The Impossibility of Trisecting an Angle with Straightedge and
... Today, the proof that straightedge–and–compass trisection is impossible is usually done using algebraic methods. The following fact, which can be found in any good abstract algebra textbook, is needed. Theorem A: If a real number α is constructible using only straightedge and compass, then α satisfi ...
... Today, the proof that straightedge–and–compass trisection is impossible is usually done using algebraic methods. The following fact, which can be found in any good abstract algebra textbook, is needed. Theorem A: If a real number α is constructible using only straightedge and compass, then α satisfi ...
4.2 The Mean Value Theorem 1. Overview
... We will use the IVT (from way back in Chapter 2!) to show the first part, and we will use Rolle’s theorem to show the second part. We let f (x) equal the LHS of the equation: f (x) = 3x + 2 cos x + 5 So then showing that there is a number c such that f (c) = 0 will show that the equation has a root. ...
... We will use the IVT (from way back in Chapter 2!) to show the first part, and we will use Rolle’s theorem to show the second part. We let f (x) equal the LHS of the equation: f (x) = 3x + 2 cos x + 5 So then showing that there is a number c such that f (c) = 0 will show that the equation has a root. ...
1 Numbers 2 Inequalities
... Remark 1. Notice, what I said the theorem said is not at all like it’s written about. It’s important to be able to think about the meaning of a theorem and be able to translate that in a more abstract statement. Likewise, one must also be comfortable with looking at the abstract stuff and translatin ...
... Remark 1. Notice, what I said the theorem said is not at all like it’s written about. It’s important to be able to think about the meaning of a theorem and be able to translate that in a more abstract statement. Likewise, one must also be comfortable with looking at the abstract stuff and translatin ...
PDF
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
![[Part 1]](http://s1.studyres.com/store/data/008795849_1-075cc9ba198c265a8a5f7353865fded1-300x300.png)
Nyquist–Shannon sampling theorem
In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals (often called ""analog signals"") and discrete-time signals (often called ""digital signals""). It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies (see Fig 1). Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are bandlimited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples.Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known. (See § Sampling of non-baseband signals below, and Compressed sensing.)In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions. The fidelity of these reconstructions can be verified and quantified utilizing Bochner's theorem.The name Nyquist–Shannon sampling theorem honors Harry Nyquist and Claude Shannon. The theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others. It is thus also known by the names Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, and cardinal theorem of interpolation.