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A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky. [ABSTRACT FROM AUTHOR]

السذكمة التي تهاجه بعض البيئات مثل شركات التأمين والسؤسدات الحكهمية حيث تتم معالجة كسية هائمة من الهثائق كل يهم ، وبالتالي ، من الزروري وجهد نظام تمقائي لمتعرف عمى الاختام. وبالتالي ، فإن استخلاص ختم عام والتعرف عميه ليس مهسة بديظة لأنه قد يحتهي عمى أشكال وأحجام وخمفيات وأنساط وألهان مختمفة ، علاوة عمى ذلك ، يسكن طباعة الختم عمى السدتشدات ذات الجهدة الديئة والتدوير بزوايا مختمفة. تقدم طريقتشا السقترحة نهجًا جديدًا لسعالجة صهر الاختام السمهنة والتعرف عميها. وتتكهن من أربع م ا رحل هي: استخلاص الاختام والسعالجة السدبقة واستخلاص السي ا زت والسظابقة. يتم استخلاص الاختام لعزل الخمفية السعقدة وإ ا زلة البيانات غير السرغهب فيها أو الزهضاء السحيظة بسشظقة الختم. مرحمة السعالجة السدبقة ضرورية لتحدين سظهع الختم ، وكذلك لمقزاء عمى الدو ا رن الذي يحدث أثشاء عسمية التختيم. في استخلاص السي ا زت ، ستسثل السعمهمات السدتخمرة متجه السي ا زت السرغهب فيها من أجل التسييز بين الظهابع مع الرسم البياني Haar Wavelet باستخدام التهزيع السحمي لمسي ا زت الإحرائية و تحهيل مهجات أخي اً ر، سيتم حفظ كل متجه ميزة مدتخمرة في قاعدة البيانات لسرحمة .Moment والمحظة Histogram السظابقة. تذير نتائج الاختبار إلى أن الشظام السقترح يهفر معدل التعرف العالي لسجسهعتين من السي ا زت )أي معدل الاعت ا رف بشدبة 99.99 ٪ لمتهزيع السحمي لمسي ا زت الإحرائية ومعدل التعرف بشدبة 90.69 ٪ لتحهيل .)Moment والمحظة Histogram مع الرسم البياني Haar Wavelet مهجات [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of widely used vibration equation with very large membrane is considered. A collocation method with Haar wavelet (HW) is applied for this purpose. The algorithm based on proposed method is very simple and easy to implement. Highly accurate approximate solutions are obtained with a reasonable number of collocation points. The method is tested upon several different exacts solutions of the vibration equation including polynomials and infinite series solutions. The numerical results confirm the accuracy, efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Highlights • A wavelet deconvolution technique for impact force reconstruction is developed by using the Haar wavelet. • The ill-posed nature of deconvolution is mitigated by controlling not only scale but also shift components. • The optimal scales and shifts for reconstruction are determined based on the compromise between the solution and residual norms. • The present technique can be deployed by considering the impulse response function explicitly or implicitly. • The validity of the present technique is verified by numerical experiment on the dynamic crushing of thin-walled columns. Abstract Deconvolution technique has become a useful solution for indirect measurement of impact force, however, a simple deconvolution technique often encounters difficulty and leads to an unsatisfactory reconstruction due to unavoidable error and the ill-posed nature of the inverse problem. This paper is concerned with a deconvolution technique using wavelet approach for reconstructing impact force. Based on the ability of wavelet transform, impact force reconstruction was formulated and implemented through its expansion coefficients of different scaled and shifted versions of wavelets. The reconstruction of impact force thus becomes the reconstruction of its expansion coefficients which are then used to reproduce the profile of impact force. Considerations of appropriate scales and shifts of wavelets in reconstruction can be regarded as a kind of regularization in order to improve the ill-posed problem. The numerical verification on reconstruction of impact force acting on a load-cell during the impact buckling of thin-walled column showed that the present technique is indeed capable to mitigate the ill-posed nature when the scale level of wavelets is controlled properly. The result also revealed that with an appropriate choice of the shifting parameter at a certain scale level, the noise in reconstructed force has been suppressed further. As a consequence, it has been verified that the present wavelet deconvolution technique can successfully reconstruct the impact force with less noise and higher accuracy in comparison with the conventional methods. [ABSTRACT FROM AUTHOR]

