2nd 7366 df-supp 7497 df-wrecs 7609 df-recs 7671 df-rdg 7709 df-1o 7763 df-2o 7764 df-oadd 7767 df-omul 7768 df-er 7946 df-map 8061 df-pm 8062 df-ixp 8113 df-en 8160 df-dom 8161 df-sdom 8162 df-fin 8163 df-fsupp 8482 df-fi 8523 df-sup 8554 df-inf 8555 df-oi 8621 df-card 9015 df-acn 9018 df-cda 9242 df-pnf 10329 df-mnf 10330 df-xr 10331 df-ltxr 10332 df-le 10333 df-sub 10521 df-neg 10522 df-div 10938 df-nn 11274 df-2 11334 df-3 11335 df-4 11336 df-5 11337 df-6 11338 df-7 11339 df-8 11340 df-9 11341 df-n0 11538 df-xnn0 11610 df-z 11624 df-dec 11740 df-uz 11886 df-q 11989 df-rp 12028 df-xneg 12145 df-xadd 12146 df-xmul 12147 df-ioo 12380 df-ioc 12381 df-ico 12382 df-icc 12383 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-fac 13264 df-bc 13293 df-hash 13321 df-shft 14093 df-cj 14125 df-re 14126 df-im 14127 df-sqrt 14261 df-abs 14262 df-limsup 14488 df-clim 14505 df-rlim 14506 df-sum 14703 df-ef 15081 df-sin 15083 df-cos 15084 df-pi 15086 df-struct 16133 df-ndx 16134 df-slot 16135 df-base 16137 df-sets 16138 df-ress 16139 df-plusg 16228 df-mulr 16229 df-starv 16230 df-sca 16231 df-vsca 16232 df-ip 16233 df-tset 16234 df-ple 16235 df-ds 16237 df-unif 16238 df-hom 16239 df-cco 16240 df-rest 16350 df-topn 16351 df-0g 16369 df-gsum 16370 df-topgen 16371 df-pt 16372 df-prds 16375 df-xrs 16429 df-qtop 16434 df-imas 16435 df-xps 16437 df-mre 16513 df-mrc 16514 df-acs 16516 df-mgm 17509 df-sgrp 17551 df-mnd 17562 df-submnd 17603 df-mulg 17809 df-cntz 18014 df-cmn 18460 df-psmet 20010 df-xmet 20011 df-met 20012 df-bl 20013 df-mopn 20014 df-fbas 20015 df-fg 20016 df-cnfld 20019 df-top 20977 df-topon 20994 df-topsp 21016 df-bases 21029 df-cld 21102 df-ntr 21103 df-cls 21104 df-nei 21181 df-lp 21219 df-perf 21220 df-cn 21310 df-cnp 21311 df-t1 21397 df-haus 21398 df-cmp 21469 df-tx 21644 df-hmeo 21837 df-fil 21928 df-fm 22020 df-flim 22021 df-flf 22022 df-xms 22403 df-ms 22404 df-tms 22405 df-cncf 22959 df-ovol 23521 df-vol 23522 df-mbf 23676 df-itg1 23677 df-itg2 23678 df-ibl 23679 df-itg 23680 df-0p 23727 df-ditg 23901 df-limc 23920 df-dv 23921

This theorem is referenced by: fourier 41011 fouriercnp 41012 fourier2 41013

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