Multiple Fourier Coefficients Research Articles

Functions of generalized bounded variation and its multiple fourier coefficients

Here, generalizing the class (Λ1, Λ2)*BV (p)([0, 2π]2) to the class (Λ1,Λ2)*BV (p,q)([0,2π]2) of functions of p, q-(Λ1,Λ2)*-bounded variation, it is observed that the class is a Banach space with respect to the pointwise operations and the generalized variation norm. Moreover, we estimate the order of magnitude of multiple Fourier coefficients of a function of this class.

Multiple Fourier coefficients and generalized Lipschitz classes in uniform metric

In this study, we give the necessary and sufficient conditions for a function f(x,y)=∑j,k∈Zcjkexp⁡(ijx+iky) to belong to the generalized Lipschitz classes Hω,m,n and hω,m,n in the uniform metric defined by the mixed difference of orders m,n∈N in terms of {cjk}j,k∈Z.

On multiple Fourier coefficients of functions of ϕ-Λ;-bounded variation

On multiple Fourier coefficients of functions of ϕ-Λ;-bounded variation

Determination of order of magnitude of multiple Fourier coefficients of functions of bounded ϕ-variation having lacunary Fourier series using Jensen's inequality

For a Lebesgue integrable complex-valued function f defined over the m -dimensional torus Tm := [0,2π)m , let f (n) denote the Fourier coefficient of f , where n = (n(1), . . . ,n(m)) ∈ Zm . Recently, in one of our papers [to appear in Mathematical Inequalities & Applications], we have defined the notion of bounded φ -variation for a complex-valued function on a rectangle [a1,b1 ]× . . .× [am,bm ] and studied the order of magnitude of Fourier coefficients of such functions on [0,2π]m . In this paper, the order of magnitude of Fourier coefficients of a function of bounded φ -variation from [0,2π]m to C and having lacunary Fourier series with certain gaps is studied and a generalization of our earlier result (Theorem in [Acta Sci. Math. (Szeged), 78, (2012), 97–109]) is proved. Interestingly, the Jensen’s inequality for integrals is used to prove the main result. Mathematics subject classification (2010): 42B05, 26B30, 26D15.

An application of Jensen's inequality in determining the order of magnitude of multiple Fourier coefficients of functions of bounded φ-variation

For a Lebesgue integrable complex-valued function f defined over the n -dimensional torus Tn := [0,2π)n , n∈N , let f (k) denote the Fourier coefficient of f , where k = (k1, . . . ,kn) ∈ Zn . The Riemann-Lebesgue lemma shows that f (k) = o(1) as |k| → 0 for any f ∈ L1(Tn) . However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive results are due to V. Fulop and F. Moricz for functions of bounded variation, and due to B. L. Ghodadra for functions of bounded p -variation. In this paper, defining the notion of bounded φ -variation for a function from [0,2π]n to C in two different ways, we prove that this is the case for Fourier coefficients of such functions also. Interestingly, in proving our main results we use the famous Jensen’s inequality for integrals. Our new results with φ(x) = xp (p 1) gives our earlier results [Acta Math. Hungar, 128 (4) (2010), 328–343]. Mathematics subject classification (2010): 42B05, 26D15, 26B30.

Efficient Identification of Objects Carrying Elements of High-Order Symmetry By Using Correlated Orbital Angular Momentum (OAM) States

A major contributions of Wigner's work was the introduction of group theory to study both the dynamics and the classification of states in quantum mechanics. The use of rotational symmetry to study the properties of angular momentum eigenstates is partic- ularly associated with him. Following along a similar path, it is shown here that advances in the study of entangled and correlated two-photon states allow the rapid detection of rotational symmetries in complex macroscopic objects, and that knowledge of this sym- metry structure can allow identification, and in some circumstances reconstruction, of the object. The potential for efficient identification of objects carrying elements of high-order sym- metry using correlated orbital angular momentum (OAM) states is demonstrated. The enhanced information capacity of this approach allows the recognition of specific spa- tial symmetry signatures present in objects with the use of fewer resources than in a conventional pixel-by-pixel imaging, representing the first demonstration of compressive sensing using OAM states. This approach demonstrates the capability to quickly evaluate multiple Fourier coefficients directly linked with the symmetry features of the object. The results suggest further application in small-scale biological contexts where symmetry and small numbers of noninvasive measurements are important.

Order of Magnitude of Multiple Fourier Coefficients

The order of magnitude of multiple Fourier coefficients of complex valued functions of generalized bounded variations like (L 1 ,���,L N )BV (p) and r− BV, over (0,2π) N , are estimated.

Order of magnitude of multiple Fourier coefficients of functions of bounded p-variation having lacunary Fourier series

Order of magnitude of multiple Fourier coefficients of functions of bounded p-variation having lacunary Fourier series

Order of magnitude of multiple Fourier coefficients of functions of bounded p-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus \( \mathbb{T}^n \):= [0, 2π)n, let \( \hat f \)(k) denote the Fourier coefficient of f, where k = (k1, … kn) ∈ ℤn. In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π]n to ℜ in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Moricz [5] and Fulop and Moricz [3].

Order of magnitude of multiple fourier coefficients of functions of bounded variation

Let f : R → C be a periodic function with period 2π in each variable and of bounded variation in the sense of Vitali over [0, 2π] with the total variation V ( f ; [0, 2π] ) . In case f is Lebesgue integrable over the n-dimensional torus [0, 2π), denote by f(k1, . . . , kn) the multiple Fourier coefficients of f , where (k1, . . . , kn) ∈ Z. We present a straightforward proof of the estimate ∣∣ f(k1, . . . , kn)∣∣ 5 V (f ; [0, 2π]n) (2π) n ∏ j=1 kj , provided kj 6= 0, j = 1, . . . , n. Both the order of magnitude and the constant in this estimate are exact. 1. Functions of bounded variation Let f = f(x1, . . . , xn) be a real or complex-valued function defined on a rectangle Q with sides parallel to the coordinate axes: Q := { (x1, . . . , xn) ∈ R : aj 5 xj 5 bj ; j = 1, . . . , n } , where −∞ < aj < bj < +∞ for each j. By a (finite) partition P of Q we mean P := P1 × . . .× Pn, where Pj : aj = xj < xj < · · · < x sj j = bj , sj = 1; j = 1, . . . , n. ∗This research was supported by the Hungarian National Foundation for Scientific Research under Grant T 046192.

Multiple Fourier analysis in rectifier problems. II

The nonlinear problem of the multiple Fourier analysis of the output from a cut-off power-law rectifier responding to a two-frequency input, reviewed in general in Part I of this study [1], is further scrutinized here for the special case of a zero-power-law device; i.e., a bang-bang device or a total limiter. Solutions for the modulation product amplitudes or multiple Fourier coefficients as in Part I appear as Bennett functions, and line graphs of the first fifteen basic functions for the problem are given. The new functions A m n ( 0 ) ( h , k ) {A_{mn}}^{\left ( 0 \right )}\left ( {h,k} \right ) studied, being based on a discontinuous device, then, together with the functions A m n ( 1 ) ( h , k ) {A_{mn}}^{\left ( 1 \right )}\left ( {h,k} \right ) studied in Part I, provide approximate solutions to the two-frequency modulation product problem for an arbitrary piecewise continuous nonlinear modulator, and the solution for this general problem is outlined. Finally, numerical tables of the zeroth-kind functions A m n ( 0 ) ( h , k ) {A_{mn}}^{(0)}\left ( {h,k} \right ) graphed have been prepared and are available separately in the United States and Great Britain. As before, the entire theory is based on the original multiple Fourier methods introduced by Bennett in 1933 and 1947.