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Meilleur makefile, Weierstrass et Fejer
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@FliiFe
FliiFe committed Dec 29, 2021
1 parent a00ae7b commit cc96817
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63 changes: 16 additions & 47 deletions Makefile
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Original file line number Diff line number Diff line change
@@ -1,9 +1,6 @@
INTEGRATION_FN := integration
ALGEBRE_FN := algebre
TOPOLOGIE_FN := topologie
PREMASTER_FN := premaster
mats := integration algebre topologie premaster

.PHONY: clean all figures help integration newchapter init release
.PHONY: clean all figures help newchapter init release $(mats)

BLACK := $(shell tput -Txterm setaf 0)
RED := $(shell tput -Txterm setaf 1)
Expand All @@ -23,22 +20,18 @@ ifndef VERBOSE
SILENT := -silent
endif
LATEXMK := latexmk $(SILENT)
INTEGRATION_TARGET := target/$(INTEGRATION_FN).pdf
ALGEBRE_TARGET := target/$(ALGEBRE_FN).pdf
TOPOLOGIE_TARGET := target/$(TOPOLOGIE_FN).pdf
PREMASTER_TARGET := target/$(PREMASTER_FN).pdf

help: ## Print available targets
@echo "${PURPLE}:: ${BOLD}${GREEN}$$(basename $$(pwd))${RESET} ${PURPLE}::${RESET}"
@echo ""
@echo "Example:"
@echo "Usage:"
@echo " | make all -j8 ${YELLOW}# Tout compiler avec 8 threads${RESET}"
@echo " | make integration ${YELLOW}# Cours d'intégration, théorie de la mesure et de probabilités${RESET}"
@echo " | make algebre ${YELLOW}# Cours d'algèbre ${RESET}"
@echo " | make topologie ${YELLOW}# Cours de topologie et de calcul différentiel${RESET}"
@echo " | make algebre ${YELLOW}# Cours d'algèbre${RESET}"
@echo ""
@echo "$(YELLOW)List of PHONY targets:$(RESET)"
@grep -E '^[a-zA-Z_0-9%-]+:.*?## .*$$' $(MAKEFILE_LIST) | sort | awk 'BEGIN {FS = ":.*?## "}; {printf " ${GREEN}${BOLD}%-22s${RESET} %s\n", $$1, $$2}'
@echo "$(YELLOW)List of courses:$(RESET)"
@echo " $(GREEN)$(BOLD)$(mats)$(RESET)"
@if ! [[ -z "$(FIGURES)" ]]; then echo "$(YELLOW)List of figures:$(RESET)"; fi
@for f in $(FIGURES); do \
echo " " $${f}; \
Expand Down Expand Up @@ -72,54 +65,30 @@ newchapter: ## Make a new chapter
release: all ## Create a new release
@gh release create "$$(date +"%Y-%m-%d-%H-%M")" target/* -t "$$(date +"%d-%m-%Y %H:%M")" -n ''

integration: $(INTEGRATION_TARGET) ## Cours d'intégration, théorie de la mesure et de probabilités
algebre: $(ALGEBRE_TARGET) ## Cours d'algèbre
topologie: $(TOPOLOGIE_TARGET) ## Cours de topologie et de calcul différentiel
premaster: $(PREMASTER_TARGET) ## Cours de prémaster
clean: ## Delete compiled documents and latex-generated files
rm -rf target build
cd src/figures && latexmk -C || true
mkdir build
ln -s ../indexstyle.ist build/

all: $(PDFS) ## Build everything

figures: $(FIGURES) ## Build every figure in src/figures/

src/figures/%.pdf: src/figures/%.tex
@echo "$(GREEN)Compiling figure $(YELLOW)$(<F)$(GREEN) into $(YELLOW)$@$(RESET)"
cd src/figures/ && $(LATEXMK) $(<F) && latexmk -silent -c $(<F)

$(INTEGRATION_TARGET): $(INTEGRATION_FN).tex src/$(INTEGRATION_FN)-*.tex src/preamble.tex src/preamble/*.tex $(FIGURES)
@echo "$(GREEN)Compiling document $(YELLOW)$(<F)$(GREEN) into $(YELLOW)$@$(RESET)"
$(LATEXMK) $<
cp build/$(@F) $@