Object detection is important in car sharing services. Accuracy, efficiency, and low memory consumption are desirable for object detection in car sharing services. This paper presents a network system that satisfies all these requirements. Our approach first divides the object detection task into multiple simpler local regression tasks. Then, we propose the generalized Haar filter-based convolutional neural network to reduce the consumption of memory and computing resource. To achieve real-time performance, we introduce a sparse window generation strategy to reduce the number of input image patches without sacrificing accuracy. We perform experiments on both vehicle and pedestrian data sets. Experimental results demonstrate that our approach can accurately detect objects under challenging conditions. Note to Practitioners—Object detection is an important part of intelligent vehicle technologies, which play an important role in car sharing services. Object detection provides metadata for collision avoidance, self-driving systems, and driver-assistance systems, which can result in better safety and consumer experiences in car sharing services. Although deep learning has achieved an excellent performance in object detection, they consume a large amount of storage and computing resource, which makes them difficult to be deployed for car sharing services. This paper suggests a novel approach which is based on the generalized Haar filter and the local regression strategy. Our approach is accurate, efficient, and light. The experimental results verify the effectiveness of the proposed approach in car sharing services. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive approach to estimate the energies of the wavelet and scale coefficients of the Haar wavelet transform used in the multifractal modeling of network traffic traces. Simulation results confirm that the estimates obtained for the modeling parameters in the wavelet domain are precise. In addition, we propose an equation to calculate the autocorrelation function of the underlying multifractal model in terms of these wavelet domain parameters. In order to enhance the prediction performance of network traffic traces, the autocorrelation function is used to update orthonormal basis functions in a fuzzy system. To validate the adaptive fuzzy prediction approach, simulations with real network traffic traces are carried out, showing that the proposed algorithm provides lower mean square errors than other algorithms in the literature. [ABSTRACT FROM AUTHOR]

The main result of this paper is a bi-parameter Tb theorem for Littlewood-Paley g-function, where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work with a product measure μ = μn × μm, where the measures μn and μm are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again. [ABSTRACT FROM AUTHOR]

Abstract: We describe representations of groupoid C ∗‐algebras on Hilbert modules over arbitrary C ∗‐algebras by a universal property. For Hilbert space representations, our universal property is equivalent to Renault's integration–disintegration theorem. For a locally compact group, it is related to the automatic continuity of measurable group representations. It implies known descriptions of groupoid C ∗‐algebras as crossed products for étale groupoids and transformation groupoids of group actions on spaces. [ABSTRACT FROM AUTHOR]

A uniqueness theorem and a recovery theorem for the coefficients of series in generalized Haar systems are proved under the assumption that the series converge in measure and satisfy a certain necessary condition on the distribution function of the majorant of partial sums. [ABSTRACT FROM AUTHOR]

It is shown that a locally compact groupoid with open range map does not always admit a Haar system. It then is shown how to construct a Haar system if the stability groupoid and the quotient by the stability groupoid both admit one. [ABSTRACT FROM AUTHOR]

We give a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problem that approximates the original MK problem for each of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem. [ABSTRACT FROM AUTHOR]

In this paper we discuss the uniqueness property of a summation method for multiple series with respect to Vilenkin and generalized Haar systems. It is proved that if the multiple series with respect to these systems is a.e. summable by that method to an integrable function on [0; 1)d and satisfies an extra condition, then it is the Fourier series of this function. [ABSTRACT FROM AUTHOR]

Security has played an essential role in human life and it has been motivated the governments of all regions in the world to make security measures tight by a higher level of safety. Although major progresses are carried out in the iris recognition domain, it will have continuous to be an open issue in the future and it will deserve additional investigation. So, as to refer these difficulties, in this paper a novel iris recognition method for feature extraction is proposed based on 2-D Haar wavelet decomposition approach. It is chosen because it causes to reduce the level of noises of the iris texture effectively. Furthermore, it accelerates the extraction process of the iris pattern and it simplifies the computing process. In order to implement this method, multilevel 2-D wavelet decomposition approach is applied on the iris images which are normalized in the pre-processing step and then, it extracts the features of these images to create a distinctive code. For finding similarity among the two iris images and carrying out the matching process, we use Hamming distance measure in order to obtain high accuracy rate in our proposed iris recognition system. Our experimental results on CASIA-Iris V3 database show the effectiveness of the proposed method in iris recognition system. [ABSTRACT FROM AUTHOR]

