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O. Kounchev

Publications and source records attributed to O. Kounchev.

14 recordsLinked to original sources
arXiv2026

On the log-concavity of the composite Bessel function $x^{\alpha}J_{\nu }\left( \beta x^{\gamma}\right) $

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

math.CA↗
arXiv2026

Fast algorithms for interpolation with clamped $L$-splines of order four

Interpolation and smoothing using cubic and generalized splines are fundamental tools in data analysis and statistical modeling. Recently, fast computational algorithms were developed for natural $L$-splines of order four, which arise as piecewise solutions to the differential operator $L_{\xi}^2 = (\frac{d^2}{dt^2} - \xi^2)^2$. In this paper, we extend this mathematical framework to the important case of clamped (or complete) boundary conditions, where the first derivatives at the interval endpoints are prescribed. We explicitly construct the governing linear system for the interpolation problem and mathematically prove that the resulting tridiagonal matrix is strictly row diagonally dominant, thereby guaranteeing its invertibility and the numerical stability of the fast algorithm. The proposed method is implemented in MATLAB. Furthermore, the developed clamped $L$-splines provide a foundation for constructing multivariate clamped polysplines, which serve as a promising alternative to Physics-Informed Neural Networks (PINNs) for solving partial differential equations in Mathematical Physics.

math.NA↗
arXiv2015

A new cubature formula with weight functions on the disc, with error estimates

We introduce a new type of cubature formula for the evaluation of an integral over the disk with respect to a weight function. The method is based on an analysis of the Fourier series of the weight function and a reduction of the bivariate integral into an infinite sum of univariate integrals. Several experimental results show that the accuracy of the method is superior to standard cubature formula on the disk. Error estimates provide the theoretical basis for the good performance of the new algorithm.

math.NA↗
arXiv2012

Regularity of generalized Daubechies wavelets reproducing exponential polynomials

We investigate non-stationary orthogonal wavelets based on a non-stationary interpolatory subdivision scheme reproducing a given set of exponentials. The construction is analogous to the construction of Daubechies wavelets using the subdivision scheme of Deslauriers-Dubuc. The main result is the smoothness of these Daubechies type wavelets.

math.NA↗
arXiv2012

Binomial Tree Model for Convertible Bond Pricing within Equity to Credit Risk Framework

In the present paper we fill an essential gap in the Convertible Bonds pricing world by deriving a Binary Tree based model for valuation subject to credit risk. This model belongs to the framework known as Equity to Credit Risk. We show that this model converges in continuous time to the model developed by Ayache, Forsyth and Vetzal [2003]. To this end, both forms of credit risk modeling, the so-called reduced (constant intensity of default model for the underlying) and the so-called synthesis (variable intensity of default model for the underlying) are considered. We highlight and quantify certain issues that arise, as transition probability analysis and threshold values of model inputs (tree step, underlying stock price, etc.). This study may be considered as an alternative way to develop the price dynamics model of Ayache et al. [2003] for convertible bonds in credit risk environment.

q-fin.PR↗
arXiv2011

Critical Analysis of the Binomial-Tree approach to Convertible Bonds in the framework of Tsiveriotis-Fernandes model

In the present paper we show that the Binomial-tree approach for pricing, hedging, and risk assessment of Convertible bonds in the framework of the Tsiveriotis-Fernandes model has serious drawbacks. Key words: Convertible bonds, Binomial tree, Tsiveriotis-Fernandes model, Convertible bond pricing, Convertible bond Greeks, Convertible Arbitrage, Delta-hedging of Convertible bonds, Risk Assessment of Convertible bonds.

q-fin.PR↗
arXiv2011

Holomorphic Continuation via Laplace-Fourier series

Let $B_{R}$ be the ball in the euclidean space $\mathbb{R}^{n}$ with center 0 and radius $R$ and let $f$ be a complex-valued, infinitely differentiable function on $B_{R}.$ We show that the Laplace-Fourier series of $f$ has a holomorphic extension which converges compactly in the Lie ball $\hat {B_{R}}$ in the complex space $\mathbb{C}^{n}$ when one assumes a natural estimate for the Laplace-Fourier coefficients.

math.FA↗
arXiv2011

Multidimensional Chebyshev spaces, hierarchy of infinite-dimensional spaces and Kolmogorov-Gelfand widths

Recently the theory of widths of Kolmogorov (especially of Gelfand widths) has received a great deal of interest due to its close relationship with the newly born area of Compressed Sensing. It has been realized that widths reflect properly the sparsity of the data in Signal Processing. However fundamental problems of the theory of widths in multidimensional Theory of Functions remain untouched, and their progress will have a major impact over analogous problems in the theory of multidimensional Signal Analysis. The present paper has three major contributions: 1. We solve the longstanding problem of finding multidimensional generalization of the Chebyshev systems: we introduce Multidimensional Chebyshev spaces, based on solutions of higher order elliptic equation, as a generalization of the one-dimensional Chebyshev systems, more precisely of the ECT--systems. 2. Based on that we introduce a new hierarchy of infinite-dimensional spaces for functions defined in multidimensional domains; we define corresponding generalization of Kolmogorov's widths. 3. We generalize the original results of Kolmogorov by computing the widths for special "ellipsoidal" sets of functions defined in multidimensional domains.

