The spectral side of Arthur's trace formula

Tobias Finis1, Erez M Lapid, Werner Müller

  • 1Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstrasse 1, D-40225 Düsseldorf, Germany.

Related Concept Videos

Attenuated Total Reflectance (ATR) Infrared Spectroscopy: Overview01:13

Attenuated Total Reflectance (ATR) Infrared Spectroscopy: Overview

Attenuated total reflectance (ATR) infrared spectroscopy is a powerful analytical technique used to study the composition of materials. It is widely employed in chemistry, materials science, forensic science, and other fields where sample characterization is required. ATR has several advantages over traditional transmission IR spectroscopy, including the requirement of little to no sample preparation and the ability to analyze a wide range of samples.
The ATR process begins by directing a beam...
Parseval's Theorem01:18

Parseval's Theorem

Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which expresses a...
Emission Spectra02:39

Emission Spectra

When solids, liquids, or condensed gases are heated sufficiently, they radiate some of the excess energy as light. Photons produced in this manner have a range of energies, and thereby produce a continuous spectrum in which an unbroken series of wavelengths is present.
Parseval's Theorem for Fourier transform01:15

Parseval's Theorem for Fourier transform

Parseval's theorem is a fundamental principle in signal processing that enables the calculation of a signal's energy in either the time domain or the frequency domain. This theorem is pivotal in demonstrating energy conservation between these two domains, ensuring that the computed energy value remains consistent regardless of the domain of analysis.
To understand Parseval's theorem, it is essential to first comprehend how signal energy is typically calculated. When considering a signal's...
Aliasing01:18

Aliasing

Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original signal...
Phasor Arithmetics01:13

Phasor Arithmetics

Phasors and their corresponding sinusoids are interrelated, offering unique insights into the behavior of alternating current (AC) circuits. One way to understand this relationship is through the operations of differentiation and integration in both the time and phasor domains.
When the derivative of a sinusoid is taken in the time domain, it transforms into its corresponding phasor multiplied by j-omega (jω) in the phasor domain, where j is the imaginary unit, and ω is the angular frequency.