We start from the question raised by Williams (Proc. Am. Math. Soc. 2016): Must a second countable, locally compact, transitive groupoid G have open range map? If the answer is positive, the topology of G is in fact locally transitive (in the sense of [Seda, 1976]). We prove that even if the answer is negative, we can replace the original topology of G with a local transitive topology so that the topologies of the r-fibres are not affected. The new topology is locally compact Hausdorff but not necessary second countable. However its full C*-algebra (introduced in [Renault, 1980]) is still isomorphic to C*(H)⊗K(L2(μ)), where H is the isotropy group at a unit u and μ is a positive Radon measure on the unit space. We also present a few remarks concerning the Haar systems on locally compact groupoids and for every locally compact groupoid having paracompact unit space and second countable r-fibres, we prove the existence of a pre-Haar system bounded on the compact sets. [ABSTRACT FROM AUTHOR]

This study is devoted to three-dimensional modeling of small falling rocks in block impact analysis in energy view using the particle flow method. The restitution coefficient of rockfall collision is introduced from the energy consumption mechanism to describe rockfall-impacting properties. Three-dimensional reconstruction of falling block is conducted with the help of spherical harmonic functions that have satisfactory mathematical properties such as orthogonality and rotation invariance. Numerical modeling of the block impact to the bedrock is analyzed with both the sphere-simplified model and the 3D reconstructed model. Comparisons of the obtained results suggest that the 3D reconstructed model is advantageous in considering the combination effects of rockfall velocity and rotations during colliding process. Verification of the modeling is carried out with the results obtained from other experiments. In addition, the effects of rockfall morphology, surface characteristics, velocity, and volume, colliding damping and relative angle are investigated. A three-dimensional reconstruction modulus of falling blocks is to be developed and incorporated into the rockfall simulation tools in order to extend the modeling results at block scale to slope scale. [ABSTRACT FROM AUTHOR]

We study a nonparametric regression model for sample data which is defined on an N -dimensional lattice structure and which is assumed to be strong spatial mixing: we use design adapted multidimensional Haar wavelets which form an orthonormal system w.r.t. the empirical measure of the sample data. For such orthonormal systems, we consider a nonparametric hard thresholding estimator. We give sufficient criteria for the consistency of this estimator and we derive rates of convergence. The theorems reveal that our estimator is able to adapt to the local smoothness of the underlying regression function and the design distribution. We illustrate our results with simulated examples. [ABSTRACT FROM AUTHOR]

The parameters of turbulence measured by means of a Doppler reflectometer at the plasma periphery in an L-2M stellarator and in atmospheric vortices (typhoons and tornadoes) are investigated using the wavelet methods with involvement of the Haar function. The periods of time taken for the transition (a bound of parameters) to occur in the L-2M stellarator plasma and in atmospheric processes are estimated. It is shown that high- and low-frequency oscillations of certain parameters, in particular, pressure, that occur in atmospheric vortices decay or increase at different moments of time, whereas the density fluctuation amplitudes that occur in plasma at different frequencies vary in a synchronous manner. [ABSTRACT FROM AUTHOR]

In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev-Triebel-Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel-Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters. [ABSTRACT FROM AUTHOR]

In this work, we focus on the buildup correction of dose calculation using Haar wavelets in the Tunisian gamma irradiation facility. The buildup effect of gamma rays was used to adjust absorbed dose calculation for different depth in the irradiated products. A buildup study with different product densities was carried out to generalize the dose adjustment approach to any product at any depth. [ABSTRACT FROM AUTHOR]

The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied. [ABSTRACT FROM AUTHOR]

Spiking neural networks (SNNs) are well suited for functions such as data/pattern classification, estimation, prediction, signal processing and robotic control applications. Whereas the real-world embedded applications are often multi-functional with orthogonal or contradicting functional requirements. The EMBRACE hardware modular SNN architecture has been previously reported as an embedded computing platform for complex real-world applications. The EMBRACE architecture employs genetic algorithm (GA) for training the SNN which offers faster prototyping of SNN applications, but exhibits a number of limitations including poor scalability and search space explosions for the evolution of large-scale, complex, real-world applications. This paper investigates the limitations of evolving real-world embedded applications with orthogonal functional goals on hardware SNN using GA-based training. This paper presents a novel, fast and efficient application prototyping technique using the EMBRACE hardware modular SNN architecture and the GA-based evolution platform. Modular design and evolution of a robotic navigational controller application decomposed into obstacle avoidance controller and speed and direction manager application subtasks is presented. The proposed modular evolution technique successfully integrates the orthogonal functionalities of the application and helps to overcome contradicting application scenarios gracefully. Results illustrate that the modular evolution of the application reduces the SNN configuration search space and complexity for the GA-based SNN evolution, offering rapid and successful prototyping of complex applications on the hardware SNN platform. The paper presents validation results of the evolved robotic application implemented on the EMBRACE architecture prototyped on Xilinx Virtex-6 FPGA interacting with the player-stage robotics simulator. [ABSTRACT FROM AUTHOR]

