Conjugate Series Research Articles

On the Approximation of Generalized Lipschitz Function by Euler Means of Conjugate Series of Fourier Series

Approximation theory is a very important field which has various applications in pure and applied mathematics. The present study deals with a new theorem on the approximation of functions of Lipschitz class by using Euler's mean of conjugate series of Fourier series. In this paper, the degree of approximation by using Euler's means of conjugate of functions belonging to Lip (ξ(t), p) class has been obtained. Lipα and Lip (α, p) classes are the particular cases of Lip (ξ(t), p) class. The main result of this paper generalizes some well-known results in this direction.

Trigonometric Approximation of Signals (Functions) Belonging to the <i>Lip</i>(<i>ξ</i>(<i>t</i>),<i>r</i>),(r＞1)-Class by (<i>E</i>,<i>q</i>) (<i>q</i>＞0)-Means of the Conjugate Series of Its Fourier Series

Various investigators such as Khan ([1-4]), Khan and Ram [5], Chandra [6,7], Leindler [8], Mishra et al. [9], Mishra [10], Mittal et al. [11], Mittal, Rhoades and Mishra [12], Mittal and Mishra [13], Rhoades et al. [14] have determined the degree of approximation of 2π-periodic signals (functions) belonging to various classes Lipα, Lip(α,r), Lip(ξ(t),r) and W(Lr,ζ(t)) of functions through trigonometric Fourier approximation (TFA) using different summability matrices with monotone rows. Recently, Mittal et al. [15], Mishra and Mishra [16], Mishra [17] have obtained the degree of approximation of signals belonging to -class by general summability matrix, which generalizes the results of Leindler [8] and some of the results of Chandra [7] by dropping monotonicity on the elements of the matrix rows (that is, weakening the conditions on the filter, we improve the quality of digital filter). In this paper, a theorem concerning the degree of approximation of the conjugate of a signal (function) f belonging to Lip(ξ(t),r) class by (E,q) summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Chandra [7] and Shukla [18].

Detecting Strength and Location of Jump Discontinuities in Numerical Data

In [1] and some following publications, Tadmor and Gelb took up a well known property of conjugate Fourier series in 1-d, namely the property to detect jump discontinuities in given spectral data. In fact, this property of conjugate series is known for quite a long time. The research in papers around the year 1910 shows that there were also other means of detecting jumps observed and analysed. We review the classical results as well as the results of Gelb and Tadmor and demonstrate their discrete case using different estimates in all detail. It is worth noting that the techniques presented are not global but local techniques. Edges are a local phenomenon and can only be found appropriately by local means. Furthermore, applying a different approach in the proof of the main estimate leads to weaker preconditions in the discrete case. Finally an outlook to a two-dimensional approach based on the work of Moricz, in which jumps in the mixed second derivative of a 2-d function are detected, is made.

Approximation of functions belonging to Lip ( ξ ( t ) , r ) class by ( N , p n ) ( E , q ) summability of conjugate series of Fourier series

In this paper, a new theorem concerning the degree of approximation of the conjugate of a function belonging to Lip ( ξ ( t ) , r ) Open image in new window class by ( N , p n ) ( E , q ) Open image in new window summability of its conjugate series of a Fourier series has been proved. Here the product of Euler ( E , q ) Open image in new window summability method and Norlund ( N , p n ) Open image in new window method has been taken.

On Degree of Approximation by Product Means of Conjugate Series of a Fourier Series

In this paper a theorem on degree of approximation of a function f ∈ Lip(α, r) by product summability (E, q)(N, pn) of conjugate series of Fourier series associated with f has been established.

Product Summability Transform of Conjugate Series of Fourier Series

A known theorem, Nigam (2010) dealing with the degree of approximation of conjugate of a signal belonging toLipξ(t)-class by(E,1)(C,1)product summability means of conjugate series of Fourier series has been generalized for the weightedW(Lr,ξ(t)),(r≥1),(t>0)-class, whereξ(t)is nonnegative and increasing function oft, byEn1Cn1~which is in more general form of Theorem 2 of Nigam and Sharma (2011).

Summation of Fourier trigonometric series and conjugate series by (C, α, β) method

The paper gives sufficient conditions for (C, α, β)-summability of Fourier trigonometric series and conjugate series.

Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator

In the present paper, an attempt is made to obtain the degree of approximation of conjugate of functions (signals) belonging to the generalized weighted W(LP, ξ(t)), (p ≥ 1)-class, by using lower triangular matrix operator of conjugate series of its Fourier series.

On the Point Behavior of Fourier Series and Conjugate Series

We investigate the point behavior of periodic functions and Schwartz distributions when the Fourier series and the conjugate series are both Abel summable at a point. In particular we show that if f is a bounded function and its Fourier series and conjugate series are Abel summable to values γ and β at the point θ_0 , respectively, then the primitive of f is differentiable at θ_0 , with derivative equal to γ , the conjugate function satisfies \lim_{θ→θ_0} \frac{3}{(θ–θ_0)^3} \int^θ_{θ_0} \tilde f(t)(θ–t)^2 dt = β , and the Fourier series and the conjugate series are both (\mathrm{C}, κ) summable at θ_0 , for any κ > 0 . We show a similar result for positive measures and L^1 functions bounded from below. Since the converse of our results are valid, we therefore provide a complete characterization of simultaneous Abel summability of the Fourier and conjugate series in terms of “average point values”, within the classes of positive measures and functions bounded from below. For general L^1 functions, we also give a.e. distributional interpretation of − \frac{1}{2π} p.v. \int^π_{–π} f (t+θ_0) \cot \frac t2 dt as the point value of the conjugate series when viewed as a distribution. We obtain more general results of this kind for arbitrary trigonometric series with coefficients of slow growth, i.e., periodic distributions.

High antiangiogenic and low anticoagulant efficacy of orally active low molecular weight heparin derivatives

Heparin, an anticoagulant that is widely used clinically, is also known to bind to several kinds of proteins through electrostatic interactions because of its polyanionic character. These interactions are mediated by the physicochemical properties of heparin such as sequence composition, sulfation patterns, charge distribution, overall charge density, and molecular size. Although this electrostatic character mediates its binding to many proteins related with tumor progression, thereby providing its antiangiogenic property, the administration of heparin for treating cancer is limited in clinical applications due to several drawbacks, such as its low oral absorption, unsatisfactory therapeutic effects, and strong anticoagulant activity which induces hemorrhaging. Here, we evaluated novel, orally active, low molecular weight heparin (LMWH) derivatives (LHD) conjugated with deoxycholic acid (DOCA) that show reduced anticoagulant activity and enhanced antiangiogenic activity. The chemical conjugate of LMWH and DOCA was synthesized by conjugating the amine group of N-deoxycholylethylamine (EtDOCA) with the carboxylic groups of heparin at various DOCA conjugation ratios. The LMWH–DOCA conjugate series (LHD1, LHD1.5, LHD2, and LHD4) were further formulated with poloxamer 407 as a solubilizer for oral administration. An in vitro endothelial tubular formation and in vivo Matrigel plug assay were performed to verify the antiangiogenic potential of LHD. Finally, we evaluated tumor growth inhibition of oral LHD administration in a SCC7 model as well as in A549 human cancer cell lines in a mouse xenograft model. Increasing DOCA conjugation ratios showed decreased anticoagulant activity, eventually to zero. LHD could block angiogenesis in the tubular formation assay and the Matrigel plug assay. In particular, oral administration of LHD4, which has 4 molecules of DOCA per mole of LMWH, inhibited tumor growth in SCC7 mice model as well as A549 mice xenograft model. LHD4 was orally absorbable, showed minimal anticoagulant activity and inhibits tumor growth via antiangiogenesis. These findings demonstrate the therapeutic potential of LHD4 as a new oral anti-cancer drug.

T · C1 summability of a sequence of Fourier coefficients

Let B n ( x) denote the nth term of the conjugate series of a Fourier series of function f. Mohanty and Nanda [R. Mohanty, M. Nanda, On the behavior of Fourier coefficients, Proc. Am. Math. Soc. 5 (1954) 79–84] were the first to establish a result for C 1-summability of the sequence { nB n ( x)}. Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795] improved it for the product summability H 1 · C 1, which was generalized by various investigators using different summability methods with different set of conditions. In this note, we extend the result of Mittal [M.L. Mittal, On the ‖ T ‖ · C 1 summability of a sequence of Fourier coefficients, Bull. Cal. Math. Soc. 81 (1989) 25–31], which in turn generalizes the results of Prasad [K. Prasad, On the ( N, p n ) · C 1 summability of a sequence of Fourier coefficients, Indian J. Pure Appl. Math. 12 (7) (1981) 874–881] and Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795].

