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In this article, we apply the Fourier transform to prove the Hyers-Ulam and Hyers-Ulam-Rassias stability for the first- and second-order nonlinear differential equations with initial conditions. Additionally, we extend the results to investigate the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of these differential equations using the proposed method.

In this study, we address a Cauchy problem within the context of the one-dimensional Timoshenko system, incorporating a distributed delay term. The heat conduction aspect is described by the Lord-Shulman theory. Our demonstration establishes that the dissipation resulting from the coupling of the Timoshenko system with Lord-Shulman’s heat conduction is sufficiently robust to stabilize the system, albeit with a gradual decay rate. To support our findings, we convert the system into a first-order form and, utilizing the energy method in Fourier space, and derive point wise estimates for the Fourier transform of the solution. These estimates, in turn, provide evidence for the slow decay of the solution.

The weighted fractional Fourier transform (WFRFT) has always been considered a development of the discrete fractional Fourier transform (FRFT). This paper points out that the WFRFT is a discrete FRFT of eigenvalue decomposition, which will change the consistent understanding of the WFRFT. Extended definitions based on the WFRFT have been proposed and widely used in information processing. This paper proposes a unified framework for extended definitions, and existing extended definitions can serve as special cases of this unified framework. In further analysis, we find that the existing extended definitions are deficient. With the help of a unified framework, we systematically analyze the reasons for the deficiencies. This has great guiding significance for the application of the WFRFT and its extended definitions.

Time series forecasting has played an important role in different industries, including economics, energy, weather, and healthcare. RNN-based methods have shown promising potential due to their strong ability to model the interaction of time and variables. However, they are prone to gradient issues like gradient explosion and vanishing gradients. And the prediction accuracy is not high. To address the above issues, this paper proposes a Fractional-order Lipschitz Recurrent Neural Network with a Frequency-domain Gated Attention mechanism (FLRNN-FGA). There are three major components: the Fractional-order Lipschitz Recurrent Neural Network (FLRNN), frequency module, and gated attention mechanism. In the FLRNN, fractional-order integration is employed to describe the dynamic systems accurately. It can capture long-term dependencies and improve prediction accuracy. Lipschitz weight matrices are applied to alleviate the gradient issues. In the frequency module, temporal data are transformed into the frequency domain by Fourier transform. Frequency domain processing can reduce the computational complexity of the model. In the gated attention mechanism, the gated structure can regulate attention information transmission to reduce the number of model parameters. Extensive experimental results on five real-world benchmark datasets demonstrate the effectiveness of FLRNN-FGA compared with the state-of-the-art methods.

Due to the lack of teachers, students’ writing can not get timely feedback, which makes the effectiveness of calligraphy teaching in higher education institutions poor. An interactive teaching system for calligraphy in higher education institutions is constructed using computer multimedia technology in this paper. Through the design of teaching links, the system fully reflects the context and interactivity of teaching content, offers a wealth of learning activities and resources and makes the calligraphy learning process more intuitive. At the same time, through the “human-computer interaction” to improve the internal drive of students to participate in the learning process, to enhance the interest of learners in learning. The denoising of the text in the picture involves the use of discrete Fourier transform, while the smoothing of the edges of the calligraphic text is achieved through binarization operation, erosion, and expansion principle. The results show that the shape similarity of the system fluctuates between 0.57922 and 0.83314, and the overall similarity reaches more than 0.85, which indicates that the system can be applied to the undergraduate classroom of Calligraphy in higher education institutions. Under the teaching environment of the system, the teacher students with different attitudes have significant variability, which also indicates that the system is effective in assisting the short-term writing practice of calligraphy teaching in higher education institutions, reflecting that the detailed evaluation provided by the interactive teaching system of calligraphy in higher education institutions has some valuable significance.

