DECAY RATE OF FOURIER TRANSFORMS OF SOME SELF-SIMILAR MEASURES

This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μλ with λ = θ-1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μλ ahaost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.

Dirichlet characters, Gauss sums and arithmetic Fourier transforms

In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic （continuous） functions is developed based on Dirichlet characters, Gauss sums and the generalized MSbius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.

SUMMATION OF TRIGONOMETRIC SERIES BY FOURIER TRANSFORMS

This paper gives the theorems concerning the summation of trigonometric series with the help of Fourier transforms. By means of the known results of Fourier transforms, many difficult and complex problems of summation of trigonometric series can be solved. This method is a comparatively unusual way to find the summation of trigonometric series, and has been used to establish the comprehensive table of summation of trigonometric series. In this table 10 thousand scries arc given, and most of them are new.

A NOTE ON DERIVATIVES OF FUNCTION AND THEIR FOURIER TRANSFORMS IN L_2

The purpose of this article is to indicate the shortcomings of a few theorems of[1]. Moreover, some interesting results are detluced.

ABOUT THE FOURIER TRANSFORMS OF NON-SMOOTH MEASURES ON HYPERSPHERES

If dμ is the Fourier transform of a smooth measure,dμ on the hypersphere Sn-1(n≥2)the there exists a constant C dependent only on n such that |dμ(y) |≤C(1+ |y |)-(n-1) /2 for all y∈Rn. In this paper, we show that the above statement is false for non-smooth measures. And we present the corresponding estimations far the Fourier transforms of certain non-smooth measures on Sn-1.

FAST FOURIER TRANSFORMS ON BINARY FIELDS

We give a unified treatment of Fast Fourier Transforms for UDMD systems which contains, as special cases, Fast Fourier algorithms for character groups of many subgroups associated with binary fields.

Wavelet-fractional Fourier transforms

This paper extends the definition of fractional Fourier transform （FRFT） proposed by Namias V by using other orthonormal bases for L^2（R） instead of Hermite-Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.

Musielak–Orlicz–Lorentz Hardy Spaces:Maximal Function,Finite Atomic,and Littlewood–Paley Characterizations with Applications to Dual Spaces and Summability of Fourier Transforms

Let q∈(0,∞]andϕbe a Musielak-Orlicz function with uniformly lower type p_(ϕ)^(−)∈(0,∞)and uniformly upper type p_(ϕ)^(−)∈(0,∞).In this article,the authors establish various realvariable characterizations of the Musielak-Orlicz-Lorentz Hardy space H^(ϕ,q)(R^(n)),respectively,in terms of various maximal functions,finite atoms,and various Little wood-Paley functions.As applications,the authors obtain the dual space of Hϕ,q(Rn)and the summability of Fourier transforms from Hϕ,q(Rn)to the Musielak-Orlicz-Lorentz space L^(ϕ,q)(R^(n))when q∈(0,∞)or from the Musielak-Orlicz Hardy space Hϕ(Rn)to Lϕ,∞(Rn)in the critical case.These results are new when q∈(0,∞)and also essentially improve the existing corresponding results(if any)in the case q=∞via removing the original assumption thatϕis concave.To overcome the essential obstacles caused by both thatϕmay not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown,the authors make full use of the obtained atomic characterization of H^(ϕ,q)(R^(n)),the corresponding results related to weighted Lebesgue spaces,and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.

Mapping Properties of Fourier Transforms,Revisited

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces B_(p)^(s)(R^(n))=B_(p,p)^(s)(R^(n)),1≤p≤∞,and between Sobolev spaces Hs p(R^(n)),1<p<∞.In contrast to the paper H.Triebel,Mapping properties of Fourier transforms.Z.Anal.Anwend.41(2022),133–152,based mainly on embeddings between related weighted spaces,we rely on wavelet expansions,duality and interpolation of corresponding(unweighted)spaces,and(appropriately extended)Hausdorff-Young inequalities.The degree of compactness will be measured in terms of entropy numbers and approximation numbers,now using the symbiotic relationship to weighted spaces.

Microstructure effect of mechanical and cracking behaviors on brittle rocks using image-based fast Fourier transform method

The internal microstructures of rock materials, including mineral heterogeneity and intrinsic microdefects, exert a significant influence on their nonlinear mechanical and cracking behaviors. It is of great significance to accurately characterize the actual microstructures and their influence on stress and damage evolution inside the rocks. In this study, an image-based fast Fourier transform (FFT) method is developed for reconstructing the actual rock microstructures by combining it with the digital image processing (DIP) technique. A series of experimental investigations were conducted to acquire information regarding the actual microstructure and the mechanical properties. Based on these experimental evidences, the processed microstructure information, in conjunction with the proposed micromechanical model, is incorporated into the numerical calculation. The proposed image-based FFT method was firstly validated through uniaxial compression tests. Subsequently, it was employed to predict and analyze the influence of microstructure on macroscopic mechanical behaviors, local stress distribution and the internal crack evolution process in brittle rocks. The distribution of feldspar is considerably more heterogeneous and scattered than that of quartz, which results in a greater propensity for the formation of cracks in feldspar. It is observed that initial cracks and new cracks, including intragranular and boundary ones, ultimately coalesce and connect as the primary through cracks, which are predominantly distributed along the boundary of the feldspar. This phenomenon is also predicted by the proposed numerical method. The results indicate that the proposed numerical method provides an effective approach for analyzing, understanding and predicting the nonlinear mechanical and cracking behaviors of brittle rocks by taking into account the actual microstructure characteristics.