$(ALGEBRE_TARGET): $(ALGEBRE_FN).tex src/$(ALGEBRE_FN)-*.tex src/preamble.tex src/preamble/*.tex $(FIGURES)
@echo "$(GREEN)Compiling document $(YELLOW)$(<F)$(GREEN) into $(YELLOW)$@$(RESET)"
$(LATEXMK) $<
cp build/$(@F) $@

$(TOPOLOGIE_TARGET): $(TOPOLOGIE_FN).tex src/$(TOPOLOGIE_FN)-*.tex src/preamble.tex src/preamble/*.tex $(FIGURES)
@echo "$(GREEN)Compiling document $(YELLOW)$(<F)$(GREEN) into $(YELLOW)$@$(RESET)"
$(LATEXMK) $<
cp build/$(@F) $@

$(PREMASTER_TARGET): $(PREMASTER_FN).tex src/$(PREMASTER_FN)-*.tex src/preamble.tex src/preamble/*.tex $(FIGURES)
@echo "$(GREEN)Compiling document $(YELLOW)$(<F)$(GREEN) into $(YELLOW)$@$(RESET)"
$(LATEXMK) $<
cp build/$(@F) $@

$(PDFS): | target

target:
mkdir target

clean: ## Delete compiled documents and latex-generated files
rm -rf target build
cd src/figures && latexmk -C || true
mkdir build
ln -s ../indexstyle.ist build/

figures: $(FIGURES) ## Build every figure in src/figures/

.SECONDEXPANSION:
PER := %
target/%.pdf: %.tex src/preamble.tex src/preamble/*.tex src/%.tex $$(patsubst $$(PER).tex,$$(PER).pdf,$$(wildcard src/figures/%-*.tex))
@echo "$(GREEN)Compiling $(YELLOW)$(<)$(GREEN) into $(YELLOW)$@$(RESET)"
target/%.pdf: %.tex src/preamble.tex src/preamble/*.tex $$(find src/%.tex,$$(wildcard src/*.tex)) $$(wildcard src/%-*.tex) $$(patsubst $$(PER).tex,$$(PER).pdf,$$(wildcard src/figures/%-*.tex))
@echo "$(GREEN)Compiling document $(YELLOW)$(<F)$(GREEN) into $(YELLOW)$@$(RESET)"
$(LATEXMK) $<
cp build/$(@F) $@

$(mats): target/$$@.pdf
70 changes: 69 additions & 1 deletion src/integration-05.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -326,8 +326,76 @@ \section{Approximation de l'unité}
de sorte que \[\|\varphi_n\star f-f\|_p^p\leq \frac{\epsilon^p}2+\int_{|y|>\delta}\varphi_n(y)\diff y (2\|f\|_p)^p\leq \epsilon^p\] pour $n$ assez grand.
\end{proof}

\begin{cor}
\begin{cor}[Weierstrass\index{Weierstrass (théorème)}]
Si $f:[a, b]\longrightarrow \C$ est continue, elle est limite uniforme de polynômes.
\end{cor}

% TODO: preuve du corrolaire

\begin{proof}
Soit \[
\varphi_n(x)=c_n(1-x^2 )^n\1_{|x|\leq 1}\geq 0 \qquad \text{ où } \qquad \frac1{c_n}=\int_{[-1,1]}(1-x^2 )^n\diff x.
\]
On a bien \[
\int_{|x|>\delta}\varphi_n\diff\lambda \xrightarrow[n\to+\infty]{}0.
\]
Supposons maintenant que $[a, b]\subset ]0,1[$ et $f:[a, b] \longrightarrow \C$ est continue. On prolonge $f$ en une fonction continue à support dans $[0,1]$. Par le théorème, $\varphi_n\star f \longrightarrow f$ uniformément sur le compact $[a, b]$. Or
\begin{align*}
\varphi_n\star f(x)&=\int_{\R} \varphi_n(x-y)f(y)\diff y\\ &=\int_{[0,1]}\varphi_n(x-y)f(y)\diff y\\&= c_n\int_{[0,1]}(1-(x-y)^2 )^n f(y)\diff y \1_{|x-y|\leq 1}
\end{align*}
et c'est un polynôme en $x$.
\end{proof}

