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Discrete Chebyshev Transform
In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind T_n (x) and the discrete Chebyshev transform on the 'extrema' grid of the Chebyshev polynomials of the first kind. Discrete Chebyshev transform on the roots grid The discrete chebyshev transform of u(x) at the points is given by: : a_m =\frac\sum_^ u(x_n) T_m (x_n) where: : x_n = -\cos\left(\frac (n+\frac)\right) : a_m = \frac \sum_^ u(x_n) \cos\left(m \cos^(x_n)\right) where p_m =1 \Leftrightarrow m=0 and p_m = 2 otherwise. Using the definition of x_n , : a_m =\frac \sum_^ u(x_n) \cos\left(\frac(N+n+\frac) \right) : a_m =\frac \sum_^ u(x_n) (-1)^m\cos\left(\frac(n+\frac) \right) and its inverse transform: : u_n =\sum_^ a_m T_m (x_n) (This so happens to the standard Chebyshev series evaluated on the roots grid.) : u_n =\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias. Transcription The surname Chebyshev has been transliterated in several different ways, like Tchebichef, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Čebyčev, Čebyšev, Chebysheff, Chebychov, Chebyshov (according to native Russian speakers, this one provides the closest pronunciation in English to the correct pronunciation in old Russian), and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Polynomials
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind T_n are defined by : T_n(\cos \theta) = \cos(n\theta). Similarly, the Chebyshev polynomials of the second kind U_n are defined by : U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big). That these expressions define polynomials in \cos\theta may not be obvious at first sight, but follows by rewriting \cos(n\theta) and \sin\big((n+1)\theta\big) using de Moivre's formula or by using the angle sum formulas for \cos and \sin repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain T_2(\cos\theta)=\cos(2\theta)=2\cos^2\theta-1 and U_1(\cos\theta)\sin\theta=\sin(2\theta)=2\cos\theta\sin\theta, which are respectively a polynomial in \cos\th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico and starte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. The DFT is the most important discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Fourier-related Transforms
This is a list of linear transformations of function (mathematics), functions related to Fourier analysis. Such transformations Map (mathematics), map a function to a set of coefficients of basis functions, where the basis functions are trigonometric function, sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component. Continuous transforms Applied to functions of continuous arguments, Fourier-related transforms include: * Two-sided Laplace transform * Mellin transform, another closely related integral transform * Laplace transform * Fourier transform, with special cases: ** Fourier series *** When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transforms
Transform may refer to: Arts and entertainment *Transform (scratch), a type of scratch used by turntablists * ''Transform'' (Alva Noto album), 2001 * ''Transform'' (Howard Jones album) or the title song, 2019 * ''Transform'' (Powerman 5000 album) or the title song, 2003 * ''Transform'' (Rebecca St. James album), 2000 * ''Transform'' (single album), by Teen Top, or the title song, 2011 *"Transform", a song by Daniel Caesar from ''Freudian'', 2017 *"Transform", a song by Your Memorial from ''Redirect'', 2012 Mathematics, science, and technology Mathematics *Tensor transformation law, a defining property of tensors *Tensor product model transformation, numerical method applied to control theory *Transformation (function), concerning functions from sets to themselves *Transform theory, theory of integral transforms **List of transforms, a list of mathematical transforms **Integral transform, a type of mathematical transform Computer graphics *Transform coding, a type of data compress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