This paper proposes a method for data acquisition of yoga movements and analyzes it using the Fourier transform. It aims to help residents choose a more scientific and effective exercise method to relieve physical fatigue, reduce body stiffness, and improve body flexibility. Initially, we transform the yoga movements into RGB and optical flow, feeding them into the STF-ResNet network to extract the temporal and spatial features of the movement data. Next, we combine the spatial and temporal flow features with residuals to compensate for the loss of high-level feature information. Finally, we add the convolutional block attention module to further filter the yoga movement features from both channel and spatial dimensions. Next, we introduce the Fourier transform to parse the gathered yoga movement features. We conducted experiments both with and without yoga movements interfering with body soft rhythms to investigate their effects. The experimental group improved the test scores of the forward and backward splitting test and the backward bending body bridge building test, with the mean value of standing forward bending being 5.11 higher after the experiment than before. It shows that yoga intervenes in the flexibility of the human body through auxiliary training, and yoga postures can have a significant beneficial effect on body flexibility.

The opera “Aleko” is a graduation work composed by Rachmaninoff, for which the opera was born in Russia in the nineteenth century and has a strong literary and artistic value. This study tries to extract the time-frequency melodic characteristics of the opera “Aleko” by using the spectral analysis technique as a research tool combined with the Fourier transform and constant Q transform. From the perspective of quantitative analysis, the innovative laws of Rachmaninoff’s melodic features are summarized with the intention of revealing the great artistic charm of this opera work. The results of the spectral analysis show that the amplitude of the first part of “Aleko” is 27.5dB, which forms its unique “Rach’s style” through its innovative “stepped” meandering tones. At the same time, the rich melodic layers of “Aleko” emphasize the drama of the opera and make the internal structure of the phrases stronger.

As computer science and technology continue to evolve and become more pervasive, their application in analyzing the audio spectrum of vocalizations offers valuable insights for vocal music education. This study introduces a method utilizing Fourier transform analysis to examine time-frequency domain signals in vocal teaching. Initially, voice frequencies are collected during vocal music instruction. Subsequently, these frequencies are processed to extract characteristic sequences, which are then reduced in scale to develop a model for voice spectrum recognition tailored to vocal music education. This model facilitates detailed spectral analysis, enabling the investigation of its auxiliary benefits in vocal music teaching, particularly in identifying prevalent instructional challenges. Our findings indicate that during training on vowels “a” and “i,” professional singers’ pitch at 4kHz declined to between −15 and −18 dB, whereas students’ pitch varied around ±6dB, trending upwards. In cases of air leakage, significant gaps were observed at frequencies of 5500Hz, 10500Hz, and 14500Hz. At the same time, students exhibited missing frequencies at 7kHz, 12kHz, and 14kHz during glottal tone production, with pronounced, abrupt peaks occurring when vocal folds were tightly constricted and devoid of excessive links. This research substantiates the theoretical and practical benefits of digital spectrum technology in enhancing vocal music education, thereby providing a scientific and supportive role.

People’s judgment of music emotion is highly subjective; how to quantify the music emotion characteristics is the key to solving the music emotion recognition problem. This paper utilizes the Fourier transform method to preprocess the input music sample signal. A digital filter accomplishes the pre-emphasis operation, and the number of frames in the music signal is determined by splitting and windowing through a convolution operation. By utilizing the Mel frequency cepstrum coefficient and cochlear frequency, emotional features of music can be extracted. Improve the multimodal model based on the RCNN algorithm, propose the TWC music emotion framework, and construct a music emotion recognition model that incorporates the improved multimodal RCNN. The proposed model’s impact on music emotion appreciation is evaluated through experiments to identify music emotions and an analysis of college music teaching practices that emphasize emotion appreciation. The results show that 1376 songs belonging to the category of “relaxation” are assigned to the category of “healing”, which is only 4 songs short of the target, and the labeling of the songs is not homogeneous, and the emotional recognition of the model is consistent with the cognition. The mean value of the empathy ability of college students in music emotion appreciation is 69.13, which is in the middle-upper level, indicating that the model proposed in this paper has a good effect on the cultivation of students’ music emotion appreciation.

