Fourier Series In Orthogonal Polynomials Research Articles

On the Convergence of Random Fourier-Jacobi Series of Continuous functions

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L 2 = L 2((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P n (x)Q n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ∈ L 2 by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ∈ L 2 in the polynomials P n (x)Q m (y), n, m = 0, 1, ..., and its “hyperbolic” partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity Ω k (A, h) of a function f ∈ L 2 are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.

Perturbation of Orthogonal Fourier Expansions

In this paper, a generalized Jacobi measure on [−1, 1] is perturbed by exponentials of functionsbof bounded mean oscillation. If we consider the Fourier series in orthogonal polynomials associated to each modification, then certain estimates (uniform inn∈N andbbelonging to some neighbourhood of the origin) are obtained. As a consequence, the partial sum operators depend analytically on the functional parameterb. The case of the Bessel series is also considered.

Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measurev on [−1, 1] are studied whenv is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in [−1, 1]. We prove some weighted norm inequalities for the partial sum operatorsS n, their maximal operatorS *, and the commutator [M b, Sn], whereM b denotes the operator of pointwise multiplication byb ∈BMO. We also prove some norm inequalites forS n whenv is a sum of a Laguerre weitht onR + and a positive mass on 0.

Weighted Norm Inequalities for Polynomial Expansions Associated to Some Measures with Mass Points

Fourier series in orthogonal polynomials with respect to a measurev on [−1, 1] are studied whenv is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in [−1, 1]. We prove some weighted norm inequalities for the partial sum operatorsS n, their maximal operatorS *, and the commutator [M b, Sn], whereM b denotes the operator of pointwise multiplication byb ∈BMO. We also prove some norm inequalites forS n whenv is a sum of a Laguerre weitht onR + and a positive mass on 0.

Weak behaviour of Fourier-Jacobi series

Necessary conditions for the weak convergence of Fourier series in orthogonal polynomials are given. It is shown that the partial sum operator associated with the Jacobi series is of restricted weak type, but not of weak type, for the endpoints of the mean convergence interval.

Necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials

Necessary conditions are found for weighted mean convergence of Fourier series in orthogonal polynomials corresponding to measures dα with support [−1, 1] for which α′ > 0 almost everywhere in [−1, 1]. Some additional properties of such orthogonal polynomials are also proved.

Convergence in the mean of the Fourier series in orthogonal polynomials

For weights p(t) and q(t) with a finite number of power-law-type singularities we obtain necessary and sufficient conditions for the inequality $$\parallel s_n^{(p)} (f)q\parallel _{L^\eta - (1,1)} \leqslant C\parallel fq\parallel _{L^\eta ( - 1,1)'}$$ to hold, where sn(p)(f) is a partial sum of the Fourier series of the function f in terms of polynomials orthogonal on [−1, 1] with weight p(t). This inequality is used to solve the problem concerning convergence in the mean and also convergence almost everywhere of the partial sums sn(p)(f).

The uniform convergence of fourier series in orthogonal polynomials

Bounds are obtained for the difference of partial sums of Fourier series for the functionf in two orthogonal systems, permitting a number of known results for Fourier-Chebyshev series to be carried over to Fourier series in certain systems of orthogonal polynomials.