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My Thesis

main.tex target

preamble.tex

references.bib

01-introduction.tex

02-background.tex

03-method.tex

04-results.tex

05-appendix.tex

main.tex

1\documentclass[12pt, oneside]{report}
2
3% — Encoding & fonts
4\usepackage[utf8]{inputenc}
5\usepackage[T1]{fontenc}
6
7% — Math environments
8\usepackage{amsmath, amssymb, amsthm}
9\newtheorem{theorem}{Theorem}[section]
10
11
12% — Layout & bibliography
13\usepackage[margin=1in]{geometry}
14\usepackage{graphicx, booktabs}
15\usepackage[numbers]{natbib}
16\usepackage{hyperref}
17
18\title{On the Convergence of Fourier Series}
19\author{Y. Mathematician}
20\date{\today}
21
22\begin{document}
23\maketitle
24\tableofcontents
25
26\input{chapters/01-introduction}
27\input{chapters/02-background}
28\input{chapters/03-method}
29\input{chapters/04-results}
30\bibliography{references}

my-thesis-main.pdf

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3.2 Convergence of Fourier Series

Let f : [−π, π] → ℝ be a square-integrable function. We denote its k-th Fourier coefficient by c_k and define the partial sum S_n(x). The following classical result establishes convergence in the L² sense.

S_n(x) = Σ_k=−nⁿ c_k e^ikx

(3.14)

Theorem 3.2 (Parseval's Identity). Let f ∈ L²([−π, π]) with Fourier coefficients {c_k}. Then the sum of squared coefficients converges, and equals the squared L² norm of f:

‖f‖₂² = Σ_k=−∞^∞ |c_k|²

(3.15)

Proof. By linearity of the inner product and orthogonality of the basis {e^ikx}, the residual satisfies ‖S_n − f‖₂² = Σ_|k|>n |c_k|², which tends to 0 as n → ∞. ▪

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my-thesis

main.tex

preamble.tex

references.bib

chapters

main.tex

\documentclass[12pt]{report}
\usepackage{amsmath, amsthm}
\newtheorem{theorem}{Theorem}
% — Math environments
\begin{theorem}
  Let $f \in L^2([-\pi, \pi])$.
\end{theorem}

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3.2 Convergence

Theorem 3.2 (Parseval). Let f ∈ L²([−π, π]). Then the Fourier coefficients {cₖ} satisfy

‖f‖₂² = Σₖ |cₖ|²

Proof. By orthogonality of the basis {eⁱᵏˣ}, the residual ‖S_n − f‖₂² collapses into the tail Σ_|k|>n |c_k|², which tends to zero. ▪

The result extends the classical Plancherel identity to the unit circle and gives a constructive convergence rate in the smooth case.

3.3 Pointwise convergence

If f is piecewise C¹ on [−π, π] and 2π-periodic, the partial sums S_n(x) converge to f(x) at every point of continuity.

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Euler AI reading theorem 3.2· chapter-03.tex

Parseval's Identity says ‖f‖₂² = Σ|cₖ|². You rely on this at eq. (3.15). I noticed \emph inside your theorem — it toggles italics instead of forcing them.

Replace \emph with \textit

Keep

Let $f \in L^2([-\pi, \pi])$. Then the Fourier

- series of \emph{f} converges to \emph{f}.

+ series of \textit{f} converges to \textit{f}.

\end{theorem}

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