In this paper, firstly, we propose to use time series analysis to observe the development and change process of social events, using the DF test and ADF test to eliminate the random autocorrelation in order to exclude the interference of white noise series. Secondly, the instantaneous frequency of non-smooth data is defined by analyzing the signal, and the constraints are modified from global constraints to local constraints. The Fourier-transformed eigenmode function is used to satisfy the four necessary conditions of completeness, orthogonality, localization, and adaptability. Finally, the development trend of ethnic folk dances and the change in communication dynamics are analyzed. The results show that the change of language on the spreading attitude of folk dance is 4.381, the change of custom on the spreading attitude is 5.943, the change of culture on the spreading attitude is 10.866, and the change of culture on the spreading attitude of folk dance is the largest, which shows that the development of folk dance away from the development of folk dance is not separated from the support of culture. The study will have a positive impact on passing on the distinctive art of ethnic folk dance and enriching Chinese traditional culture and art.

In this paper, we first use the short-time Fourier transform method to extract statistical features in the frequency domain of vocal music. The extracted features are fused using D − S -evidence theory. The fused vocal features are inputted into the improved deep learning network to construct a vocal singing style classification model. Secondly, the requirements of vocal music resources according to the classification of song styles are constructed for the vocal singing resource library system. Finally, the vocal music resource library system undergoes testing in all directions to ensure it meets both functional and performance requirements. The results show that under the respective optimal threads of the vocal music resource library, the number of DM7 network reads and writes remains between 200 and 300 kb, and the random read performance of HBase reaches 8340 TPS, indicating that the resource library provides users with a fast and convenient way to retrieve multidimensional resources. This paper provides a long-term reference for the preservation and use of vocal singing resources.

Pipa fretting and pitch measurement are crucial aspects of playing. In this paper, Pipa’s string vibration is simplified to bounded string vibration, and a simplified structural model of Pipa is built. Transverse free vibration is used to establish the vibration equations of Pipa strings, and differential analysis is used to analyze the standing waves. Meanwhile, the vibration of the resonance panel of the Pipa was abstracted into a dynamic model, and the modal analysis was performed using coordinate transformation. In the actual test, the sound signal generated by the vibration was also processed with the help of the Fourier transform, and the pitch of the Pipa was measured by analyzing the time and frequency domains. The vibration of the Pipa panel is complex but regular, with the vibration pitch line at 32.5 hz being a central axis and the vibration pitch line at the frequency of 85 hz being on the upper side, while the vibration corresponding to the frequencies of 32.5 hz and 85 hz can be regarded as a synthesis of the vibration corresponding to the frequency of 175.5 hz. Under the viewpoint of vibration modal analysis, the performance effect of the Pipa has a more refined expression.

In order to further reduce the errors arising in synchronous sampling, this paper designs a multi-channel hybrid sampling electrical parametric feature modeling method based on an AIOT sensing module in the process of design. The AIOT sensing module is used to process the received RF signals using a transconductance amplifier to improve the digitization and reconfigurability of the RF front-end circuit. The signal is transformed and calculated using the Fast Fourier Transform to obtain electrical parameters such as the amplitude of each harmonic and the power factor. The results show that the variation amplitude of the window function of the design method in this paper is generally maintained between -1.5 and 1.5, the amplitude accuracy of the active power rate is within 3.17555%, and the relative error accuracy is within 0.000951%. It can be seen that the AIOT-based sensing module can significantly improve the calculation accuracy of electrical parameters and can be applied to the process of sampling digital measurements of other power signal characteristics.

