Approximation Theory is defined in a way that includes approximation of functions by simpler functions. But what about the Taylor expansion, the Fourier series, orthogonal polynomials? Where do they fit in this definition?
Your contribution is highly appreciated.
http://en.wikipedia.org/wiki/Approximation_theory
Issue should be considered from several points of view:
- Definition of approximation theory. The definition states that you need to use a simple function, but not necessarily.
- Methods of approximation theory and application of approximation theory. Here it is necessary to consider the possibility of constructing and studying such functions for different applications.
An approximation method requires a set of approximating
functions.
Approximation Theory is the mathematical study of how given quantities can be approximated by other (usually simpler) ones under appropriate conditions. Approximation theory also studies the size and properties of the error introduced by approximation. Approximations are often obtained by power series expansions in which the higher order terms are dropped.
In other words, the Approximation Theory is the branch of mathematical analysis studying methods for approximating some mathematical objects by others and also studying questions related to the research and estimation of the error that arises here.
The main contents of approximation theory concern the approximation of functions. Its foundations are laid by the work of P.L. Chebyshev (1854–1859) on best uniform approximation of functions by polynomials and by K. Weierstrass, who in 1885 established that in principle it is possible to approximate a continuous function on a finite interval by polynomials with arbitrary pre-given error. The development of approximation theory was to a large extent determined by the fundamental work of H. Lebesgue, Ch.J. de la Vallée-Poussin, S.N. Bernstein [S.N. Bernshtein], D. Jackson, J. Favard, A.N. Kolmogorov and S.M. Nikol'skii on the approximation of functions and function classes.
Engineering, finance, science, and many areas of mathematics itself make use of quantities that are too complicated, too difficult, and even too abstract to work with directly. A major goal of approximation theory is to discover and analyze simple, easy to work with, concrete quantities that can do a good, efficient job in their place - for example, splines to fit messy curves, wavelets to analyze noisy signals and to compress large images, and radial basis functions to fit scattered data and serve as the ``approximation engine'' of neural networks. Areas of faculty interest include approximations by orthogonal polynomials, radial basis functions, and wavelets; further topics of interest include scattered data surface fitting, rates of approximation, constrained approximation, polynomial inequalities, orthogonal polynomials, wavelets, splines, non-linear approximation. There are significant overlapping interests between groups studying Approximation Theory and the groups studying partial differential equations and numerical analysis.
Dear Abedallah
I believe you cant find more informations in The Journal of Approximation Theory Open Archive.
Dear @Nelson, thank you for the tip. Journal of Approximation Theory is a very good resource, however, each journal is devoted to a narrow field.
Dear @Sylantyev, thank you for the contribution.
Dear @Jorge, thank you for the contribution.
In a way out of the manuals, approximation methods are a relatively accurate way to approach results that are not yet described efficiently by any applicable theory.
Students always enjoy these explanations "out of the manuals".
Approximation Theory helps us to anticipate an ideal result. Nature is complicated and multi-dependenced. In Physics, there are many approximations that permit us to anticipate specific characteristics. Its simulate the ideal response, offer a domain value of results. And ... that is very useful!
Dears @Alexandre and @Cristina, thank you for the contribution. Yes, indeed, Approximation Theory is used in almost all fields of sciences.
For a function that is infinitely differentiable at a point, Taylor series is exact. Usaully we take only a finite number of term. So what we useally call a Taylor series is an approximation. The good thing about Taylor series is that if the other is close the accuray is high. If it is not high enough, one can involves high-order terms. So Taylor series is very useful and widely used.
Its limitation is that it deals with a function and the derivatives up to a certain order must exist.
If it is a graph, one cannot approaximate it by a Taylor series.
Issue should be considered from several points of view:
- Definition of approximation theory. The definition states that you need to use a simple function, but not necessarily.
- Methods of approximation theory and application of approximation theory. Here it is necessary to consider the possibility of constructing and studying such functions for different applications.
Dear Abedallah. Thanks, but I only made a small contribution to the debate.
According to my opinion, there is a Theory of Approximation that study how given quantities can be approximated by other (usually simpler) ones under appropriate conditions and the size and properties of the error introduced by such approximation.
Dear @Huajiang , yes indeed the Taylor expansion is a strong mathematical tool that is used to derive many numerical methods.
Dear @Jorge, your contribution is highly appreciated.
Dear @Henk, Thank you for the contribution. The question is really open. "Is there a general theory of approximation? "
Dear @Vyacheslav, Sure, The Issue should be considered from several points of view: Norm, Form, Exactness, unique, characterization, and methods of finding the solution.