math.AP↗
arXiv2009

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We study the existence and shape preserving properties of a generalized Bernstein operator $B_{n}$ fixing a strictly positive function $f_{0}$, and a second function $f_{1}$ such that $f_{1}/f_{0}$ is strictly increasing, within the framework of extended Chebyshev spaces $U_{n}$. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator $B_{n}:C[a,b]\to U_{n}$ with strictly increasing nodes, fixing $f_{0}, f_{1}\in U_{n}$. If $U_{n}\subset U_{n + 1}$ and $U_{n + 1}$ has a non-negative Bernstein basis, then there exists a Bernstein operator $B_{n+1}:C[a,b]\to U_{n+1}$ with strictly increasing nodes, fixing $f_{0}$ and $f_{1}.$ In particular, if $% f_{0},f_{1},...,f_{n}$ is a basis of $U_{n}$ such that the linear span of $% f_{0},..,f_{k}$ is an extended Chebyshev space over $[ a,b] $ for each $k=0,...,n$, then there exists a Bernstein operator $B_{n}$ with increasing nodes fixing $f_{0}$ and $f_{1}.$ The second main result says that under the above assumptions the following inequalities hold B_{n}f\geq B_{n+1}f\geq f for all $(f_{0},f_{1})$-convex functions $f\in C[ a,b] .$ Furthermore, $B_{n}f$ is $(f_{0},f_{1})$-convex for all $(f_{0},f_{1})$% -convex functions $f\in C[ a,b] .$ In the specific case of exponential polynomials we give alternative proofs of shape preserving properties by computing derivatives of the generalized Bernstein polynomials.

math.CA↗
arXiv2009

On a new method for controlling exponential processes

Unlike the classical polynomial case there has not been invented up to very recently a tool similar to the Bernstein-Bezier representation which would allow us to control the behavior of the exponential polynomials. The exponential analog to the classical Bernstein polynomials has been introduced in a recent authors' paper which appeared in Constructive Approximations, and this analog retains all basic properties of the classical Bernstein polynomials. The main purpose of the present paper is to contribute in this direction, by proving some important properties of the "Bernstein exponential operator" which has been introduced. We also fix our attention upon some special type of exponential polynomials which are particularly important for the further development of theory of representation of Multivariate data.

math.NA↗
arXiv2008

Bernstein operators for exponential polynomials

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $λ_{0},...,λ_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex conjugation. If the length of the interval $[ a,b] $ is smaller than $π/M_{n}$, where $M_{n}:=\max \left\{| \text{Im}% λ_{j}| :j=0,...,n\right\} $, then there exists a basis $p_{n,k}$%, $k=0,...n$, of the space $U_{n}$ with the property that each $p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b,$ and each $% p_{n,k}$ is positive on the open interval $(a,b) .$ Under the additional assumption that $λ_{0}$ and $λ_{1}$ are real and distinct, our first main result states that there exist points $% a=t_{0}<t_{1}<...<t_{n}=b$ and positive numbers $α_{0},..,α_{n}$%, such that the operator \begin{equation*} B_{n}f:=\sum_{k=0}^{n}α_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies $B_{n}e^{λ_{j}x}=e^{λ_{j}x}$, for $j=0,1.$ The second main result gives a sufficient condition guaranteeing the uniform convergence of $B_{n}f$ to $f$ for each $f\in C[ a,b] $.

math.CA↗
arXiv2008

Bernstein Operators for Extended Chebyshev Systems

Let $U_{n}\subset C^{n}[ a,b] $ be an extended Chebyshev space of dimension $n+1$. Suppose that $f_{0}\in U_{n}$ is strictly positive and $% f_{1}\in U_{n}$ has the property that $f_{1}/f_{0}$ is strictly increasing. We search for conditions ensuring the existence of points $% t_{0},...,t_{n}\in [ a,b] $ and positive coefficients $α_{0},...,α_{n}$ such that for all $f\in C[ a,b]$, the operator $B_{n}:C[ a,b] \to U_{n}$ defined by $% B_{n}f=\sum_{k=0}^{n}f(t_{k}) α_{k}p_{n,k}$ satisfies $% B_{n}f_{0}=f_{0}$ and $B_{n}f_{1}=f_{1}.$ Here it is assumed that $% p_{n,k},k=0,...,n$, is a Bernstein basis, defined by the property that each $% p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b.$

math.CA↗