Abstract Harmonic Analysis

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Ocean tides and vibrating strings are common and simple examples. The theoretical approach is often to try to describe the system by a differential equation or system of equations to predict the essential features, including the amplitude, frequency, and phases of the oscillatory components. The specific equations depend on the field, but theories generally try to select equations that represent major principles that are applicable. The experimental approach is usually to acquire data that accurately quantifies the phenomenon.


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For example, in a study of tides, the experimentalist would acquire samples of water depth as a function of time at closely enough spaced intervals to see each oscillation and over a long enough duration that multiple oscillatory periods are likely included. In a study on vibrating strings, it is common for the experimentalist to acquire a sound waveform sampled at a rate at least twice that of the highest frequency expected and for a duration many times the period of the lowest frequency expected.

The waveform appears oscillatory, but it is more complex than a simple sine wave, indicating the presence of additional waves. The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the Fourier transform , the result of which is shown in the lower figure.


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One of the most modern branches of harmonic analysis, having its roots in the midth century, is analysis on topological groups. The core motivating ideas are the various Fourier transforms , which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups. The theory for abelian locally compact groups is called Pontryagin duality. Harmonic analysis studies the properties of that duality and Fourier transform and attempts to extend those features to different settings, for instance, to the case of non-abelian Lie groups.

For general non-abelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations. For compact groups, the Peter—Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations. This choice of harmonics enjoys some of the useful properties of the classical Fourier transform in terms of carrying convolutions to pointwise products, or otherwise showing a certain understanding of the underlying group structure.

See also: Non-commutative harmonic analysis. If the group is neither abelian nor compact, no general satisfactory theory is currently known "satisfactory" means at least as strong as the Plancherel theorem. However, many specific cases have been analyzed, for example SL n. In this case, representations in infinite dimensions play a crucial role.

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From Wikipedia, the free encyclopedia. Excursion: Two half day tours to local attractions are planned.

One is in the afternoon of Wednesday, June 27, to visit the Cijin Island for seafood lunch and the light tower. Participants are expected to share the cost. These two hotels are door to door, and close to the O1 metro station. See maps. Travel Information:. There are frequent scheduled flights connecting Kaohsiung airport code: KHH to other cities. If you want to fly to Taipei airport code: TPE , though not advised , and plan to take a high speed train 90 minute one way from Taipei to Kaohsiung, please arrange your flight to arrive in Taipei before 7pm to catch the last train departing at pm, and arriving at Zuoying HSR station at pm.

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You need to take an airport metro train train or an air shuttle bus connecting the airport and the high speed train station at Taoyuan for approximately 40 minutes one way. The metro operates until midnight. Sun Yat-sen.

Trevor Wooley - A translation-invariant perspective on arithmetic (and) harmonic analysis

Paying tips is not necessary in Taiwan. We suggest you to make money exchange at the KHH airport. The rate is same as in downtown banks there. Or you can take the No. You can also use the metro and stop at the O1 station. The campus is in 10 minute walking distance from the O1 Orange One metro station. Then turn right at the first traffic light and you will be on Chung Cheng Road. Follow Chung Cheng Road until the end for about 20 minutes.

Soon after you pass the Love River Bridge, make a right turn at the loop. Make a left turn there. Drive straight and make a right turn at Lin Hai 2nd Road. After four traffic lights you will arrive at the University Tunnel Entrance for bicycles and walkers only. Tell the guards there you are attending an international mathematics workshop at the International Research Building, and then you will know how to drive to the venue.

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Abstract harmonic analysis

Visas and other information:. Foreign nationals may obtain tourist visas to enter Taiwan if they hold foreign passports or travel documents valid for more than six months.

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A brief history of Taiwan, as well as travel information, is available there, too. The temperature of Kaohsiung in July is about degrees C.

Abstract Harmonic Analysis | SpringerLink

No heavy clothing is in need. Currency Exchange:. It is best to make exchange at Kaohsiung Airport right after you walk out from the custom.

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You can also make exchange at banks in downtown or at the hotel. Almost no service charge or surplus applies when you do it in a bank or at the airport.

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You can change it back to US dollars or Euros or other currencies at the airport again before you leave the country. Credit cards are very popular here. We will provide pickup service in the Kaohsiung International Airport for those participants who have already given us their arrival flight information. Shuttle bus pickups at Kingship Hotel at and am every morning, and back after the last talk every day.

It is a 30 minute or 2km walk between Kingship Hotel and the venue. The last talk will finish at am, June 29, Two half day tours to local attractions are planned. Group photo being shot at am, Monday, June

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