Determination of jumps via advanced concentration factors

If a periodic function f with period 2 π has a discontinuity at ξ ∈ [ − π , π ) , then the partial sums of its Fourier conjugate series diverge to + or − infinity at ξ. It was shown in [F. Lukács, Über die Bestimmung des Sprunges einer Function aus iher Fourierreihe, J. Reine Angew. Math. 150 (1920) 107–122] that, if we divide these partial sums at n by ln n , however, we obtain essentially the length of the jump in the limit, as n tends to infinity. But the convergence rate in this way is very slow. This result has been improved if these partial sums are replaced by other sequences, similar to appropriate summability sequences (see [N.S. Banerjee, J. Geer, Exponentially accurate approximations to piecewise smooth periodic functions, ICASE report 95-17, NASA Langley Research Center, 1995; B.I. Golubov, Determination of jump of a function of bounded p-variation by its Fourier series, Math. Notes 12 (1972) 444–449; G. Kvernadge, Determination of jumps of a bounded function by its Fourier series, J. Approx. Theory 92 (1998) 107–122]). In the present paper we obtain still stronger results along these lines. We will prove that there are concentration factors { μ n , k } such that the condition of Dini-type introduced in [A. Gelb, E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999) 101–135] can be canceled and for piecewise smooth functions the convergence speed keeps the same rate as A. Gelb and E. Tadmor's method shown in [A. Gelb, E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999) 101–135].

More on determination of jumps

We generalize some old and new results on the determination of jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of f in [1], and in terms of the Abel-Poisson means in [2], to some more general linear operators which satisfy some certain conditions. The linear operators in discussion include the Fejer means, de la Vallee-Poussin means, and Bernstein-Rogosinski sums.

Detection of edges from spectral data: New results

We are concerned with the problem of recovering edges of piecewise smooth functions with finitely many jump discontinuities. In a series of papers, Gelb and Tadmor presented computationally simple methods for this task that are based on the conjugate Fourier series with different concentration kernels. In this article we present experimental results comparing conjugate series based methods with a new approach based on polynomial filters and suitable approximations. This new approach proves to be more accurate and stable.

On the strong Nörlund summability of conjugate Fourier series

In continuation of a recent work by Mittal [M.L. Mittal, On strong Nörlund summability of Fourier series, J. Math. Anal. Appl. 314 (2006),75–84], the present authors obtain a sufficient condition for the summability [ N , p n ( 1 ) , 2 ] of the conjugate Fourier series. In conjunction with the known Tauberian theorem on the strong Nörlund summability, which was also considered earlier by Mittal [M.L. Mittal, A Tauberian theorem on strong Nörlund summability, J. Indian Math. Soc. (N.S.) 44 (1980), 369–377], our result gives a sufficient condition for the summability [ C, 1 , 2] of the conjugate Fourier series. Our main theorem generalizes the results given earlier by Prasad [G. Prasad, On Nörlund Summability of Fourier Series, Ph.D. thesis, University of Roorkee, Roorkee, 1967] and Singh [U.N. Singh, On the strong summability of a Fourier series and its conjugate series, Proc. Nat. Inst. Sci. India Part A 13 (1947), 319–325].

Approximation of signals (functions) belonging to the weighted W(Lp, ξ(t))‐class by linear operators

Mittal and Rhoades (1999–2001) and Mittal et al. (2006) have initiated the studies of error estimates En(f) through trigonometric Fourier approximations (TFA) for the situations in which the summability matrix T does not have monotone rows. In this paper, we determine the degree of approximation of a function , conjugate to a periodic function f belonging to the weighted W(Lp, ξ(t))‐class (p ≥ 1), where ξ(t) is nonnegative and increasing function of t by matrix operators T (without monotone rows) on a conjugate series of Fourier series associated with f. Our theorem extends a recent result of Mittal et al. (2005) and a theorem of Lal and Nigam (2001) on general matrix summability. Our theorem also generalizes the results of Mittal, Singh, and Mishra (2005) and Qureshi (1981‐1982) for Nörlund (Np)‐matrices.