Childhood is a critical stage for the development of perceptual and motor abilities, and strengthening the training of children with incomplete mental development at this stage will affect the development of motor skills during their growth. In this paper, we take VR technology as a starting point to build a perceptual model and introduce a convolutional sparse representation algorithm. First, a sparse representation with few non-zero elements is found to optimize a function consisting of a data fidelity term and a sparse induced penalty function. Then, the sum of the convolution of the filter and the convolution sparse feature map, i.e., the convolution operation, is computed to generate the translation invariants. Then the convolutional sparse coding method is introduced to the traditional unsupervised problem by calculating the minimization objective function and solving it in an iterative manner alternatively. Finally, the constituents of the signal are analyzed and the discrete equivalence of the convolution is derived based on the Fourier transform to derive the intervening variables. The experimental results showed that the mean value of the post-test of motor ability compared with the pre-test increased by 4.6 through an eight-week VR sports game training intervention study test on different children with incomplete mental development. Therefore, it is of great theoretical and practical significance to understand the characteristics of perceptual and motor abilities of children with incomplete intellectual development and to develop corresponding programs for VR sports game training according to their characteristics.

The principle of the Fourier transform is explored in this paper to compare protein structure similarity, transforming protein structure into a distance sequence, and performing spectral analysis on a fast Fourier transform. Secondly, protein fraction classification and similarity analysis of whole-seed wheat protein fractions were performed using the fast Fourier transform. Fourier transform infrared spectra were analyzed using two parameters: diatomic vibrations and molecular leaps. Finally, the whole seed wheat protein fraction content test analysis was analyzed experimentally. The results showed that the spectral range of whole seed wheat protein fractions was selected from 10539 cm−1 to 6080 cm−1, the fraction of fractions was determined as 9, the parameter R2 val was 0.9614. This paper provides practical reference material for researching whole-seed wheat protein fields.

In this paper, the Skip-gram model is used to process the main melody information of contextualized music, combining feature extraction and Schenkel analysis to extract the main melody note sets and vectors. By applying the short-time Fourier transform (STFT) to the audio signal, the spectral center of mass and irregularities can be calculated to represent the emotional features of the music. To obtain music features based on wavelet coefficients, the music signal is decomposed, and each scale’s signal features are counted individually. Finally, by taking the musical elements as independent variables and the perceived results of pleasure and activation in the contextual model as dependent variables, we compared the variability of the dynamic associations between emotional perceptions and the musical elements among different contextualized music pieces. The results showed that the overall differences in emotional cognition across musical performances ranged between ±0.5, with the mean short-term energy amplitude of relatively cheerful music ranging between ±0.2, which was smaller than that of angry (±0.3) and sad (±0.5) music. In this study, musical works were accurately characterized, and their expressive and infectious power was effectively enhanced through contextual construction.

In this paper, the weighted class fractional Fourier transform is redefined by means of the weighted superposition of state functions, and the Fourier transform is applied to dissect the intrinsic relationship between the M-WFRFT and the 4-WFRFT and to obtain the weighting coefficients of the electronic communication signals. On the basis of the WFRFT discrete algorithm, the electronic communication signal is expanded to include M a term after the M -WFRFT and the hybrid carrier modulation method based on the WFRFT is given. According to the design idea of the electronic communication system, the hardware part and software part of the electronic communication system are designed, and the electronic communication signal modulation recognition system is experimentally analyzed by using the simulation experiment method. The results show that when the signal-to-noise ratio is -8dB and the optimization algorithms are the Sgd algorithm, Sgdm algorithm, and Adam algorithm, the modulation identification accuracy of OFDM signal is 39.3%, 57.5% and 43.5%, respectively, which indicates that the Sgdm optimization algorithm is the most suitable one compared with other algorithms, and it is more effective in the modulation identification of OFDM signals at low signal-to-noise ratio. The number of scans of the three-layer multi-parameter WFRFT signal by the non-cooperative party is 6.4 × 1013 times, and the increase in the number of scans leads to an increase in the cost paid by the non-cooperative party, which confirms that the secure communication system based on three-layer weighted fractional-order Fourier transform has good resistance to parameter scanning characteristics and high confidentiality. This study further improves the a priori parameter setting of the electronic communication system and enhances its complexity, resulting in better anti-interception identification features.

In this paper, in the process of music teaching, the music notes characterized in the time domain are Fourier transformed to the frequency domain, and the resulting notes in the frequency domain are analyzed and processed to obtain the inverted spectral domain features of the notes. On the basis of the cepstrum features, the music recognition model based on the depth confidence network is constructed, and after the training of the depth confidence network, the overfitting phenomenon often occurs for the depth confidence network, and the optimization is carried out by embedding the Dropout method between the implicit layers. From the perspective of music recognition and informationized music teaching, the research on traditional music culture integration of informationized music teaching is designed, and statistical analysis and simulation analysis methods are constructed. The results show that better recognition performance than using music features from different convolutional layers can be obtained using Deep Confidence Networks, with values of 77.5%, 78.8%, and 78.2%, respectively, and that music recognition research based on Deep Confidence Networks is able to better explore and pass on traditional music culture. In the linear regression analysis between the factors of incorporating folk songs into music teaching, the Sig. F and Sig values are 0, which are smaller than the significance level of 0.01 and 0.05, indicating that there is a significant relationship between the factors of students’ gender, age, whether they like folk songs, and the channels of exposure to folk songs and whether folk songs are incorporated into the music teaching program of colleges and universities. This study takes folk songs as representative of traditional music culture, raises people’s awareness of the value of folk songs, and enhances their understanding of the importance of music culture inheritance.

This paper presents a study on the determination of synthetic pigments using high performance liquid chromatography (HPLC) method combined. A retention value qualitative approach, combined with an uncertainty assessment algorithm for the determination of pigment content, was used for the simultaneous determination of nine synthetic pigments, namely, lemon yellow, amaranthine red, indigo, carmine, sunset yellow, brilliant blue, seductive red, erythrosine, and seductive red, in foodstuffs by reversed-phase high-performance liquid chromatography (RP-HPLC). The sample pretreatment method was optimized, and the chromatographic conditions were set to investigate the UV determination wavelength, mobile phase, column temperature, and wavelength of synthetic pigments. Under the same mobile phase and column conditions, the results indicated that the components’ retention times did not significantly fluctuate with the change in column temperature. The results showed that the chromatographic response signals of lemon yellow, sunset yellow and seductive red were higher at the UV determination wavelength of 254 nm, so 254 nm was finally determined as the UV detection wavelength in this experiment. Mobile phase A: methanol, B: ammonium acetate (0.02mo/L) using gradient elution, the separation degree R>1.5, and the symmetry and stability of the chromatogram were better. The separation of the components was good, and the peak shape was sharp and symmetric when the column temperature was 35 ℃, so the column temperature was chosen to be 35 ℃, and the establishment of the chromatographic conditions was thus completed.

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions.

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation.

This study examines the pace at which solutions to a Bresse system in combination with the Cattaneo law of heat conduction and the dispersed delay term degradation. We establish our major finding utilizing the energy approach in the Fourier space.

In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with thermal effect and damping term. The heat conduction is given by the theory of Lord-Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord-Shulman's theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow.

In this paper, we introduce a convolution-type product for strongly integrable Hilbert $C^*$-module valued maps on locally compact groups. We investigate various properties of this product related to uniform continuity, boundless, etc. For instance, we prove a convolution theorem. Also, we study the boundless of the related convolution operator in various settings.

Calcium is an essential element in our body and plays a vital role in moderating calcium signalling. Calcium is also called the second messenger. Calcium signalling depends on cytosolic calcium concentration. In this study, we focus on cellular calcium fluctuations with different buffers, including calcium-binding buffers, using the Hilfer fractional advection-diffusion equation for cellular calcium. Limits and start conditions are also set. By combining with intracellular free calcium ions, buffers reduce the cytosolic calcium concentration. The buffer depletes cellular calcium and protects against toxicity. Association, dissociation, diffusion, and buffer concentration are modelled. The solution of the Hilfer fractional calcium model is achieved through utilizing the integral transform technique. To investigate the influence of the buffer on the calcium concentration distribution, simulations are done in MATLAB 21. The results show that the modified calcium model is a function of time, position, and the Hilfer fractional derivative. Thus the modified Hilfer calcium model provides a richer physical explanation than the classical calcium model.

This investigation theoretically describes the exact solution of an unsteady fractional a second-grade fluid upon a bottom plate constrained by two walls at the sides which are parallel to each other and are normal to the bottom plate. The flow in the fluid is induced by the time dependent motion of the bottom plate. Initially the flow equation along with boundary and initial conditions are considered which are then transformed to dimensionless notations using suitable set of variables. The Laplace as well as Fourier transformations have been employed to recover the exact solution of flow equation. The time fractional differential operator of Caputo-Fabrizio has been employed to have constitutive equations of fractional order for second-grade fluid. After obtaining the general exact solutions for flow characteristics, three different cases at the surface of bottom plate are discussed; namely (i) Stokes first problem (ii) Accelerating flow (iii) Stokes second problem. It has noticed in this study that, for higher values of Reynolds number the flow characteristics have augmented in all the three cases. Moreover, higher values of time variable have supported the flow of fractional fluid for impulsive and constantly accelerated motion and have opposeed the flow for sine and cosine oscillations.

To further enhance the security of image encryption, a new chaos-based image encryption algorithm (IEA) based on discrete Fourier transform and Joseph traversal is proposed to encrypt the plain image in both the frequency domain and space domain simultaneously. In the proposed IEA, the logistic map is used to generate the appropriate chaotic sequence, and the improved Joseph traversal is used to scramble the image in different starting positions and variable step sizes. Then, block diffusion is performed at the end. The main finding concerning the proposed IEA is that the combination of discrete Fourier transform and Joseph traversal can enhance the security of the image information, which has been validated by measuring the performance in resisting the common types of attacks.

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations.

Background. Singular integral equations in degenerate cases describe many processes in natural science and technology. The theory of these equations has been studied quite well, but as far as the authors know, there are currently no analytical methods for solving them. In this regard, there is a need to construct approximate methods for solving singular integral equations in degenerate cases. The article is devoted to the construction of such methods, which determines its relevance. Materials and methods. When constructing approximate methods, iteration-projection methods are used. Results and conclusions. A spline-collocation method for solving a degenerate singular characteristic equation is constructed. A two-stage approximate method is proposed for solving complete singular integral equations in degenerate cases and their characteristic equations.

This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.

In Electromagnetics, the field scattered by an ensemble of particles—of arbitrary size, shape, and material—can be obtained by solving the Lippmann–Schwinger equation. This singular vectorial integral equation is generally formulated in the direct space Rn (typically n=2 or n=3). In the article, we rigorously computed the Fourier transform of the vectorial Lippmann–Schwinger equation in the space of tempered distributions, S′(R3), splitting it in a singular and a regular contribution. One eventually obtains a simple equation for the scattered field in the Fourier space. This permits to draw an explicit link between the shape of the scatterer and the field through the Fourier Transform of the body indicator function. We compare our results with accurate calculations based on the T-matrix method and find a good agreement.

Let $ X $ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of nonpositive curvature, and in particular all known examples of non-compact harmonic manifolds except for the flat spaces. We use the Fourier transform from [1] to investigate the dynamics on $ L^p(X) $ for $ p > 2 $ of certain bounded linear operators $ T : L^p(X) \to L^p(X) $ which we call "$ L^p $-multipliers" in accordance with standard terminology. Examples of $ L^p $-multipliers are given by the operator of convolution with an $ L^1 $ radial function, or more generally convolution with a finite radial measure. In particular elements of the heat semigroup $ e^{t\Delta} $ act as multipliers. Given $ 2 < p < \infty $, we show that for any $ L^p $-multiplier $ T $ which is not a scalar multiple of the identity, there is an open set of values of $ \nu \in {\mathbb C} $ for which the operator $ \frac{1}{\nu} T $ is chaotic on $ L^p(X) $ in the sense of Devaney, i.e., topologically transitive and with periodic points dense. Moreover such operators are topologically mixing. We also show that there is a constant $ c_p > 0 $ such that for any $ c \in {\mathbb C} $ with $ \operatorname{Re} c > c_p $, the action of the shifted heat semigroup $ e^{ct} e^{t\Delta} $ on $ L^p(X) $ is chaotic. These results generalize the corresponding results for rank one symmetric spaces of noncompact type and harmonic $ NA $ groups (or Damek-Ricci spaces).

