Wavelet Transforms and Localization Operators

Localization Operators Via Time-Frequency Analysis

“…The result (i) is well known and first appeared in [34]; here we sketch a different and shorter proof of it. More details are contained in [4].…”

“…For instance, localization operators could be used to filter out noise from given (noisy) signals. Since then, they have been extensively investigated as an important mathematical tool in the applications mentioned above [11,16,27,33,34]. Therefore, let us come deeper into this subject.…”

“…Depending on the field of application, these operators are known under the names of Wick, anti-Wick or Toeplitz operators, as well as wave packets, Gabor or short time Fourier transform multipliers. There is a host of articles on the subject written by mathematicians, physicists and engineers; we refer, for instance, to [2,9,10,11,12,16,24,26,30,34].…”

“…We shall start considering symbols in L p -spaces, with 1 ≤ p ≤ ∞, we go to potential Sobolev spaces and we end up with the the so-called modulation spaces. For symbols in L p -spaces, we recall results contained in [34] and we show how to get them by means of shorter proofs [4]. However, sometimes in the applications one needs rougher symbols, even tempered distributions rather than functions.…”

Generalized Spectrograms and t -Wigner Transforms

“…is a way to represent the energy of a signal f simultaneously with respect to time and frequency; V g f is the short-time Fourier transform, or Gabor transform, with window g, see for reference [13], [16], [10]. In [3], the two window spectrogram has been introduced and studied: it depends on two windows and it is defined by the skew-linear form…”