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Basis Functions
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points). Examples Monomial basis for ''Cω'' The monomial basis for the vector space of analytic functions is given by \. This basis is used in Taylor series, amongst others. Monomial basis for polynomials The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n for some n \in \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Square-integrable Function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty,+\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions (with respect to Lebesgue measure) forms the ''Lp'' space with p=2. Among the ''Lp'' spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Biorthogonal Wavelet
A Biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in .... Designing biorthogonal wavelets allows more degrees of freedom than orthogonal wavelets. One additional degree of freedom is the possibility to construct symmetric wavelet functions. In the biorthogonal case, there are two scaling functions \phi,\tilde\phi, which may generate different multiresolution analyses, and accordingly two different wavelet functions \psi,\tilde\psi. So the numbers ''M'' and ''N'' of coefficients in the scaling sequences a,\tilde a may differ. The scaling sequences must satisfy the following biorthogonality condition :\sum_ a_n \tilde a_=2\cdot\delta_. Then the wavelet sequences can be de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Orthogonal Wavelet
An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal. That is, the inverse wavelet transform is the adjoint of the wavelet transform. If this condition is weakened one may end up with biorthogonal wavelets. Basics The scaling function is a refinable function. That is, it is a fractal functional equation, called the refinement equation (twin-scale relation or dilation equation): :\phi(x)=\sum_^ a_k\phi(2x-k), where the sequence (a_0,\dots, a_) of real numbers is called a scaling sequence or scaling mask. The wavelet proper is obtained by a similar linear combination, :\psi(x)=\sum_^ b_k\phi(2x-k), where the sequence (b_0,\dots, b_) of real numbers is called a wavelet sequence or wavelet mask. A necessary condition for the ''orthogonality'' of the wavelets is that the scaling sequence is orthogonal to any shifts of it by an even number of coefficients: :\sum_ a_n a_=2\delta_, where \delta_ is the Kronecker delta. In this case there is the same num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term " harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fourier Analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis ( quadrature rules), probability theory, representation theory (of Lie groups, quantum groups, and related objects), enumerative combinatorics, algebraic combinatorics, mathematical physics (the theory of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Inner-product Space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the picture); ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Orthonormal Basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of \R^n under dot product. Every finite-dimensional inner product space has an orthonormal basis, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Biorthogonal System
In mathematics, a biorthogonal system is a pair of indexed families of vectors \tilde v_i \text E \text \tilde u_i \text F such that \left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_, where E and F form a pair of topological vector spaces that are in duality, \langle \,\cdot, \cdot\, \rangle is a bilinear mapping and \delta_ is the Kronecker delta. An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct. A biorthogonal system in which E = F and \tilde v_i = \tilde u_i is an orthonormal system. Projection Related to a biorthogonal system is the projection P := \sum_ \tilde u_i \otimes \tilde v_i, where (u \otimes v) (x) := u \langle v, x \rangle; its image is the linear span of \left\, and the kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dual Basis
In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with the same index set ''I'' such that ''B'' and ''B''∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span ''V''∗. If it does span ''V''∗, then ''B''∗ is called the dual basis or reciprocal basis for the basis ''B''. Denoting the indexed vector sets as B = \_ and B^ = \_, being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in ''V''∗ on a vector in the original space ''V'': : v^i\cdot v_j = \delta^i_j = \begin 1 & \text i = j\\ 0 & \text i \ne j\text \end where \delta^i_j is the Kronecker delta symbol. Introduction To perform operations with a vector, we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |