Search Results (19)

This paper explores the structural representation and Fourier analysis of elements in Horváth distribution spaces ${\mathcal{S}}_k^{\prime }$, for $k<-n$. We prove that any element in ${\mathcal{S}}_k^{\prime }$ can be expressed as a finite sum of derivatives of continuous $L^1\left({\mathbb{R}}^n\right)$-functions acting on Schwartz test functions. This representation leads to an explicit expression for their distributional Fourier transform in terms of classical Fourier transforms. Additionally, we present a distributional representation for the convolution of two such elements, showing that the convolution is well-defined over $\mathcal{S}$. These results deepen our understanding of non-tempered distributions and extend Fourier methods to a broader functional framework. Full article

Probability models are instrumental in a wide range of applications by being able to accurately model real-world data. Over time, numerous probability models have been developed and applied in practical scenarios. This study introduces the AAP-X family of distributions—a novel, flexible framework for continuous data analysis named after authors Aadil Ajaz and Parvaiz. The proposed family effectively accommodates both symmetric and asymmetric characteristics through its shape-controlling parameter, an essential feature for capturing diverse data patterns. A specific subclass of this family, termed the “AAP Exponential” (AAPEx) model is designed to address the inflexibility of classical exponential distributions by accommodating versatile hazard rate patterns, including increasing, decreasing and bathtub-shaped patterns. Several fundamental mathematical characteristics of the introduced family are derived. The model parameters are estimated using six frequentist estimation approaches, including maximum likelihood, Cramer–von Mises, maximum product of spacing, ordinary least squares, weighted least squares and Anderson–Darling estimation. Monte Carlo simulations demonstrate the finite-sample performance of these estimators, revealing that maximum likelihood estimation and maximum product of spacing estimation exhibit superior accuracy, with bias and mean squared error decreasing systematically as the sample sizes increases. The practical utility and symmetric–asymmetric adaptability of the AAPEx model are validated through five real-world applications, with special emphasis on cancer survival times, COVID-19 mortality rates and reliability data. The findings indicate that the AAPEx model outperforms established competitors based on goodness-of-fit metrics such as the Akaike Information Criteria (AIC), Schwartz Information Criteria (SIC), Akaike Information Criteria Corrected (AICC), Hannan–Quinn Information Criteria (HQIC), Anderson–Darling (A*) test statistic, Cramer–von Mises (W*) test statistic and the Kolmogorov–Smirnov (KS) test statistic and its associated p-value. These results highlight the relevance of symmetry in real-life data modeling and establish the AAPEx family as a powerful tool for analyzing complex data structures in public health, engineering and epidemiology. Full article

Harmonic synthesis describes translation invariant linear spaces of continuous complex valued functions on locally compact abelian groups. The basic result due to L. Schwartz states that such spaces on the reals are topologically generated by the exponential monomials in the space; in other words, the locally compact abelian group of the reals is synthesizable. This result does not hold for continuous functions in several real variables, as was shown by D.I. Gurevich’s counterexamples. On the other hand, if two discrete abelian groups have this synthesizability property, then so does their direct sum, as well. In this paper, we show that if two locally compact abelian groups have this synthesizability property and at least one of them is discrete, then their direct sum is synthesizable. In fact, more generally, we show that any extension of a synthesizable locally compact abelian group by a synthesizable discrete abelian group is synthesizable. This is an important step toward the complete characterization of synthesizable locally compact abelian groups. Full article

This paper is devoted to the study of a multi-parameter subsequential version of the “Wiener–Wintner” ergodic theorem for the noncommutative Dunford–Schwartz system. We establish a structure to prove “Wiener–Wintner”-type convergence over a multi-parameter subsequence class $\Delta$ instead of the weight class case. In our subsequence class, every term of $\underline{\mathbf{k}}\in \Delta$ is one of the three kinds of nonzero density subsequences we consider. As key ingredients, we give the maximal ergodic inequalities of multi-parameter subsequential averages and obtain a noncommutative subsequential analogue of the Banach principle. Then, by combining the critical result of the uniform convergence for a dense subset of the noncommutative $L_p\left(\mathcal{M}\right)$ space and the noncommutative Orlicz space, we immediately obtain the main theorem. Full article

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. Full article

In this work, we pose and solve, in tempered distribution spaces, an open problem proposed by Schrödinger in 1925. In particular, on the Schwartz distribution spaces, we define the linear continuous quantum operators associated with relativistic Hamiltonians of massive particles—particles with rest mass different from 0 and evolving in the four-dimensional Minkowski vector space ${\mathbb{M}}_4$. In other words, upon the tempered distribution state-space ${\mathcal{S}}^{\prime }({\mathbb{M}}_4,\mathbb{C})$, we have found the most natural way to introduce the free-particle relativistic Hamiltonian operator and its corresponding Schrödinger equation (together with its conjugate equation, standing for antiparticles). We have found the entire solution space of our relativistic linear continuous evolution equation by completely solving a division problem in tempered distribution space. We define the Hamiltonian (Schwartz diagonalizable) operator as the principal square root of a strictly positive, Schwartz diagonalizable second-order differential operator (linked with the “Klein–Gordon operator” on the tempered distribution space ${\mathcal{S}}_4^{\prime }$). The principal square root of a Schwartz nondefective operator is defined in a straightforward way—following the heuristic fashion of some classic and greatly efficient quantum theoretical approach—in the paper itself. Full article

Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued analytic functions in tubes $T^C={\mathbb{R}}^n+iC$ obtain vector-valued tempered distributions as boundary values. In a certain vector-valued case, we study the structure of this boundary value, which is shown to be the Fourier transform of the distributional derivative of a vector-valued continuous function of polynomial growth. A set of vector-valued functions used to show the structure of the boundary value is shown to have a one–one and onto relationship with a set of vector-valued distributions, which generalize the Schwartz space ${\mathcal{D}}_{L^2}^{\prime }\left({\mathbb{R}}^n\right)$; the tempered distribution Fourier transform defines the relationship between these two sets. By combining the previously stated results, we obtain a Cauchy integral representation of the vector-valued analytic functions in terms of the boundary value. Full article

This paper presents the characteristics of air void systems in hardened concrete with the method of digital image analysis (DIA) coupled with Schwartz-Saltykov (SS) conversion. The results indicate that the DIA method coupled with SS conversion estimates the air content with more accuracy than it would without SS conversion; the correlation between air content obtained from the DIA method, and that from the thin section (TS) method is as good as the correlation observed between the pressure saturation (PS) method and the TS method. It was also found that the DIA method shows a better correlation with the TS method when the spacing factor without SS conversion is considered, while both methods show poor correlations when the corresponding specific surface is considered. In addition, it indicates that the peak of three-dimensional size distribution (3-DSD) of air voids after SS conversion falls in smaller voids, and 3-DSD of air voids shifts to a narrow size range, in comparison with the 2-DSD without SS conversion; the shape of the 3-DSD air voids remains constant irrespective of the class widths. Increasing the number of classes can minimise the standard deviation in the estimation, however, it also results in a leap in voids volume density, which will influence the estimation of air content. Full article

In this paper we first introduce multilinear fractional wavelet transform on ${\mathbb{R}}^n\times {\mathbb{R}}_{+}^n$ using Schwartz functions, i.e., infinitely differentiable complex-valued functions, rapidly decreasing at infinity. We also give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. We then study boundedness of the multilinear fractional wavelet transform on Lebesgue spaces and Lorentz spaces. Full article

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form $-\Delta u=0$ in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric provided $n\ne 4$. Using then separation of variables, the solutions u can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space $SL(n,{\mathbb{R}}^{})/SO\left(n\right)$. Since one can define a Schwartz space and tempered distributions in $SL(n,{\mathbb{R}}^{})/SO\left(n\right)$ as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator. Full article

The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expression is better than the other depending on the situation. Likewise, the Cartesian form and the polar form can be defined in a complex fuzzy number set. First, the complex fuzzy numbers (CFNs) are categorized into two types, the polar form and the Cartesian form, as type I and type II. The properties of the complex fuzzy number set of those two expressions are discussed, and how the expressions can be used practically is shown through an example. Second, we study the complex fuzzy inner product structure in each category and find the non-existence of an inner product on CFNs of type I. Several properties of the fuzzy inner product space for type II are proposed from the modulus that is newly defined. Specfically, the Cauchy-Schwartz inequality for type II is proven in a compact way, not only the one for fuzzy real numbers. In fact, it was already discussed by Hasanhani et al; however, they proved every case in a very complicated way. In this paper, we prove the Cauchy-Schwartz inequality in a much simpler way from a general point of view. Finally, we introduce a complex fuzzy scalar product for the generalization of a complex fuzzy inner product and propose to study the condition for its existence on CFNs of type I. Full article

Background and objectives: Dental crowding is more pronounced in the mandible than in the maxilla. When exceeding a significant amount, the creation of new space is required. The mandibular expansion devices prove to be useful even if the increase in the lower arch perimeter seems to be just ascribed to the vestibular inclination of teeth. The aim of the study was to compare two activation protocols of the Schwartz appliance in terms of effectiveness, particularly with regard to how quickly crowding is solved and how smaller is the increasing of vestibular inclination of the mandibular molars. Materials and Methods: We compared two groups of patients treated with different activation’s protocols of the lower Schwartz appliance (Group 1 protocol consisted in turning the expansion screw half a turn twice every two weeks and replacing the device every four months; Group 2 was treated by using the classic activation protocol—1/4 turn every week, never replacing the device). The measurements of parameters such as intercanine distance (IC), interpremolar distance (IPM), intermolar distance (IM), arch perimeter(AP), curve of Wilson (COW), and crowding (CR) were made on dental casts at the beginning and at the end of the treatment. Results: A significant difference between protocol groups was observed in the variation of COWL between time 0 and time 1 with protocol 1 with protocol 1 subjects showing a smaller increase in the parameter than protocol 2 subjects. The same trend was observed also for COWR, but the difference between protocol groups was slightly smaller and the interaction protocol-by-time did not reach the statistical significance. Finally, treatment duration in protocol 1 was significantly lower than in protocol 2. Conclusion: The results of our study suggest that the new activation protocol would seem more effective as it allows to achieve the objective of the therapy more quickly, and likely leading to greater bodily expansion. Full article

First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwartz inequality is also proved. Full article

In this work, we define the ultrahyperbolic Klein-Gordon operator of order $\alpha$ on the function f by $T^{\alpha }(f)=W_{\alpha }\ast f,$ where $\alpha \in \mathbb{C},W_{\alpha }$ is the ultrahyperbolic Klein-Gordon kernel, the symbol ∗ denotes the convolution, and $f\in \mathcal{S},\mathcal{S}$ is the Schwartz space of functions. Our purpose of this work is to study the convolution of $W_{\alpha }$ and obtain the operator $L^{\alpha }={\left[T^{\alpha }\right]}^{-1}$ such that if $T^{\alpha }(f)=\phi$ , then $L^{\alpha }\phi =f.$ Full article

In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution $f\in S^{{}^{\prime }}\left({\mathbb{R}}^n\right)$ with wavelet kernel $\psi \in S\left({\mathbb{R}}^n\right)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{{}^{\prime }}\left({\mathbb{R}}^n\right)$ . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution. Full article