Composite fermions in a negative effective magnetic field: A Monte Carlo study

The method of Jain and Kamilla [J. K. Jain and R. K. Kamilla, Phys. Rev. B 55, R4895 (1997)] allows numerical generation of composite-fermion trial wave functions for large numbers of electrons in high magnetic fields at filling fractions of the form $\ensuremath{\nu}=p∕(2mp+1)$ with $m$ and $p$ positive integers. In the current paper we generalize this method to the case where the composite fermions are in an effective (mean) field with opposite sign from the actual physical field, i.e., when $p$ is negative. We examine both the ground-state energies and the low-energy neutral excitation spectra of these states. Using particle-hole symmetry we can confirm the correctness of our method by comparing results for the series $m=1$ with $p>0$ (previously calculated by others) to our results for the conjugate series $m=1$ with $p<0$. Finally, we present similar results for ground-state energies and low-energy neutral excitations for the states with $m=2$ and $p<0$, which were not previously addressable, comparing our results to the $m=1$ case and the $p>0$, $m=2$ cases.

Oxoazabenzo[de]anthracenes Conjugated to Amino Acids: Synthesis and Evaluation as DNA-Binding Antitumor Agents

We report the synthesis of an original series of oxoazabenzo[de]anthracenes conjugated to an amino acid: Ala, Phe, Pro, Lys, or Gly (4a-e, respectively). The compounds, derived from 1,8-dihydroxyanthracene-9,10-dione, were studied for DNA binding and cytotoxicity. Melting temperature, fluorescence quenching, and surface plasmon resonance methods all indicated that the lysine derivative 4d binds to DNA much more strongly that the Pro, Ala, and Gly conjugates whereas the Phe analogue showed the lowest DNA binding capacity. These compounds form intercalation complexes with DNA, as judged from electric linear dichroism and topoisomerase I-based DNA unwinding experiments. Preferential binding of 4d to defined sequences such as 5'-CTAAAGG and 5'-ATGC was evidenced by DNase I footprinting. This Lys conjugate was found to be over 20 times more cytotoxic to CEM human leukemia cells than the other conjugates, with an IC50 in the submicromolar range. A high antiproliferative activity, likely attributable to the enhanced DNA binding capacity, is maintained despite the incapacity of the compound to stabilize topoisomerase-DNA covalent complexes. The cell cycle effects of 4d consisted in an S phase accumulation of cells coupled with a pro-apoptotic action (appearance of hypodiploid sub-G1 cells) which were confirmed by measuring the inhibition of BrdU incorporation into DNA and labeling of phosphatidylserine residues with annexin V-FITC by means of flow cytometry. Altogether, the work provides interesting structure-activity relationships in the oxoazabenzo[de]anthracene-amino acid conjugate series and identifies the lysine derivative 4d as a promising candidate for further in vivo evaluation and drug design.

Extrapolation Algorithms for Filtering Series of Functions, and Treating the Gibbs Phenomenon

In this paper, we study the application of some convergence acceleration methods to Fourier series, to orthogonal series, and, more generally, to series of functions. Sometimes, the convergence of these series is slow and, moreover, they exhibit a Gibbs phenomenon, in particular when the solution or its first derivative has discontinuities. It is possible to circumvent, at least partially, these drawbacks by applying a convergence acceleration method (in particular, the e-algorithm) or by approximating the series by a rational function (in particular, a Pade approximant). These issues are discussed and some numerical results are presented. We will see that adding its conjugate series as an imaginary part to a Fourier series greatly improves the efficiency of the algorithms for accelerating the convergence of the series and reducing the Gibbs phenomenon. Conjugacy for series of functions will also be considered.

Degree of approximation of conjugate of Lip $ \alpha$ class function by $ {K^\lambda}$-summability means of conjugate series of a Fourier series

In this paper the degree of approximation of conjugate of a function belonging to Lip $ \alpha$ class by $ K^\lambda$-summability means of conjugate series of its Fourier series has been determined.