Dear Abedallah,
Fourier series can be counted as approximations to periodic functions. In fact the Fourier series is the best trigonometric series to approximate a period function in the sense that it provides the least error. Depending on the required accuracy , one can use a finite number of terms.
Dear @Behrouz, thank you for the contribution. For you as a scientist, what is a compact definition of Approximation Theory?
Dear Abedallah,
A compact definition:"
In approximation theory we use simpler functions to approximate a function. For a Taylor series. For instance in Taylor series we use polynomials and in Fourier series for periodic functions we use Sin and Cos functions
Thank you for explaining the difference between the Taylor(Polynomial) and Fourier(Trigonometric) expansions.
By my opinion, the first impulse in solving a problem should be intuition. Approximations are the next attempts to reach the expected results by simpler means. At last, the final aim is to find the most exact and fine approach to the problem.
Dear Professor Ustinova, the first attempt will be sure trying to find the analytical solution if possible; otherwise approximation is used. Moreover, some approximation methods, like what @Behrouz mentioned, are used to develop farther numerical methods.
Well the definition I have is based on the one on Wikipedia: In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby, where "best" and "simpler" depend on the form and norm.
Surely, term "approximation'" may be used in various meanings. Thank you for your specification. But why?
Dear Professor Ustinova, Thank you for bringing this issue under light. The kind of approximation depends on the norm we are using. For example, using the L_\infity norm leads to the approximation polynomials of least deviation that are characterized by the Chebyshev (Пафну́тий Льво́вич Чебышёв,) orthogonal polynomials.
OK, I understand all the specifications. I don't understand why do you need a general definition of approximation? Everyone can use "approximation" in a case, which is interested him this time: approximated functions, approximated polynomials, approximated results, etc.
Dear Professor Ustinova, I am working in the field of Approximation Theory and looking forward to having a compact definition of it.
Dear Professor Ustinova, thank you for the participation and widen the horizon of our thoughts.
The following link from amazon contains 2 books on approximation Theory:
1. Approximation Theory: From Taylor Polynomials to Wavelets (Applied and Numerical Harmonic Analysis) by Ole Christensen.
2. An Introduction to the Approximation of Functions (Dover Books on Mathematics) by Theodore J. Rivlin.
http://www.amazon.com/Approximation-Theory-Polynomials-Wavelets-Numerical/dp/0817636005
Scientists from the fields of engineering have experience with the Fourier series and do prefer Fourier series in approximation.
I am not an expert in mathematics, but this definition seems to me as good
"The primary aim of a general approximation is to represent non-arithmetic quantities by arithmetic quantities so that the accuracy can be ascertained to a desired degree"
http://www4.ncsu.edu/~mtchu/Teaching/Lectures/MA530/chapter7.pdf
I hope that I am giving a constructive contribution which shows the importance of "approximation", but you judge because you have more knowledge than me in the approaches of approximating:
In chemistry, we have the "Born-Oppenheimer approximation": a way to simplify the complicated Schrödinger equation (which is a partial differential equation that describes how the wave function of a physical system evolves over time). The nucleus & electrons are attracted to each other with the same magnitude of charge, thus they exert the same force & momentum. While exerting the same kind of momentum, the nucleus, with a much larger mass than that of the electron, will have a very small velocity that is almost negligible. Born-Oppenheimer takes advantage of this and assumes that the nucleus's motion can be ignored while solving Schrödinger equation; i.e., the nucleus is assumed to be stationary while electrons move around it. The motion of the nuclei & the electrons can be separated and the electronic & nuclear problems can be solved with independent wave functions.
Dear Abedallah. Thank very much for important as well as interesting topic. At present, we have a lot of "method for approximation" for various purposes and situations (for example, for the solutions of differential equations, integral equations, linear algebraic equations and so on). As Dr. Nizar pointed out "Born-oppenheimer approximation" is also important. In my situation for wave problems for classical mechanics, asymptotic expansion for far-field solution has been had an important role. Therefore, If we restrict the problem of approximation for a certain specified area, the definition of "approximation" will be easier. If we seek generalized concept of "approximation", I feel the problem is not very easy to solve. Maybe, before that, we have to review historical work in vast areas for various situations. But this might be the work that somebodies, some research groups have to do for our future.
Dears Professors @Nizar and Terumi, Thank you for the feedback about the use of some approximation methods in your fields; the Schrödinger PDE occurs often in the applications in Physics and Chemistry and is simplified using the Born-Oppenheimer approximation.
Dear Professors
I think the best definition of approximation theory is "replace something complicated by something simple"
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