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Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group. Authors. Valery V. Volchkov; Vitaly V. Volchkov. Book.
Table of contents
- ספרים חדשים יולי - ספטמבר 2013
- Harmonic Analysis
- Product | Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group
- The Case of Compact Symmetric Spaces
Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle.
The critical case for this principle is the Gaussian function , of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution e. The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer , where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral , making it an integral transform , although this definition is not suitable for many applications requiring a more sophisticated integration theory. This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics , where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both.
In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. The latter is routinely employed to handle periodic functions. Many other characterizations of the Fourier transform exist. In , Joseph Fourier showed that some functions could be written as an infinite sum of harmonics.
One motivation for the Fourier transform comes from the study of Fourier series. In the study of Fourier series, complicated but periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity. Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral.
This has the advantage of simplifying many of the formulas involved, and provides a formulation for Fourier series that more closely resembles the definition followed in this article. Re-writing sines and cosines as complex exponentials makes it necessary for the Fourier coefficients to be complex valued.
The usual interpretation of this complex number is that it gives both the amplitude or size of the wave present in the function and the phase or the initial angle of the wave. These complex exponentials sometimes contain negative "frequencies".
ספרים חדשים יולי - ספטמבר 2013
Hence, frequency no longer measures the number of cycles per unit time, but is still closely related. There is a close connection between the definition of Fourier series and the Fourier transform for functions f that are zero outside an interval. For such a function, we can calculate its Fourier series on any interval that includes the points where f is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval in which we calculate the Fourier series, then the Fourier series coefficients begin to resemble the Fourier transform and the sum of the Fourier series of f begins to resemble the inverse Fourier transform.
Then, the n th series coefficient c n is given by:. Under appropriate conditions, the Fourier series of f will equal the function f. In other words, f can be written:. This second sum is a Riemann sum. Under suitable conditions, this argument may be made precise. In the study of Fourier series the numbers c n could be thought of as the "amount" of the wave present in the Fourier series of f.
Similarly, as seen above, the Fourier transform can be thought of as a function that measures how much of each individual frequency is present in our function f , and we can recombine these waves by using an integral or "continuous sum" to reproduce the original function. The following figures provide a visual illustration how the Fourier transform measures whether a frequency is present in a particular function.
The second factor in this equation is an envelope function that shapes the continuous sinusoid into a short pulse. Its general form is a Gaussian function. This function was specially chosen to have a real Fourier transform that can be easily plotted.
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The first image contains its graph. The second image shows the plot of the real and imaginary parts of this function. Therefore, in this case, the integrand oscillates fast enough so that the integral is very small and the value for the Fourier transform for that frequency is nearly zero. The Fourier transform has the following basic properties: . In particular the Fourier transform is invertible under suitable conditions. These equalities of operators require careful definition of the space of functions in question, defining equality of functions equality at every point?
These are not true for all functions, but are true under various conditions, which are the content of the various forms of the Fourier inversion theorem. This approach is particularly studied in signal processing , under time—frequency analysis. So these are two distinct copies of the real line, and cannot be identified with each other.
Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition. See the article on linear algebra for a more formal explanation and for more details. This point of view becomes essential in generalisations of the Fourier transform to general symmetry groups, including the case of Fourier series.
That there is no one preferred way often, one says "no canonical way" to compare the two copies of the real line which are involved in the Fourier transform—fixing the units on one line does not force the scale of the units on the other line—is the reason for the plethora of rival conventions on the definition of the Fourier transform.
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The various definitions resulting from different choices of units differ by various constants. In probability theory, and in mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because so many random variables are not of continuous type, and do not possess a density function, and one must treat not functions but distributions , i. From the higher point of view of group characters , which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on a locally compact Abelian group.
The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties. By the Riemann—Lebesgue lemma , . For example, the Fourier transform of the rectangular function , which is integrable, is the sinc function , which is not Lebesgue integrable , because its improper integrals behave analogously to the alternating harmonic series , in converging to a sum without being absolutely convergent.
It is not generally possible to write the inverse transform as a Lebesgue integral. But if f is continuous, then equality holds for every x. The Plancherel theorem , which follows from the above, states that .
Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem. See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.
The Poisson summation formula PSF is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. The Poisson summation formula says that for sufficiently regular functions f ,. It has a variety of useful forms that are derived from the basic one by application of the Fourier transform's scaling and time-shifting properties. The formula has applications in engineering, physics, and number theory.
Product | Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group
The frequency-domain dual of the standard Poisson summation formula is also called the discrete-time Fourier transform. Poisson summation is generally associated with the physics of periodic media, such as heat conduction on a circle. The fundamental solution of the heat equation on a circle is called a theta function. It is used in number theory to prove the transformation properties of theta functions, which turn out to be a type of modular form , and it is connected more generally to the theory of automorphic forms where it appears on one side of the Selberg trace formula.
Then the Fourier transform of the derivative is given by. More generally, the Fourier transformation of the n th derivative f n is given by. By applying the Fourier transform and using these formulas, some ordinary differential equations can be transformed into algebraic equations, which are much easier to solve.
The Fourier transform translates between convolution and multiplication of functions. Since the complete set of Hermite functions provides a resolution of the identity, the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed. This approach to define the Fourier transform was first done by Norbert Wiener. These operators do not commute, as their group commutator is. Denote the Heisenberg group by H 1.
This J can be extended to a unique automorphism of H 1 :. This operator W is the Fourier transform. Many of the standard properties of the Fourier transform are immediate consequences of this more general framework. The Paley—Wiener theorem says that f is smooth i. This theorem has been generalised to semisimple Lie groups.
The Case of Compact Symmetric Spaces
It may happen that a function f for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the complex plane. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis. This theorem implies the Mellin inversion formula for the Laplace transformation, .
Dirichlet-Dini theorem , the value of f at t is taken to be the arithmetic mean of the left and right limits, and provided that the integrals are taken in the sense of Cauchy principal values.