The main purpose of this paper is to study a class of the $ p $-ary functions $ f_{\lambda, u, v}(x) = Tr_1^k(\lambda x^{p^k+1})+Tr^n_1(ux)Tr_1^n(vx) $ for any odd prime $ p $ and $ n = 2k, \lambda\in GF(p^k)^*, u, v\in GF(p^n)^*. $ With the help of Fourier transforms, we are able to subdivide the class of all $ f_{\lambda, u, v} $ into sublcasses of bent, near-bent and 2-plateaued functions. It is shown that the choice of $ \lambda, u $ and $ v $, ensuring that $ f $ is bent, 2-plateaued or near-bent, is directly related to finding the subset $ A\subset GF(p)^3 $. The efficient method for defining the set $ A\subset GF(p)^3 $ is described in detail.

The multi-resolution analysis (MRA) associated with quadratic phase Fourier transform (QPFT) serves as a tool to construct orthogonal bases of the L2(R). Consequently, it assumes a pivotal role in facilitating potential applications of QPFT. Inspired by the sampling theorem applicable to band-limited signals in the QPFT domain, this paper formulates the development of the MRA linked with QPFT. Subsequently, we develop a method for constructing orthogonal bases for L2(R), followed by some examples.

Edge detection is a technique in digital image processing that detects the contours of objects based on changes in brightness. Edges can be used to determine the size, orientation, and properties of the object of interest within an image. There are different techniques employed for edge detection, one of them being phase congruency, a recently developed but still relatively unknown technique due to its mathematical and computational complexity compared to more popular methods. Additionally, it requires the adjustment of a greater number of parameters than traditional techniques. Recently, a unique formulation was proposed for the mathematical description of phase congruency, leading to a better understanding of the technique. This formulation consists of three factors, including a quantification function, which, depending on its characteristics, allows for improved edge detection. However, a detailed study of the characteristics had not been conducted. Therefore, this article proposes the development of a generalized function for quantifying phase congruency, based on the family of functions that, according to a previous study, yielded the best results in edge detection.

In this study, the Wigner–Ville distribution is associated with the one sided Clifford–Fourier transform over Rn, n = 3(mod 4). Accordingly, several fundamental properties of the WVD-CFT have been established, including non-linearity, the shift property, dilation, the vector differential, the vector derivative, and the powers of τ∈Rn. Moreover, powerful results on the WVD-CFT have been derived such as Parseval’s theorem, convolution theorem, Moyal’s formula, and reconstruction formula. Eventually, we deduce a directional uncertainty principle associated with WVD-CFT. These types of results, as well as methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

In the present work, the main objective is to find the solution of the generalized heat and generalized Laplace equations using the fractional Fourier transform, which is a general form of the solution of the heat equation and Laplace equation using the classical Fourier transform. We also formulate its solution using a sampling formula related to the fractional Fourier transform. The fractional Fourier transform is introduced, and related theorems and essential properties are collected. Several results related to the sampling formula are derived. A few examples are presented to illustrate the effectiveness and powerfulness of the proposed method compared to the classical Fourier transform method.

In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,\theta}(\mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,\theta}B(\mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,\theta}(\mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(\mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 \leqslant p \leqslant q \leqslant \infty$, $(p,q)\neq \{(1,1), (\infty, \infty)\}$, $d\geq 1$. In the case $2

As one of the main directions of applied mathematics research, inverse Fourier transform (FT) has been widely used in image speech analysis and other fields in recent decades of development. FT is the basic content of digital image processing technology. In practical analysis, image design and interaction can be realised by using time-space domain and frequency domain, which can accurately obtain image information characteristics and achieve the expected application goals. In this paper, based on the understanding of FT and inverse transform, an improved algorithm is used to lay the foundation for the realisation of image design and interactive technology.

