2일전에 올라온 조회수 0짜리,

아무 대사도 없고 소리도 없고 자막도 없는,

그래프 값만 변화하는 영상 찾아서 설명해달라 함.


값의 변화에 따른 푸리에 변환 그래프를 걍 쭉 보여주는 31초짜리 영상이였는데, 

제목이 '푸리에 변환' 이여서 제목보고 때려맞출수도 있겠다 싶어서


20초부터 나오는 값에 대한 설명도 해달라고 덧붙였더니

20초의 값을 정확히 보고 프롬프트에 언급함... 미쳤네 진짜



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'푸리에 변환' 이라는 키워드로 찾은 2일전 영상.

내가 보기 전까지 '조회수 없음' 이였음.


https://www.youtube.com/watch?v=mCrCp4AFy8E

푸리에 급수푸리에 급수www.youtube.com



그리고 맨 밑에가 16.9초 동안 생각하고 내놓은 답.


다 읽어볼 필요는 없고, 영상의 내용과 그래프 결과에 대해서 제대로 된 답을 내놓음.

(그래프의 x,y축 해설, 임계근사값 (0.59) 추측 등)


그리고 20초 이후의 값에 대한 설명을 해달라는 말에 summary에서

20초부터 영상 끝까지의 값 (700~1000)을 짚어냄.


정확히는 20초 언저리에선 740 좀 넘은 정도긴 한데, 유의미한 수치는 아님.


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질문


영상의 내용과, 특히나 20초 이후로 나오는 값들에 대한 자세한 해설해줘


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Okay, let's break down the video and what's happening, especially after the 20-second mark.


**Overall Video Explanation:**


1.  **The Goal:** The video is visually demonstrating how a **Fourier series** approximates a **square wave**. A square wave is a function that abruptly jumps between two constant values (in this case, roughly +0.5 and -0.5).

2.  **The Approximation (Red Line):** The red, oscillating line represents the sum of a *finite* number of sine and cosine waves (the terms of the Fourier series). It's trying to "build" the shape of the square wave.

3.  **The Parameter `n`:** The text `n = [value, value]` (e.g., `n=[9,9]` at the start, `n=[1000,1000]` at the end) indicates the **number of terms** being used in the Fourier series approximation. As the video progresses, `n` increases, meaning more and more sine/cosine waves (with higher frequencies) are being added together.

4.  **The Observation:**

    *   Initially (low `n`), the approximation is quite rough, with large, slow oscillations.

    *   As `n` increases, the approximation gets much better in the "flat" parts of the square wave (between 0 and 1, and between 1 and 2). The red line becomes flatter and closer to the +0.5 and -0.5 levels.

    *   *However*, near the points where the square wave jumps (at x=0, x=1, and x=2), distinct "overshoots" (like at point A) and "undershoots" (like at point B) appear and persist.


**Explanation from 20 Seconds Onward:**


1.  **High Number of Terms (`n`):** From 20 seconds onwards, the value of `n` becomes very large (hundreds, eventually reaching 1000). This means the Fourier series approximation is using a huge number of terms.

2.  **Gibbs Phenomenon:** What you are witnessing is a famous mathematical effect called the **Gibbs phenomenon**.

    *   Even as you add an *infinite* number of terms to the Fourier series (`n` -> ∞), the approximation near a jump discontinuity **never perfectly settles down**.

    *   Instead, it always **overshoots** the target value just before the jump and **undershoots** it just after the jump.

    *   The amount of this overshoot/undershoot (relative to the size of the jump) does **not** go to zero as `n` increases. It converges to a specific value.

3.  **Points A and B:**

    *   Point **A** highlights the peak of the **overshoot** just before the square wave drops from +0.5 to -0.5 at x=1.

    *   Point **B** highlights the lowest point of the **undershoot** just after the square wave drops at x=1.

4.  **The Magnitude of Overshoot/Undershoot:** The Gibbs phenomenon predicts that the maximum overshoot (and undershoot) will be approximately **9%** of the total jump height.

    *   In this graph, the jump is from +0.5 to -0.5, so the total jump height is 1.0.

    *   Therefore, the overshoot peak (A) gets closer and closer to `0.5 + (0.09 * 1.0) ≈ 0.59`.

    *   Similarly, the undershoot valley (B) gets closer and closer to `-0.5 - (0.09 * 1.0) ≈ -0.59`.

    *   Notice in the video, even when `n=1000`, the peaks at A and B are still significantly far from the +0.5 and -0.5 levels, consistent with this phenomenon.

5.  **Narrowing Oscillations:** While the *height* of the overshoot/undershoot doesn't decrease, the *width* of the region where these oscillations occur gets narrower and narrower, pushed closer and closer to the exact point of the jump (x=1) as `n` increases.


**In Summary (After 20s):**


The video shows that even with a very large number of terms (`n` = 700 to 1000) in the Fourier series approximation of a square wave, the approximation exhibits persistent overshoots (A) and undershoots (B) near the discontinuities. This is the Gibbs phenomenon, demonstrating that while the approximation improves overall, the convergence is not uniform near jumps, and the maximum error near the jump does not tend to zero. The peaks stabilize around ±0.59 instead of settling at ±0.5.


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