For systems of functions F = { f ∈ K : n ∈ ℕ} and G = { g ∈ K : n ∈ ℕ} , we consider an F-polynomial $$ f={\sum}_{k=1}^n{\uplambda}_k{f}_k $$ , a G-polynomial $$ g={\sum}_{k=1}^n{\uplambda}_k{g}_k $$ , and an F ⊗ G-polynomial $$ h={\sum}_{k,j=1}^n{\uplambda}_{k,j}{f}_k\otimes {g}_j $$ , where ( f ⊗ g )( x, y) = f ( x) g ( y) . By using the well-known Haar's condition from the approximation theory, we study the following problem: Under what assumptions every function h : X × Y → K, such that all x-sections h = h( x, ·) are G-polynomials and all y -sections h = h( ·, y) are F-polynomials, is an F ⊗ G-polynomial? A similar problem is investigated for functions of n variables. [ABSTRACT FROM AUTHOR]

In this paper, we applied the adaptive grid Haar wavelet collocation method (AGHWCM) for the numerical solution of parabolic partial differential equations (PDEs). The approach of AGHWCM for the numerical solution of parabolic PDEs is mentioned, the obtained numerical results, error analysis are presented in figures and tables. This shows that, the AGHWCM gives better accuracy than the HWCM and FDM. Some of the test problems are taken for demonstrating the validity and applicability of the AGHWCM. [ABSTRACT FROM AUTHOR]

A new collocation method based on Haar wavelet is presented for numerical solution of three-dimensional elliptic partial differential equations with Dirichlet boundary conditions. An important advantage of the method is that it can be applied to both linear as well as nonlinear problems. The algorithm based on this new method is simple and can be easily implemented in any programming language. Experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results. The proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations. The numerical experiments confirm the accuracy and diverse applicability of the method. [ABSTRACT FROM AUTHOR]

A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in O ( N log N ) operations under some mild conditions, where N is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method. [ABSTRACT FROM AUTHOR]

In this paper, we obtain five tests (three of which are symmetric) of pointwise convergence of Fourier series with respect to generalized Haar systems; the tests are similar to the Dini convergence tests. It is shown that the Dini convergence tests for Price systems are also valid for generalized Haar systems. It is also shown that the classicalDini convergence test does not apply, in general, even to generalized Haar systems, although the classical symmetric Dini test for generalized Haar systems is valid. Also upper bounds for the Dirichlet kernels for generalized Haar systems are obtained. [ABSTRACT FROM AUTHOR]

Mitotic event recognition is a crucial and challenging task in biomedical applications. In this paper, we introduce the slow feature analysis and propose a fully-automated mitotic event recognition method for cell populations imaged with time-lapse phase contrast microscopy. The method includes three steps. First, a candidate sequence extraction method is utilized to exclude most of the sequences not containing mitosis. Next, slow feature is learned from the candidate sequences using slow feature analysis. Finally, a hidden conditional random field (HCRF) model is applied for the classification of the sequences. We use a supervised SFA learning strategy to learn the slow feature function because the strategy brings image content and discriminative information together to get a better encoding. Besides, the HCRF model is more suitable to describe the temporal structure of image sequences than nonsequential SVM approaches. In our experiment, the proposed recognition method achieved 0.93 area under curve (AUC) and 91% accuracy on a very challenging phase contrast microscopy dataset named C2C12. [ABSTRACT FROM AUTHOR]

We consider the classical ill-posed problem of recovering of a function from noisy Fourier coefficients with respect to the Haar system. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound. [ABSTRACT FROM PUBLISHER]

Given a dynamical system, we say that a performance function has property P if its time averages along orbits are maximized at a periodic orbit. It is conjectured by several authors that for sufficiently hyperbolic dynamical systems, property P should be typical among sufficiently regular performance functions. In this paper we address this problem using a probabilistic notion of typicality that is suitable to infinite dimension: the concept of prevalence as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two symbols, we prove that property P is prevalent in spaces of functions with a strong modulus of regularity. Our proof uses Haar wavelets to approximate the ergodic optimization problem by a finite-dimensional one, which can be conveniently restated as a maximum cycle mean problem on a de Bruijin graph. [ABSTRACT FROM AUTHOR]

Two sharp results for best approximations of periodic functions are established in this paper. We prove the sharpness of the step of the modulus of continuity in Jackson's inequality with least possible constant for approximations by trigonometric polynomials. We also prove the sharpness of the constants in a Jackson-type inequality for approximations by Haar polynomials in several variables. [ABSTRACT FROM AUTHOR]