In this paper, we introduce the notion of periodic wavelet bi-frames on local fields and establish the theory for the construction of periodic Bessel sequences and periodic wavelet bi-frames on local fields.

In this paper, we introduce the structure of nonuniform super wavelets over local fields. We shall also provide the characterization of nonuniform parseval frame, nonuniform semi-orthogonal pareseval multiwavelets, and nonuniform super wavelets over local fields.

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

We establish the existence of global solutions and $L^q$ time-decay of a three dimensional chemotaxis system with chemoattractant and repellent. We show the existence of global solutions by the energy method. We also study $L^q$ time-decay for the linear homogeneous system by using Fourier transform and finding Green's matrix. Then, we find $L^q$ time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimate.

In this paper, we generalize the N-dimensional Heisenberg’s inequalities for the windowed linear canonical transform (WLCT) of a complex function. Firstly, the definition for N-dimensional WLCT of a complex function is given. In addition, the N-dimensional Heisenberg’s inequality for the linear canonical transform (LCT) is derived. It shows that the lower bound is related to the covariance and can be achieved by a complex chirp function with a Gaussian function. Finally, the N-dimensional Heisenberg’s inequality for the WLCT is exploited. In special cases, its corollary can be obtained.

The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations.

The recovery of bandlimited signals with high dynamic range is a hot issue in sampling research. The unlimited sampling theory expands the recordable range of traditional analog-to-digital converters (ADCs) arbitrarily, and the signal is folded back into a low dynamic range measurement, avoiding the saturation problem. Since the non-bandlimited signal in the Fourier domain cannot be directly applied to its existing theory, the non-bandlimited signal in the Fourier domain may be bandlimited in the fractional Fourier domain. Therefore, this brief report studies the unlimited sampling problem of high dynamic non-bandlimited signals in the Fourier domain based on the fractional Fourier transform. Firstly, a mathematical signal model for unlimited sampling is proposed. Secondly, based on this mathematical model, the annihilation filtering method is used to estimate the arbitrary folding time. Finally, a novel fractional Fourier domain unlimited sampling theorem is obtained. The theory proves that, based on the folding characteristics of the self-reset ADC, the number of samples is not affected by the modulo threshold, and any folding time can be handled.

A mixed signal with several unknown modes is common in the industry and is hard to decompose. Variational Mode Decomposition (VMD) was proposed to decompose a signal into several amplitude-modulated modes in 2014, which overcame the limitations of Empirical Mode Decomposition (EMD), such as sensitivity to noise and sampling. We propose an improved VMD, which is simplified as iVMD. In the new algorithm, we further study and improve the mathematical model of VMD to adapt to the decomposition of the broad-band modes. In the new model, the ideal flattest response is applied, which is derived from the mathematical integral form and obtained from different-order derivatives of the improved modes’ definitions. The harmonics can be treated via synthesis in our new model. The iVMD algorithm can decompose the complex harmonic signal and the broad-band modes. The new model is optimized with the alternate direction method of multipliers, and the modes with adaptive broad-band and their respective center frequencies can be decomposed. the experimental results show that iVMD is an effective algorithm based on the artificial and real data collected in our experiments.

The Hilbert transform is essentially the only singular operator in one dimension. This undoubtedly make it one of the most important linear operator in harmonic analysis. This is an expository paper about the Hilbert transform aimed to anyone that has even scratched the surface of the theory of integration, and functional analysis as well as a basic rudiments of Fourier transform. We provide a systematic (Although by no means complete) account of the basic results on the Hilbert transform. We want to point out that we present a friendly proof of the remarkable result due to Stein and Weiss [Math Mech. 8, 1959] and we use it combined with the Cavalieri principle to obtain an exact formula for the Lp-norm of H(χE).

Let A and B be Borel subsets of the Euclidean n-space with dimA+dimB>n. This is a survey on the following question: what can we say about the Hausdorff dimension of the intersections A∩(g(B)+z) for generic orthogonal transformations g and translations by z?