\begin{thm}
L'ensemble $\mathcal C^\infty(\R^d, \C)$ est dense dans $\mathcal L^p(\R^d, \mathcal B(\R^d), \lambda)$ pour tout $p \in [1, \infty[$.
\end{thm}

\begin{proof}[Idée de preuve]
Choisir $\varphi_n$ approximation de $\delta_0$, $\mathcal C^\infty$ et utiliser le fait que \[
\forall \alpha, \quad \frac{\partial^\alpha}{\partial x^\alpha}(\varphi_n\star f)= \frac{\partial^\alpha \varphi_n}{\partial x^\alpha}\star f
\]
\end{proof}

\begin{thm}[Féjer\index{Féjer (théorème)}]
Soit $f:\R\longrightarrow \C$ une fonction continue $2\pi$-périodique. Soit \[
\hat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx}\diff x, \quad n \in \Z
\]
Alors \[
\frac{1}{N+1}\sum_{n=0}^N\sum_{k=-n}^n \hat{f}(k)e^{ikx} \xrightarrow[N\to +\infty]{}f(x)
\]
uniformément sur $[0, 2\pi]$ ou sur $\R$.
\end{thm}

\begin{proof}
On va se placer sur l'espace $\mathcal L^2([0,2\pi[, \frac\lambda{2\pi})$, qu'on voit comme contenant les fonctions continues $2\pi$-périodiques (en restreignant à une période). On adopte dans ce contexte la définition de la convolution suivante: \[
f\star g(x)= \frac{1}{2\pi}\int_0^{2\pi}f(y)g(x-y)\diff y
\]
On a \[
\sum_{k=-n}^n \hat{f}(k)e^{ikx} = \frac{1}{2\pi}\int_0^{2\pi} \sum_{k=-n}^n f(y)e^{ik(x-y)}\diff y=D_n\star f(x)
\]
où $D_n$ est le noyau de Dirichlet \[
D_n(x)= \frac{e^{i(n+1)x}-e^{-inx}}{e^{ix}-1}= \frac{\sin((n+\sfrac12)x)}{\sin(\sfrac x2)}
\]
Remarquons que \[
\frac{1}{2\pi}\int_0^{2\pi} D_n(x)\diff x=\sum_{k=-n}^n\frac{1}{2\pi}\int_0^{2\pi} e^{inx}\diff x=1
\]
On a alors \[
\frac{1}{N+1}\sum_{n=0}^N D_n\star f(x)=\frac{1}{2\pi}\int_0^{2\pi}\left(\frac{1}{N+1}\sum_{n=0}^ND_n(x-y)\right) f(y)\diff y=F_n\star f(x)
\]
où $F_n$ est le noyau de Féjer \[
F_N(x)=\frac{1}{N+1}\sum_{n=0}^N D_n(x)
\]
qui est tel que \[
\frac{1}{2\pi}\int_0^{2\pi}F_n\diff \lambda=1.
\]
Comme \begin{align*}
F_N(x) &= \frac{1}{(N+1)\sin(\sfrac{x}{2})} \sum_{n=0}^{N} \Im\left(e^{i(n+\sfrac12)x}\right)\\
&= \frac{1}{(N+1)\sin(\sfrac{x}{2})}\Im \left( \frac{e^{i(N+\sfrac32)x}-e^{i\sfrac{x}{2}}}{e^{ix}-1}\right),
\end{align*}
on a
\begin{align*}
F_N(x) &= \frac{1}{(N+1)\sin(\sfrac{x}{2})}\Im \left( \frac{e^{i(N+1)x}-1}{2i\sin(\sfrac{x}{2})}\right)\\
&= \frac{1}{(N+1)\sin(\sfrac{x}{2})^2}\Im \left( e^{i \sfrac{N+1}{2}x}\frac{e^{i \sfrac{N+1}{2}x}-e^{i \sfrac{N+1}{2}x}}{2i}\right)\\
&= \frac{\sin(\sfrac{N+1}{2})x}{(N+1)\sin(\sfrac{x}{2})^2}\geq 0
\end{align*}
\end{proof}

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