A review on Harmonic Wavelets and their fractional extension
---
paper_title: A Mathematical Introduction to Wavelets

paper_content:

1. A small sample 2. General constructions 3. Some important wavelets 4. Compactly supported wavelets 5. Multivariable wavelets 6. Function spaces 7. Unconditional convergence 8. Wavelet bases in Lp and H1 9. Wavelets and smoothness of functions.
---
paper_title: Multiscale Analysis of Wave Propagation in Composite Materials

paper_content:

Abstract The multiscale solution of the Klein‐Gordon equations in the linear theory of (two‐phase) materials with microstructure is defined by using a family of wavelets based on the harmonic wavelets. The connection coefficients are explicitly computed and characterized by a set of differential equations. Thus the propagation is considered as a superposition of wavelets at different scale of approximation, depending both on the physical parameters and on the connection coefficients of each scale. The coarse level concerns with the basic harmonic trend while the small details, arising at more refined levels, describe small oscillations around the harmonic zero‐scale approximation.
---
paper_title: Galerkin trigonometric wavelet methods for the natural boundary integral equations

paper_content:

A simply supported boundary value problem of biharmonic equations in the unit disk is reduced into an equivalent second kind natural boundary integral equation (NBIE) with hypersingular kernel (in the sense of Hadamard finite part). Trigonometric Hermite interpolatory wavelets introduced by Quak [Math. Comput. 65 (1996) 683-722] as trial functions are used to its Galerkin discretization with 2^J^+^1 degrees of freedom on the boundary. It is proved that the stiffness matrix is a block diagonal matrix and its diagonal elements are some symmetric and block circulant submatrices. The simple computational formulae of the entries in stiffness matrix are obtained. These show that we only need to compute 2(2^J+J+1) elements of a 2^J^+^2x2^J^+^2 stiffness matrix. The error estimates for the approximation solutions are established. Finally, numerical examples are given.
---
paper_title: Fractional Calculus and Shannon Wavelet

paper_content:

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for any function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grunwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.
---
paper_title: A Multilevel Wavelet Collocation Method for Solving Partial Differential Equations in a Finite Domain

paper_content:

Abstract A multilevel wavelet collection method for the solution of partial differential equations is developed. Two different approaches of treating general boundary conditions are suggested. Both are based on the wavelet interpolation technique developed in the present research. The first approach uses wavelets as a basis and results in a differential-algebraic system of equations, where the algebraic part arises from boundary conditions. The second approach utilizes extended wavelets, which satisfy boundary conditions exactly. This approach results in a system of coupled ordinary differential equations. The method is tested on the one-dimensional Burgers equation with small viscosity. The solutions are compared with those resulting from the use of other numerical algorithms. The present results indicate that the method is competitive with well-established numerical algorithms.
---
paper_title: Shannon wavelets theory.

paper_content:

Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction of functions. The differential properties ::: of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets are -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the -functions.
---
paper_title: Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

paper_content:

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the MittagLeffler function.
---
paper_title: Sinc-Fractional Operator on Shannon Wavelet Space

paper_content:

In this paper the sinc-fractional derivative is extended to the Hilbert space based on Shannon wavelets. Some new fractional operators based on wavelets are defined. One of the main task is to investigate the localization and compression properties of wavelets when dealing with the non-integer order of a differential operator.
---
paper_title: Compactly supported wavelet and the numerical solution of the Vlasov equation

paper_content:

A new scheme for solving the Vlasov equation using a compactly supported wavelets basis is proposed. We use a numerical method which minimizes the numerical diffusion and conserves a reasonable time computing cost. So we introduce a representation in a compactly supported wavelet of the derivative operator. This method makes easy and simple the computation of the coefficients of the matrix representing the operator. This allows us to solve the two equations which result from the splitting technique of the main Vlasov equation. Some numerical results are exposed using different numbers of wavelets.
---
paper_title: Harmonic wavelet analysis

paper_content:

A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.
---
paper_title: Harmonic wavelets towards the solution of nonlinear PDE

paper_content:

In this paper, harmonic wavelets, which are analytically defined and band limited, are studied, together with their differentiable properties. Harmonic wavelets were recently applied to the solution of evolution problems and, more generally, to describe evolution operators. In order to consider the evolution of a solitary profile (and to focus on the localization property of wavelets), it seems to be more expedient to make use of functions with limited compact support (either in space or in frequency). The connection coefficients of harmonic wavelets are explicitly computed (in the following) at any order, and characterized by some recursive formulas. In particular, they are functionally and finitely defined by a simple formula for any order of the basis derivatives.
---
paper_title: Harmonic wavelet analysis

paper_content:

A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.
---
paper_title: Harmonic wavelets towards the solution of nonlinear PDE

paper_content:

In this paper, harmonic wavelets, which are analytically defined and band limited, are studied, together with their differentiable properties. Harmonic wavelets were recently applied to the solution of evolution problems and, more generally, to describe evolution operators. In order to consider the evolution of a solitary profile (and to focus on the localization property of wavelets), it seems to be more expedient to make use of functions with limited compact support (either in space or in frequency). The connection coefficients of harmonic wavelets are explicitly computed (in the following) at any order, and characterized by some recursive formulas. In particular, they are functionally and finitely defined by a simple formula for any order of the basis derivatives.
---
paper_title: Compactly supported wavelet and the numerical solution of the Vlasov equation

paper_content:

A new scheme for solving the Vlasov equation using a compactly supported wavelets basis is proposed. We use a numerical method which minimizes the numerical diffusion and conserves a reasonable time computing cost. So we introduce a representation in a compactly supported wavelet of the derivative operator. This method makes easy and simple the computation of the coefficients of the matrix representing the operator. This allows us to solve the two equations which result from the splitting technique of the main Vlasov equation. Some numerical results are exposed using different numbers of wavelets.
---
paper_title: Fractional Calculus and Shannon Wavelet

paper_content:

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for any function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grunwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.
---
paper_title: Harmonic wavelets towards the solution of nonlinear PDE

paper_content:

In this paper, harmonic wavelets, which are analytically defined and band limited, are studied, together with their differentiable properties. Harmonic wavelets were recently applied to the solution of evolution problems and, more generally, to describe evolution operators. In order to consider the evolution of a solitary profile (and to focus on the localization property of wavelets), it seems to be more expedient to make use of functions with limited compact support (either in space or in frequency). The connection coefficients of harmonic wavelets are explicitly computed (in the following) at any order, and characterized by some recursive formulas. In particular, they are functionally and finitely defined by a simple formula for any order of the basis derivatives.
---
paper_title: Compactly supported wavelet and the numerical solution of the Vlasov equation

paper_content:

A new scheme for solving the Vlasov equation using a compactly supported wavelets basis is proposed. We use a numerical method which minimizes the numerical diffusion and conserves a reasonable time computing cost. So we introduce a representation in a compactly supported wavelet of the derivative operator. This method makes easy and simple the computation of the coefficients of the matrix representing the operator. This allows us to solve the two equations which result from the splitting technique of the main Vlasov equation. Some numerical results are exposed using different numbers of wavelets.
---
paper_title: Fractional Calculus and Shannon Wavelet

paper_content:

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for any function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grunwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.
---
paper_title: Sinc-Fractional Operator on Shannon Wavelet Space

paper_content:

In this paper the sinc-fractional derivative is extended to the Hilbert space based on Shannon wavelets. Some new fractional operators based on wavelets are defined. One of the main task is to investigate the localization and compression properties of wavelets when dealing with the non-integer order of a differential operator.
---
paper_title: Harmonic wavelet analysis

paper_content:

A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.
---
paper_title: A Mathematical Introduction to Wavelets

paper_content:

1. A small sample 2. General constructions 3. Some important wavelets 4. Compactly supported wavelets 5. Multivariable wavelets 6. Function spaces 7. Unconditional convergence 8. Wavelet bases in Lp and H1 9. Wavelets and smoothness of functions.
---
paper_title: Harmonic wavelet analysis

paper_content:

A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.
---
paper_title: Sinc-Fractional Operator on Shannon Wavelet Space

paper_content:

In this paper the sinc-fractional derivative is extended to the Hilbert space based on Shannon wavelets. Some new fractional operators based on wavelets are defined. One of the main task is to investigate the localization and compression properties of wavelets when dealing with the non-integer order of a differential operator.
---
paper_title: On a fractal LC-electric circuit modeled by local fractional calculus

paper_content:

Abstract A non-differentiable model of the LC-electric circuit described by a local fractional differential equation of fractal dimensional order is addressed in this article. From the fractal electrodynamics point of view, the relaxation oscillator, defined on Cantor sets in LC-electric circuit, and its exact solution using the local fractional Laplace transform are obtained. Comparative results among local fractional derivative, Riemann–Liouville fractional derivative and conventional derivative are discussed. Local fractional calculus is proposed as a new tool suitable for the study of a large class of electric circuits.
---
paper_title: Harmonic wavelet analysis

paper_content:

A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.
---
paper_title: Shannon wavelets theory.

paper_content:

Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction of functions. The differential properties ::: of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets are -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the -functions.
---

Title: A Review on Harmonic Wavelets and Their Fractional Extension

Section 1: Introduction
Description 1: Provide an overview of harmonic wavelets, their definition, properties, and applications in numerical approximation of differential problems. Introduce the concept of fractional derivatives, particularly focusing on local fractional derivatives.

Section 2: Wavelets
Description 2: Describe harmonic wavelets (Newland wavelets), including their parameters for scale and localization, and their unique features in function approximation.

Section 3: Harmonic Scaling Function
Description 3: Define the harmonic scaling function and introduce its modified form using the Mittag-Leffler function. Include plots and further descriptions of the real and imaginary parts.

Section 4: Scaling Function in Fourier Domain
Description 4: Describe the Fourier transform of the harmonic scaling function and its properties in the frequency domain, including the fractional scaling function.

Section 5: Harmonic Wavelet Function
Description 5: Define the harmonic (Newland) wavelet function and provide its Fourier transform. Explain the procedure to derive the corresponding wavelet function and its multiresolution analysis.

Section 6: Dilated and Translated Instances
Description 6: Explain the family of dilated and translated harmonic wavelet functions, and compute their Fourier transforms.

Section 7: Multiscale Harmonic Wavelet Reconstruction of Functions
Description 7: Detail the inner product space structure, orthonormality of harmonic wavelets, and how functions are reconstructed using harmonic wavelet series.

Section 8: Examples of Harmonic Wavelet Reconstruction
Description 8: Provide examples of harmonic wavelet reconstruction and computation of wavelet coefficients in the Fourier domain, including fractional harmonic scaling and wavelet.

Section 9: Some Properties of the Harmonic Wavelets in Fourier Domain
Description 9: Discuss the properties of harmonic wavelets in the Fourier domain, including the criteria for function reconstruction and scenarios where wavelet coefficients vanish.

Section 10: Local Fractional Calculus
Description 10: Define the local fractional derivative and apply this operator to fractional wavelets. Describe the importance of this knowledge for solving fractional differential equations.

Section 11: Conclusion
Description 11: Summarize the main properties of complex harmonic wavelets and the definition of fractional harmonic wavelets. Highlight how these wavelets can be used for modeling solutions to fractional differential problems.