Functions of Minimal Norm with the Given Set of Fourier Coefficients

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 651
Author(s):  
Pyotr Ivanshin
Keyword(s):  
Fourier Series ◽  
Existence And Uniqueness ◽  
Fourier Coefficients ◽  
Nonnegative Function ◽  
Trigonometric Fourier Series ◽  
Minimum Norm ◽  
Minimal Norm ◽  
Uniqueness Of The Solution ◽  
The Given

We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .

scholarly journals Reconstruction of Piecewise Smooth Multivariate Functions from Fourier Data

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 88
Author(s):  
David Levin
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Order Approximation ◽  
High Accuracy ◽  
Rectangular Domain ◽  
High Order ◽  
Piecewise Smooth ◽  
Multidimensional Case ◽  
Order Spline ◽  
The Given

In some applications, one is interested in reconstructing a function f from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to f using a Padé-like method. Namely, we do this by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of f. Given the Fourier series coefficients of a function on a rectangular domain in Rd, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2D case, we find high accuracy approximation to the curves separating between smooth segments of f. Secondly, simultaneously we find the approximations of all the different segments of f. We start by developing and demonstrating a high accuracy algorithm for the 1D case, and we use this algorithm to step up to the multidimensional case.

scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

scholarly journals Identification of the initial temperature from the given temperature data at the left end of a rod

2019 ◽  
Vol 4 (2) ◽  
pp. 469-474
Author(s):  
Yesim Sarac ◽  
S. Sule Sener
Keyword(s):  
Existence And Uniqueness ◽  
Initial Temperature ◽  
Adjoint Problem ◽  
Temperature Data ◽  
Minimizing Sequence ◽  
Fréchet Differential ◽  
Uniqueness Of The Solution ◽  
The Cost ◽  
The Given ◽  
Theoretical Results

AbstractThis study aims to investigate the problem of determining the unknown initial temperature in a variable coefficient heat equation. We obtain the existence and uniqueness of the solution of the optimal control problem considered under some conditions. Using the adjoint problem approach, we get the Frechet differential of the cost functional. We construct a minimizing sequence and give the convergence rate of this sequence. Also, we test the theoretical results in a numerical example by using the MAPLE® program.

On the Absolute Convergence of Fourier Series

1997 ◽  
Vol 4 (4) ◽  
pp. 333-340
Author(s):  
T. Karchava
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Sufficient Conditions ◽  
Trigonometric Fourier Series ◽  
Periodic Functions ◽  
Continuity Modulus ◽  
The Absolute ◽  
Necessary And Sufficient

Abstract The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2π-periodic functions which in [0, 2π] have a finite number of intervals of convexity, and whose 𝑛th Fourier coefficients are O(ω(1/𝑛; 𝑓)/𝑛), where ω(δ; 𝑓) is the continuity modulus of the function 𝑓.

NONLOCAL BOUNDARY VALUE PROBLEM FOR ONE MIXED THIRD ORDER EQUATION

2021 ◽  
pp. 18-24
Author(s):  
Alena G. Ezaova ◽  
Liana V. Kanukoeva ◽  
Gennady V. Kupovykh
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
The Third ◽  
Uniqueness Of The Solution ◽  
The Given

The paper considers a nonlocal boundary value problem for a mixed hyperbolic-parabolic equation of the third order. The equation is considered in a finite simply connected domain consisting of a hyperbolic and a parabolic part. The solution to the problem posed is considered for various cases of the parameter λ, which is in the original equation. In the case when (1-2m)/2< <λ<1, the solution of the problem is reduced to a singular integral equation, which is reduced by the well-known Carleman-Vekua method to the Fredholm integral equation of the third kind. In the case when λ=(1-2m)/2, a theorem on the existence and uniqueness of a solution to the problem posed is formulated and proved. To prove the uniqueness of the solution, the method of energy integrals is used and inequalities of the type are derived on the given functions that are in the boundary condition. It is shown that the homogeneous problem corresponding to the original problem, under the conditions of the uniqueness theorem, has only a trivial solution in the entire considered domain. From which we can conclude that the original problem has only a single solution. If the obtained conditions for the given functions are violated, the problem posed does not have a unique solution. When investigating the question of the existence of a solution to the problem posed, a system of two equations is considered, consisting of the basic functional relations between the trace of the desired function and the traces of the derivative of the desired function, brought to the line of degeneration y = 0. Eliminating from the system the function τ (x) - the trace of the desired solution on the line of degeneration, we arrive at an equation for the trace of the derivative of the desired function. Under the condition of the existence and uniqueness theorem, the problem posed is equivalently reduced to the Fredholm integral equation of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problem posed.

scholarly journals Well-posedness and numerical approximation of tempered fractional terminal value problems

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Maria Luísa Morgado ◽  
Magda Rebelo
Keyword(s):  
Numerical Methods ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Numerical Approximation ◽  
Shooting Method ◽  
Well Posedness ◽  
Numerical Examples ◽  
Uniqueness Of The Solution ◽  
The Given ◽  
Theoretical Results

AbstractFor a class of tempered fractional terminal value problems of the Caputo type, we study the existence and uniqueness of the solution, analyze the continuous dependence on the given data, and using a shooting method we present and discuss three numerical schemes for the numerical approximation of such problems. Some numerical examples are considered in order to illustrate the theoretical results and evidence the efficiency of the numerical methods.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

scholarly journals The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 389
Author(s):  
Jeong-Gyoo Kim
Keyword(s):  
Hilbert Space ◽  
Fourier Series ◽  
Fourier Coefficients ◽  
Double Sequence ◽  
Intermediate Stage ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Abstract Space ◽  
Double Sequences ◽  
Two Parameter

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

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