We obtain a complete characterization of the weak-type (1, 1) for Haar shift operators in terms of generalized Haar systems adapted to a Borel measure µ in the operator-valued setting. The main technical tool in our method is a noncommutative Calderon-Zygmund decomposition valid for arbitrary Borel measures. [ABSTRACT FROM AUTHOR]

In this paper, we present a robust and fast online object tracking algorithm, in which object tracking is achieved by combining L2-regularized least square (L2-RLS) and compressed Haar-like features matching in a Bayesian inference framework. Firstly, the extent of occlusion can be evaluated by L2 tracker. Secondly, the compressed features matching method is used to locate the target object if the extent of occlusion satisfies two inequality constraints. Finally, most of the insignificant samples are removed before computing the compressed features, which makes the computational load of our fused algorithm be only slightly higher than L2 tracker. Both qualitative and quantitative evaluations on numerous challenging image sequences demonstrate that the proposed method is more robust and stable than L2 tracker when the target object undergoes pose variation or rotation, and a paired T-test verifies that it significantly outperforms other state-of-the-art algorithms in terms of accuracy. In addition, our tracker meets the requirement of real-time tracking. [ABSTRACT FROM AUTHOR]

For weighted symmetric (or rearrangement-invariant) spaces with nontrivial Boyd indices and weights from suitable Muckenhoupt classes, the basis property of the Haar system in these spaces and two versions of the direct theorem on the approximation by polynomials in the Haar system are established. [ABSTRACT FROM AUTHOR]

Space-time spectral analysis of tropical cloudiness data shows strong evidence that convectively coupled n = 0 mixed Rossby-gravity waves (MRGs) and eastward inertio-gravity waves (EIGs) occur primarily within the western/central Pacific Ocean. Spectral filtering also shows that MRG and EIG cloudiness patterns are antisymmetric with respect to the equator, and they propagate coherently toward the west and east, respectively, with periods between 3 and 5 days, in agreement with Matsuno's linear shallow-water theory. In contrast to the spectral approach, in a companion paper it has been shown that empirical orthogonal functions (EOFs) of 2-6-day-filtered cloudiness data within the tropical Pacific Ocean also suggest an antisymmetric pattern, but with the leading EOFs implying a zonally standing but poleward-propagating oscillation, along with the associated tropospheric flow moving to the west. In the present paper, these two views are reconciled by applying an independent approach based on a tracking method to assess tropical convection organization. It is shown that, on average, two-thirds of MRG and EIG events develop independently of one another, and one-third of the events overlap in space and time. This analysis also verifies that MRG and EIG cloudiness fields tend to propagate meridionally away from the equator. It is demonstrated that the lack of zonal propagation implied from the EOF analysis is likely due to the interference between eastward- and westward-propagating disturbances. In addition, it is shown that the westward-propagating circulation associated with the leading EOF is consistent with the expected theoretical behavior of an interference between MRGs and EIGs. [ABSTRACT FROM AUTHOR]

For second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence. [ABSTRACT FROM AUTHOR]

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened. [ABSTRACT FROM AUTHOR]

The article focuses on the convergence of Fourier series with respect to general orthonormal systems. Topics discussed include Lebesgue function of the orthonormal system; implying the existence of a function through Banach-Steinhaus theorem; and Walsh system, or a Haar system for the implying the equivalence classes.

Object detection has been attracting much interest due to the wide spectrum of applications that use it. It has been driven by an increasing processing power available in software and hardware platforms. In this work we present a developed application for multiple objects detection based on OpenCV libraries. The complexity-related aspects that were considered in the object detection using cascade classifier are described. Furthermore, we discuss the profiling and porting of the application into an embedded platform and compare the results with those obtained on traditional platforms. The proposed application deals with real-time systems implementation and the results give a metric able to select where the cases of object detection applications may be more complex and where it may be simpler. [ABSTRACT FROM AUTHOR]

We investigate the rearrangement of the Haar system induced by the postorder on the set of dyadic intervals in [0, 1] with length greater than or equal to 2-N. By means of operator norms on BMON we prove that the postorder has maximal distance to the usual lexicographic order. [ABSTRACT FROM AUTHOR]

The article focuses on development of uniqueness theorems involved in the Franklin system. Topics discussed include evolution of first Orthonormal basis by Ph. Franklin, establishment of uniqueness theory for trigonometric series by use of Cantor uniqueness theorem, achievement of mathematical proofs for Haar series by Contor theorem, limited studies on problems related to franklin system series and several theorems are mentioned of Fourier-Franklin series for estimation of Dirichlet kernel.