Fourier Transforms of Bounded Bilinear Forms on C＊（S1）×C＊（S2） of Foundation ＊-semigroups S1 and S2

The author gives a characterization of the Fourier transforms of bounded bilinear forms on C＊（S1）×C＊（S2） of two foundation semigroups S1 and S2 in terms of Jordan ＊-representations, hemimogeneous random fields, and as well as weakly harmonizable random fields of S1 and S2 into Hilbert spaces.

Rotation-invariant texture analysis using Radon and Fourier transforms

Texture analysis is a basic issue in image processing and computer vision, and how to attain the rotationinvariant texture characterization is a key problem. This paper proposes a rotation-invariant texture analysis technique using Radon and Fourier transforms. This method uses Radon transform to convert rotation to translation, then utilizes Fourier transform and takes the moduli of the Fourier transform of these functions to make the translation invariant. A k-nearest-neighbor rule is employed to classify texture images. The proposed method is robust to additive white noise as a result of summing pixel values to generate projections in the Radon transform step. Experiment results show the feasibility of the proposed method and its robustness to additive white noise.

Restricted Summability of Fourier Transforms and Local Hardy Spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.

A Deepfake Detection Algorithm Based on Fourier Transform of Biological Signal

Deepfake-generated fake faces,commonly utilized in identity-related activities such as political propaganda,celebrity impersonations,evidence forgery,and familiar fraud,pose new societal threats.Although current deepfake generators strive for high realism in visual effects,they do not replicate biometric signals indicative of cardiac activity.Addressing this gap,many researchers have developed detection methods focusing on biometric characteristics.These methods utilize classification networks to analyze both temporal and spectral domain features of the remote photoplethysmography(rPPG)signal,resulting in high detection accuracy.However,in the spectral analysis,existing approaches often only consider the power spectral density and neglect the amplitude spectrum—both crucial for assessing cardiac activity.We introduce a novel method that extracts rPPG signals from multiple regions of interest through remote photoplethysmography and processes them using Fast Fourier Transform(FFT).The resultant time-frequency domain signal samples are organized into matrices to create Matrix Visualization Heatmaps(MVHM),which are then utilized to train an image classification network.Additionally,we explored various combinations of time-frequency domain representations of rPPG signals and the impact of attention mechanisms.Our experimental results show that our algorithm achieves a remarkable detection accuracy of 99.22%in identifying fake videos,significantly outperforming mainstream algorithms and demonstrating the effectiveness of Fourier Transform and attention mechanisms in detecting fake faces.

An application of Chen's theorem to convergence of Fourier multiplier transforms

Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.

Calculation of local Fourier transforms for formal connections

We calculate the local Fourier transforms for formal connections. In particular, we verify some formulas analogous to a conjecture of Laumon and Malgrange for-adic local Fourier transforms.

A property of Pisot numbers and Fourier transforms of self-similar measures

For any Pisot number β it is known that the set F（β） ={t ： limn→∞‖tβn‖ = 0} is countable, where ‖α‖ is the distance between a real number a and the set of integers. In this paper it is proved that every member in this set is of the form cβn, where n is a nonnegative integer and e is determined by a linear system of equations. Furthermore, for some self-similar measures μ associated with β, the limit at infinity of the Fourier transforms limn→μ（tβn）≠0 if and only if t is in a certain subset of F（β）. This generalizes a similar result of Huang and Strichartz.

Domains of holomorphy for Fourier transforms of solutions to discrete convolution equations

We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in C^n, and we determine possible domains in terms of the properties of the convolution operator.

Design and implementation of code acquisition using sparse Fourier transform

Code acquisition is the kernel operation for signal synchronization in the spread-spectrum receiver.To reduce the computational complexity and latency of code acquisition,this paper proposes an efficient scheme employing sparse Fourier transform(SFT)and the relevant hardware architecture for field programmable gate array(FPGA)and application-specific integrated circuit(ASIC)implementation.Efforts are made at both the algorithmic level and the implementation level to enable merged searching of code phase and Doppler frequency without incurring massive hardware expenditure.Compared with the existing code acquisition approaches,it is shown from theoretical analysis and experimental results that the proposed design can shorten processing latency and reduce hardware complexity without degrading the acquisition probability.

Mid-infrared Optical Frequency Comb-based Fourier Transform Spectrometer for Broadband Molecular Spectroscopy

Optical frequency combbased Fourier transform spectroscopy has the features of broad spectral bandwidth,high sensitivity,andmultiplexed trace gas detection,which has valuable application potential in the fields of precision spectroscopy and trace gas detection.Here,we report the development of a mid-infrared Fourier transform spectrometer based on an optical frequency comb combined with a Herriott-type multipass cell.Using this instrument,the broadband absorption spectra of several important molecules,including methane,acetylene,water molecules and nitrous oxide,are measured by near real-time data acquisition in the 2800-3500 cm^(-1)spectral region.The achieved minimum detectable absorption of the instrument is 4.4×10^(-8)cm^(-1)·Hz^(-1/2)per spectral element.Broadband spectra of H_(2)0 are fited using the Voigt profile multispectral fitting technique and the consistency of the concentration inversion is 1%.Our system also enables precise spectroscopic measurements,and it allows the determination of the spectral line positions and upper state constants of N_(2)O in the(0002)-(1000)band,with results in good agreement with those reported by Toth[Appl.Opt.30,5289(1991)].